Ubiquitous Dynamical Time Asymmetry in Measurements on Materials and Biological Systems
UUbiquitous Dynamical Time Asymmetry in Measurements on Materials andBiological Systems
Alessio Lapolla, Jeremy C. Smith,
2, 3 and Aljaˇz Godec ∗ Mathematical bioPhysics Group, Max Planck Institute for Biophysical Chemistry, G¨ottingen 37077, Germany Center for Molecular Biophysics, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA Department of Biochemistry and Cellular and Molecular Biology,University of Tennessee, Knoxville, Tennessee 37996, USA
Many measurements on soft condensed matter (e.g., biological and materials) systems track low-dimensional observables projected from the full system phase space as a function of time. Examplesare dynamic structure factors, spectroscopic and rheological response functions, and time series ofdistances derived from optical tweezers, single-molecule spectroscopy and molecular dynamics simu-lations. In many such systems the projection renders the reduced dynamics non-Markovian and theobservable is not prepared in, or initially sampled from and averaged over, a stationary distribution.We prove that such systems always exhibit non-equilibrium, time asymmetric dynamics. That is,they evolve in time with a broken time-translation invariance in a manner closely resembling agingdynamics. We identify the entropy associated with the breaking of time-translation symmetry thatis a measure of the instantaneous thermodynamic displacement of latent, hidden degrees of freedomfrom their stationary state. Dynamical time asymmetry is a general phenomenon, independent ofthe underlying energy surface, and is frequently even visible in measurements on systems that havefully reached equilibrium. This finding has fundamental implications for the interpretation of manyexperiments on, and simulations of, biological and materials systems.
INTRODUCTION
Relaxation refers to the dynamics of approaching a sta-tionary state (e.g. thermodynamic equilibrium) and isa hallmark of non-equilibrium physics, from condensedmatter [1–3] to single-molecule systems [4] initially per-turbed near [3, 5–13] or far [14–19] from equilibrium. Inextreme cases the non-stationary behavior of a systemextends over all experimentally accessible time-scales –a phenomenon often referred to as “aging” [20–24]. Ag-ing is typically assumed to occur in systems whose energylandscapes contain a large number (scaling exponentiallywith the system size) of meta-stable states [20–25]. Ithas been observed in polymeric [26, 27], spin [28, 29] andcolloidal glasses [30, 31], supercooled liquids [32–35] andrecently in protein internal dynamics [36–39], where itmay also affect biological function [40–43].Typical manifestations of aging are a complex, non-exponential relaxation spectrum and non-stationary cor-relation and response functions [26–34, 36–39, 44, 45]that depend strongly and systematically on the timeelapsed since the system was prepared [15, 26, 46–48]or, when derived from time-series measurements, on theduration of the observation [39, 45]. The temporal extentof apparent aging dynamics in experimental systems (e.g.spin glass materials), although very long, may be finite[29]. Throughout we will refer to aging systems withexperimentally observable equilibration as “transientlyaging” irrespective of the precise manner in which therelaxation time depends on the system size.Theoretical studies on aging have focused mainly ∗ [email protected] on non-stationary correlations and responses [24, 45–52] as well as generalizations to aging systems of thefluctuation-dissipation relation [14, 15, 53–55]. Aging dy-namics has frequently been associated with the existenceof deep traps with unbounded depth in the potential en-ergy function [21, 23], fractal properties of the underlyingfree energy landscape [36, 37, 56], the presence of disorder[48, 53], and other effects [25, 46, 47, 57, 58].Recent efforts in understanding relaxation dynamicsthat are not limited to systems with unobservable sta-tionary states focus on diverse aspects of the thermo-dynamics of relaxation, e.g. the rˆole of initial condi-tions in the context of the so-called “Mpemba effect” (i.e.the phenomenon where a system can cool down fasterwhen initiated at a higher temperature) [16, 17], asym-metries in the kinetics of relaxation from thermodynami-cally equidistant temperature quenches [19], a spectralduality between relaxation and first-passage processes[59, 60], so-called “frenetic” concepts [12, 13], and thestatistics of the ’house-keeping’ heat [61, 62] and entropyproduction [63]. Important advances in understandingtransients of relaxation also include information-theoreticbounds on the entropy production during relaxation farfrom equilibrium [18] and the so-called “thermodynamicuncertainty relation” for non-stationary initial conditionsthat bounds transient currents by means of the total en-tropy production [64].Here, we look at non-stationary physical observablesfrom a more general, “first principles” perspective. By di-rectly analyzing the mathematical structure of the under-lying multi-point probability density functions we revealthe universality of a broken time-translation invariancethat we coin as dynamical time asymmetry (DTA). Weprove the established linear aging correlation functionsto be ambiguous indicators of broken time-translation in- a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b variance. DTA has many of the properties commonly as-sociated with aging but, unlike theoretical models of ag-ing [20–25, 65], does not require any particular functionalform of the dependence on the aging time nor that therelaxation time increases exponentially with system sizeand is therefore experimentally unobservable. Moreover,we here show that specific properties, such as deep trapsin the potential energy function [21, 23], fractal proper-ties of the underlying free energy landscape [36, 37, 56],or the presence of disorder [48, 53] that are often requiredfor aging to occur, are not required for DTA dynamics, al-though they can amplify the breaking of time-translationinvariance. In fact, DTA typically implies (transient) ag-ing but the converse is not true. Instead, we prove DTAto emerge whenever (i) a physical observable correspondsto a lower-dimensional projection in configuration spacethat renders the reduced dynamics non-Markovian, and(ii) the projected physical observable is not prepared in,or initially sampled from and averaged over, a stationarydistribution i.e., a distribution that does not change intime.Most measurements on condensed matter correspondto projections of type (i), examples being structure fac-tors in scattering experiments [30, 31, 33, 34, 56], spectro-scopic response functions (e.g. magnetization [28, 29, 53]and dielectric responses [27, 32, 49]), the rheology ofsoft materials [66, 67], diverse empirical order parameters[45–47] and measurements of mechanical responses [26].These projections also inevitably arise in single-particletracking [34, 45, 56] and measurements of various reac-tion coordinates in all single-molecule experiments (e.g.internal distances) and simulations (e.g. projections ontodominant principal modes in Principal Component Anal-ysis) [37–39, 41–43, 68–71].In these measurements (i) applies as soon as the latentdegrees of freedom (DOF) (those being effectively inte-grated out) evolve on a time-scale similar to the moni-tored observable. In contrast, (i) does not apply whenthe latent DOF relax much faster than the observable,for example when neglecting inertia and integrating outsolvent degrees of freedom of a colloidal particle in a lowReynolds number environment. Condition (ii) applieswhenever the observable evolves from a non-stationaryinitial condition. This includes all experiments involv-ing an instantaneous perturbation of the observable inequilibrium (e.g. magnetization or dielectric, rheologicaland mechanical response), and all experiments involvingevolution from a quench, such as in temperature, pres-sure, or volume (which inter alia includes scattering ex-periments on supercooled liquids). Condition (ii) alsoholds in situations where the observable is neither per-turbed nor quenched but is initially under-sampled fromequilibrium, that is, when it is sampled from equilibriumwith a limited number of repetitions (say 1 − ) suchas in single-molecule FRET, AFM or optical tweezersexperiment, as well as particle-based computer simula-tions. This yields a distribution that does not convergeto the invariant measure. In fact, as regards DTA we prove quenching and the under-sampling of equilibriumto be qualitatively equivalent. Whenever both conditions(i) and (ii) are fulfilled, DTA emerges irrespective of thedetails of the dynamics.In the main text and in the examples we focus on sys-tems whose dynamics obey detailed balance and, as awhole, are initially prepared at equilibrium. The moni-tored lower-dimensional observable is assumed to evolvefrom some non-equilibrium initial distribution (i.e. notthe marginalized equilibrium distribution [72]). Gener-alizations to a non-equilibrium preparation of the fullsystem (e.g. by a temperature quench) are discussed indetail the Appendix. THEORY
We consider a mechanical system at least weakly cou-pled to a thermal reservoir, such that the full system’sdynamics (i.e. all degrees of freedom; Fig. 1a, red tra-jectory) obeys a time-homogeneous Markovian stochas-tic equation of motion [73] (for details see Appendix),which generates ergodic dynamics in phase space. Thatis, starting from any initial condition the system is as-sumed to evolve to a unique stationary distribution in afinite, but potentially extremely long, time that may ormay not be reached during an observation. This assump-tion is true for a vast majority of soft matter and biologi-cal systems of interest and also includes glassy materials.To impose only the mildest of assumptions we considerthat the full system is prepared in an equilibrium state at t = 0, i.e. the full system was created at a time t = −∞ and the initiation of an experiment or phenomenon im-poses a time origin at t = 0, whereas the actual obser-vation starts after some time t a ≥ t a is the so-called aging (or waiting) time and the mea-surement time-window is the time delay τ = t − t a . Themore restrictive assumption of a non-stationary prepa-ration (e.g. a temperature quench [14, 19]) is treatedin the Appendix B 2. In practice, a stationary prepara-tion means that at t = 0 the full system’s configurationis distributed according to a stationary, invariant prob-ability measure. This refers either to the initial statis-tical ensemble of configurations in a bulk system or tothe repeated sampling of individual initial configurations(say in a single molecule experiment), which are drawnrandomly from the invariant probability measure. Weassume that only the projected observable is being mon-itored at all times t ≥
0. The assumptions stated abovesuffice to prove our claims (for details see Appendix).For simplicity we use (cid:104)·(cid:105) interchangeably to denote theaverage over an ensemble of trajectories at a given timeand over time along a given trajectory, respectively, keep-ing in mind that they are identical only when the tra-jectory is much longer than the longest relaxation time t rel . The state of the observable is denoted by q ( t ) ∈ Ξ(Fig. 1a, black trajectory), which we assume, without lossof generality, to be one dimensional (for the general case
Figure 1.
Schematics of projected observ-ables, multi-point propagation and modelsystems. a) A physical observable correspond-ing to a simple lower-dimensional projection(shadow trajectory) of the full system’s trajec-tory (red line), defining projected and latent(hidden) degrees of freedom. b) Trajectories oflength t evolving from preparation, through anaging (or pre-evolution) period of length t a , fol-lowed by the observation of duration τ = t − t a .c) Optical tweezers experimental set-up prob-ing DNA-hairpin dynamics; d) Structure of theyeast PGK protein with the reduced coordinaterepresented by the arrow. e) Rouse model ofa polymer chain, comprising Hookean springswith a zero rest-length immersed in a heat bath.The reduced coordinate corresponds to the end-to-end distance. f) Single file model with thetracer particle depicted in red. see the Appendix). Theoretically, each repetition of theexperiment/process leads to an initial condition for q ( t )drawn randomly from the reduced stationary probabilitydensity p inv ( q ). In practice, however, this is not neces-sarily the case. For example, supercooled liquids [32–34]as well as polymeric [26, 27], spin [28, 29], and colloidal[30, 31] glasses are prepared by a quench in an externalparameter (typically temperature) [26–31], such that theobservable q ( t ) nominally attains a non-stationary initialcondition. A process may also start with the observ-able internally constrained to a subdomain of p inv ( q ),e.g. a chaperone stabilizing a particular configurationof a folded protein, with the biological process startingupon unbinding of the chaperone [74]. In another exam-ple single-molecule enzyme experiments may monitor thestatistics of substrate turnover, where q ( t ) reflects the ge-ometry of the catalytic site of an enzyme that is reactiveonly for a specific sub-ensemble of configurations [41–43]. Binding of a substrate molecule enforces an initialconstraint on q ( t ) thereby imposing non-stationary ini-tial conditions on the chemical reaction. Alternatively,we may simply choose to initialize the experiment (i.e.reset our clock) a posteriori, such that q (0) has a pre-set value, or we are dealing with a single, or a limitednumber of time-series [39] which do not sample p inv ( q )sufficiently. In all these cases the observable is effectivelynot prepared in a stationary state, i.e. p ( q ) (cid:54) = p inv ( q ).The dynamics in aging systems is conventionally an-alyzed via the normalized two-time correlation function[15, 28, 45–48] C t a ( τ ) = (cid:104) q ( τ + t a ) q ( t a ) (cid:105) − (cid:104) q ( τ + t a ) (cid:105)(cid:104) q ( t a ) (cid:105)(cid:104) q ( t a ) (cid:105) − (cid:104) q ( t a ) (cid:105) . (1)A system is often said to be aging if C t a ( τ ) strongly de-pends on t a in the sense that the relaxation of a systemtakes place on time-scales that grow with the age of thesystem t a , and continue to do so beyond the largest times accessible within an experiment or simulation [21, 23–25, 65].However, the analysis and interpretation of time-seriesof physical observables that show DTA require a fun-damentally different approach irrespective of whetherequilibrium is attainable in an experiment or not. Weprove below that C t a ( τ ) cannot conclusively indicatewhether time-translation invariance is broken (see Ap-pendix C, Lemma 2); in particular it cannot disentanglebroken time-translation invariance from “trivial” corre-lations with a non-stationary initial condition (i.e. from“weak” or “second order” non-stationarity [75]). This isparticularly problematic if one uses Eq. (1) as a “defini-tion of DTA” to infer whether a complex experimentalsystem, such as an individual protein molecule [38, 39],evolves with broken time-translation invariance or not.Eq. (1) is nevertheless reasonable, albeit sub-optimal, forquantifying DTA in materials that are known to possesa broken time-translation invariance.Our aim is to conclusively and unambiguously in-fer whether relaxation evolves with a broken time-translation invariance that is encoded in G ( q, t a + τ | q (cid:48) , t a , q ∈ Ω ), the probability density for the observ-able to be found in an infinitesimal volume element cen-tered at q at time τ + t a given that it was at q (cid:48) attime t a and started at t = 0 somewhere in a subdomain q ∈ Ω ⊂ Ξ (Fig. 1b) with probability p ( q ). Ω isstrictly non-empty and may be a point, an interval or aunion of intervals.The dynamics of an observable q ( t ) is generally said tobe time-translation invariant (mathematically referred toas “strictly stationary” [75, 76] or “well-aged” [77]) if theunderlying (effective) equations of motion that govern theevolution of q ( t ) do not explicitly depend on time. Thatis, the probability of a path { q ( t ) } for t ∈ [ t a , t a + τ ]does not depend on t a . This is the case, e.g. in Newto-nian dynamics or Langevin dynamics driven by Gaussianwhite noise [75] as well as generalized Langevin dynam-ics driven by stationary Gaussian colored noise [78–80].Here q ( t ) is said to be time-translation invariant if andonly if (see also Definition 1 in Appendix C) G ( q, t a + τ | q (cid:48) , t a , q ∈ Ω ) = G ( q, t (cid:48) + τ | q (cid:48) , t (cid:48) , q ∈ Ω ) , (2)holds for any τ and t (cid:48) [81]. Conversely, if time-translationinvariance is broken we say that the system is dynami-cally time asymmetric. That is, time-translation invari-ance is broken if and only if the two-point conditionedGreen’s function G ( q, t a + τ | q (cid:48) , t a , q ∈ Ω ) depends on t a (see also Definition 2 in Appendix C). The two-pointconditioned Green’s function is defined as G ( q, t a + τ | q (cid:48) , t a , q ∈ Ω ) ≡ P ( q, t a + τ, q (cid:48) , t a , q ∈ Ω ) P ( q (cid:48) , t a , q ∈ Ω ) , (3)where P ( q, t a + τ, q (cid:48) , t a , q ∈ Ω ) denotes the joint densityof q ( t ) to be found initially within Ω and to pass q (cid:48) attime t a and to end up in q at time t a + τ , and P ( q (cid:48) , t a , q ∈ Ω ) the joint density of q ( t ) to be found initially withinΩ and to pass q (cid:48) at time t a .Note that there seems to be some relation betweenDTA and aging. A system is typically said to be aging if C t a ( τ ) in Eq. (1) depends on t a (i.e. that q ( t ) is weaklynon-stationary) but in a specific manner, e.g. the so-called “slow”, non-stationary component of C t a ( τ ) mustscale for all large t a as some power of τ /t a [21, 23, 24](for a rigorous discussion see [65]). However, this doesnot require that time-translation invariance (i.e. Eq. (2))is broken [21, 23, 24]. So-called kinetically constrainedmodels [25] and the spherical p-spin model [52, 53, 82],for example, have correlation functions Eq. (1) that showaging, but, when fully observed and not averaged overdisorder (and only then), satisfy Eq. (2. Clearly, if time-translation invariance is broken (see also Definition 1 inthe Appendix C) then C t a ( τ ) automatically depends on t a . If the dynamics is furthermore such that C t a ( τ ) de-pends on t a as some power of τ /t a (see Eq. (8) belowas well as Eqs. (C7) and (C10) as well as [45, 83, 84])and, in addition, equilibrium cannot be attained duringan observation then DTA also implies aging dynamics.However, the converse is not true.To connect the aging correlation function in Eq. (1)with Eq. (2) we note that the numerator in Eq. (1) in-volves averages (cid:104) q ( t ) (cid:105) ≡ (cid:90) Ξ qG ( q, t | q ∈ Ω ) dq (4) (cid:104) q ( τ + t a ) q ( t a ) (cid:105) ≡ (cid:90) Ξ (cid:90) Ξ qq (cid:48) G ( q, τ + t a , q (cid:48) , t a | q ∈ Ω ) dqdq (cid:48) where the conditional density of the projected observable G ( q, t | q ∈ Ω ) is discussed in [19, 85] and in Appendix B 2(see Eq. (B2)). The three-point conditional probabilitydensity G ( q, τ + t a , q (cid:48) , t a | q ∈ Ω ) – the probability densityfor the observable to pass through an infinitesimal volumeelement centered at q (cid:48) at time t a and end up in q at time τ + t a having started at t = 0 in a subdomain q ∈ Ω ⊂ Ξ with probability p ( q ), is defined as (for details seeAppendix B 2, Eq. (B20)) G ( q, t a + τ, q (cid:48) , t a | q ∈ Ξ ) ≡ P ( q, t a + τ, q (cid:48) , t a , q ∈ Ω ) P ( q ∈ Ω ) . (5)Based on the mathematical properties of G ( q, t a + τ | q (cid:48) , t a , q ∈ Ω ) and G ( q, t a + τ, q (cid:48) , t a | q ∈ Ξ ) weprove in the Appendix C (see Theorem 1, Corollary 1.1and, Lemma 2) that C t a ( τ ) in Eq. (1) can show a t a -dependence even if Eq. (2) is satisfied, i.e. when the sys-tem is time-translation invariant. That is, if the system isdynamically time asymmetric then C t a ( τ ) depends on t a ,whereas the converse is not necessarily true. In turn thisimplies that one cannot determine on the basis of C t a ( τ )derived from a time-series q ( t ) whether time-translationinvariance is broken, and a definitive and unambiguousindicator must be sought for.We demonstrate this using the cleanest and most ele-mentary example of a time-translation invariant system– a Brownian particle confined to a box of unit length(i.e. L = 1) evolving from a a point Ω = x and froma uniform distribution within an interval Ω = [ a, b ] forsome 0 < a < b <
1. For this example the denominatorin Eq. (3) is defined as P ( q (cid:48) , t, q ∈ Ω ) ≡ (cid:82) ba Q ( q (cid:48) , t | q ) dq and the numerator as P ( q, t a + τ, q (cid:48) , t a , q ∈ Ω ) ≡ Q ( q, τ + t a | q (cid:48) ) P ( q (cid:48) , t a , q ∈ Ω ), where Q ( x, t | x ) denotesthe propagator of the confined Brownian particle. Plug-ging into Eq. (3) confirms the validity of Eq. (2) andhence time-translation invariance. Nevertheless, the verysame system exhibits a t a -dependence of the aging auto-correlation function defined in Eq. (1) over more thantwo orders of magnitude in time measured in units ofthe relaxation time t rel = L /Dπ as depicted explic-itly in Fig. 2. Note that by allowing the box to becomemacroscopic in size (i.e. L → ∞ ) the relaxation timeand thereby the extent of the t a -dependence can becomearbitrarily large when expressed in absolute units.A general mathematical analysis (see Appendix B 2)therefore necessarily ties DTA to the three-point (non-Markovian) conditional probability density, G ( q, τ + t a , q (cid:48) , t a | q ∈ Ω ). If the projected dynamics is Marko-vian it is in turn fully described by two-point con-ditional densities G Markov ( q, τ + t a , q (cid:48) , t a | q ∈ Ω ) = G ( q, τ | q (cid:48) , G ( q (cid:48) , t a | q ∈ Ω ). If, on the other hand, thereduced dynamics is non-Markovian but the initial con-dition is sampled from the full (invariant) stationary den-sity p ( q ) → p inv ( q ) (or equivalently, Ω = Ξ), we have G ( q, τ + t a , q (cid:48) , t a | q ∈ Ξ) = G ( q, τ | q (cid:48) , p inv ( q (cid:48) ). In bothcases there is no DTA (see Appendix C). To quantify bro-ken time-translation invariance on the level of reducedphase space probability densities we therefore define the . .
661 10 − − x = 0 . a) C t a ( τ ) τ [ λ − ] t a = 0 . t a = 0 . t a = 0 . t a → ∞ . .
661 10 − − Ω = [0 . , . b) C t a ( τ ) τ [ λ − ] t a = 0 t a = 0 . t a = 0 . t a → ∞ Figure 2.
Aging correlation functions display fictitiousdynamical time asymmetry in time-translation invari-ant systems.
Analytical results for the aging correlationfunction C t a ( τ ) defined in Eq. (1) for a Brownian particleconfined to a unit box evolving from a) the point Ω = 0 . = [0 . , .
35] for several values of theaging time t a . Time τ is expressed in units of the relaxationtime λ − . time asymmetry index asΥ Ω ( t a , τ ) ≡ (cid:90) Ω dq (cid:90) Ω dq (cid:48) (cid:20) G ( q, τ + t a , q (cid:48) , t a | q ∈ Ω ) × ln G ( q, τ + t a , q (cid:48) , t a | q ∈ Ω ) G ( q, τ | q (cid:48) ) G ( q (cid:48) , t a | q ∈ Ω ) (cid:21) (6)where for notational convenience we henceforth drop theexplicit dependence on Ω , i.e. Υ Ω ( t a , τ ) ≡ Υ( t a , τ ).The time asymmetry index measures the relative entropybetween the actual evolution of the observable and a cor-responding “fictitious” dynamics that has the same prob-ability density of the intermediate point q at time t a butwhere at time t a the latent degrees of freedom are instan-taneously quenched to equilibrium. Broken time trans-lation invariance reflects that the effective equations ofmotion that govern the evolution of q ( t ) change in timeas a result of the relaxation of the hidden DOF the ob-servable is coupled to. That is, if one were e.g. to derivean effective generalized Langevin equation for q ( t ) thelatter would contain a memory kernel and noise that de-pend explicitly on the time elapsed since the preparationof the system (see e.g. [86]).A broken time-translation invariance is evidently aclear signature of non-equilibrium dynamics and there-fore intimately related to entropy production. Υ maythus also be given a thermodynamic interpretation as an entropy associated with the breaking of time-translationinvariance in analogy to the “instantaneous excess freeenergy” – the relative entropy between G ( q, t | q ∈ Ω )and p inv ( q ) [19, 87–89]. Therefore it appears that the en-tropy of breaking time-translation invariance measures the instantaneous thermodynamic displacement of latentdegrees of freedom at time t a from their stationary state.Note that Υ( t a , τ ) > q ( t ) (fordetails see Appendix D 4).The relative entropy is a pseudo-metric and thereforethe absolute value of the time asymmetry index (otherthan Υ( t a , τ ) = 0 implying time-translation invarianceand Υ( t a , τ ) > detect andquantify conclusively broken time-translation invarianceaccording to Eq. (2). It effectively measures the instan-taneous relaxation of the latent degrees of freedom andis unaffected by spurious non-stationarity due to correla-tions between the value of the observable at time t a + τ and the particular “initial” value at time t a . These cor-relations are spurious because they exist for any t a andrelax as a function of τ irrespective of whether a systemis time-translation invariant or not.By construction Υ( t a , τ ) ≥ t a and τ if and only if q ( t ) is time-translationinvariant. In turn, the observable q ( t ) is time-translationinvariant if and only if it is Markovian and/or q ( t = 0)is sampled from a distribution converging in law to theinvariant measure (the proof is presented in the Ap-pendix C, Theorem 2 and Corollary 1.1). As a resultΥ( t a , τ ) is identically zero for all τ and t a for the time-translation invariant dynamics of a confined Brownianparticle evolving from a non-equilibrium initial condi-tion (see, however, the fictitious DTA due to weak non-stationarity that is implied by the aging autocorrelationfunction in Fig. 2). Moreover, the extent of DTA is lim-ited by the relaxation time t rel such that Υ( t a , τ ) → t a (cid:29) t rel or τ (cid:29) t rel . Obviously, if the full sys-tem is initially quenched into any non-stationary initialcondition (see e.g. [19]), then Υ( t a , τ ) > t a , τ ) (cid:54) = 0 for some values t a and τ smaller than t rel , the dynamics is time asymmetric, inspecific cases with a self-similar scaling (see Appendix C,Propositions 1 & 2). In addition the following genericstructure emerges: C t a ( τ ) = (1 − ϕ ) g ( τ ) + ϕg ( τ, t a ) , (7)with 0 < ϕ < g , depending on the details of thedynamics (see Appendix C, Theorem 3) in agreementwith the properties of aging systems [15, 28, 45–49, 54].These results are universal – they are independent of de-tails of the dynamics, and, in particular, the underlyingenergy landscape.Microscopically reversible dynamics in general allowsfor a spectral expansion of propagators and thus correla-tion and response functions (see e.g. Appendix B). More-over, in specific cases the projection renders the observeddynamics self-similar with parameter α , that is, a changeof time-scale merely effects an α -dependent renormalizionof the spectrum (for details see Definition 4 in the Ap-pendix B 2). This arises, for example, when the observ-able corresponds to an internal distance within a singlepolymer molecule [92] (studied here in Figs. 3a and 4)or within individual protein molecules [93, 94], as well asin diffusion on fractal objects [95]. The aging correlationfunction in Eq. (1) then displays a power-law scaling for α > α = 0 a logarithmic behavior (as observed in[38]; see also Eq. (C8) in the Appendix C). The latteris mathematically equivalent to the logarithmic relax-ation found in [96]. For more details see Propositions 1and 2 in the Appendix C, respectively. In particular for τ /t a (cid:29)
1, in the glassy literature referred to as the “fullaging” [20, 49, 96, 97] regime, we find (see Appendix C,Eqs. (C7) and (C10)) C t a ( τ ) (cid:39) A + (cid:40) B α (cid:0) t a τ (cid:1) α , α > ,B α (cid:0) t a τ (cid:1) , α = 0 . (8)with constants A and B α that depend on the detailsof the dynamics. On a transient time-scale the asymp-totic results in Eq. (8) agree with predictions of min-imalistic “trap” models [21, 23, 24] as well as frac-tional dynamics and random walks with diverging wait-ing times [45, 58, 98] (for more details see also Remark 2.1in the Appendix C). Fractional dynamics and randomwalks with long waiting times (that as well display DTA[83, 84, 98]) were in fact explicitly shown to arise as tran-sients in projected dynamics when the latent degrees offreedom are orthogonal to q ( t ) [85] and in the spatialcoarse-graining of continuous dynamics on networks [99].The phenomenology of systems displaying an algebraicscaling of C t a ( τ ) as in Eq. (8) is therefore by no meansunique, and represents only a specific class of dynamicalsystems with a broken time-translation invariance. Dy-namical time asymmetry is much more general. EXAMPLES
It is not difficult to verify the above claims in practiceas all corresponding quantities can readily be obtainedfrom experimental or simulation-derived time-series. Tothat end we analyze DTA in four very different systems(see Fig. 1c-e): DNA hairpin dynamics measured by dualoptical tweezers experiments, where q ( t ) reflects the end- to-end distance (Fig. 1c and Appendix D 4 a) [68, 69],extensive MD simulations of internal motions of yeastPGK, where q ( t ) corresponds to the inter-domain dis-tance (Fig. 1d and Appendix D 4 b) [39], as well as twotheoretical examples: the end-to-end distance fluctua-tions of a Rouse polymer chain [100] (Fig. 1e and Ap-pendix D8) and tracer particle dynamics in a single fileof impenetrable diffusing particles, where q ( t ) reflects theposition of the tracer particle [19, 85, 101] (Fig. 1f andAppendix D 3). The underlying energy landscapes ofthese four systems are fundamentally very different; theDNA-hairpin exhibits two well-defined metastable con-formational states/ensembles [68, 69], the yeast PGK hasa very rugged and apparently fractal energy landscape[39], that of the Rouse polymer is perfectly smooth andexactly parabolic, and that of the single file is flat withthe tracer motion confined to a hyper-cone as a resultof the non-crossing condition between particles. Yet, de-spite these striking differences, all systems display thesame qualitative time asymmetric behavior, consistentwith the proven universality of DTA.The aging correlation functions C t a ( τ ) and time asym-metry indices Υ( t a , τ ) are shown in Fig. 3. With the ex-ception of the PGK protein, which does not equilibratewithin the duration of the trajectory, in agreement withprevious findings [39], DTA is manifested as a transientphenomenon. The precise form of C t a ( τ ) depends on thedetails of the dynamics, which naturally vary betweenthe systems. Moreover, the dependence of C t a ( τ ) on t a is non-monotonic. The generic form of Υ( t a , τ ) displaysan initial increase towards a plateau, followed by a long-time decay to zero, which can be understood as follows.Irrespective of the details a finite time is required in orderto allow for a build-up of memory, that is, of correlationsbetween the instantaneous state of the projected observ-able and the initial condition of the latent variables. Thememory at some point reaches a maximum. Afterwards,the memory of the preparation of the system is progres-sively lost as a result of the mixing of trajectories in fullphase space during relaxation. Due to a relatively highersampling frequency and sufficiently long sampling timesthat extend beyond the relaxation time all these effectsare resolved in the experimental DNA-hairpin data butnot in the case of the PGK simulation.Moreover, a hallmark of aging is that at least part ofthe relaxation of a system takes place on time-scales thatgrow with the age of the system t a , and continue to do soup to the largest times accessible within an experimentor simulation. Interestingly, Figs. 3 and 4 show that therelaxation time increases (at least transiently) with theaging time, i.e. Υ( t a , τ ) decays with t more slowly as t a grows at least up to a threshold time. If an experiment orsimulation does not reach this threshold time the break-ing of time-translation invariance would seemingly takeplace on timescales that grow indefinitely, somewhat sim-ilar to the aging phenomenon. Note that the thresholdtime may become arbitrarily large in large systems (e.g.the relaxation time and thus the threshold time in natu- . c ) DNA Hairpin C t a ( τ ) t a = 02 . . . d ) C t a ( τ ) t a = 06 . . a ) Rouse Chain C t a ( τ ) t a = 0110 00 . b ) Single File C t a ( τ ) t a = 00 . . − − τ [ ms ]10 − − t a [ m s ] . . . t a , τ ) 10 τ [ ps ]10 t a [ p s ] . . . t a , τ )10 − τ − t a . . . . t a , τ ) 10 − τ − t a . . . . t a , τ ) Figure 3.
Aging two-point correlation function and time asymmetry index. C t a ( τ ) for different values of aging time t a and corresponding Υ( t a , τ ) for: a) the Rouse polymer chain with 50 beads with initial end-to-end distance in dimensionlessunits equal to q = 9 .
85, which corresponds to the most likely end-to-end distance (the dimensionless relaxation time herecorresponds to t rel (cid:39) . N = 5 confined to a box of unit length, taggingthe central particle particle with initial condition q = 0 . t rel (cid:39) . . · ms sampled at 400kHz.The initial condition was taken at the absolute maximum of equilibrium probability density q = 2 . ± q refersto deviations from the mean distance (cid:104) d (cid:105) , i.e. q ( t ) = d ( t ) − (cid:104) d (cid:105) (the relaxation time is t rel ≈
15 ms); The statistical error indetermining Υ( t a , τ ) from the hairpin data is less than 1% (see Fig. D6 in the Appendix D 4 a); d) inter-domain motion betweenthe centers of mass of the N-terminal (residues 1-185) and C-terminal domains (residues 200-389) in yeast PGK determinedfrom a 200 ns atomistic MD simulation sampled every 150 ps. The initial condition was q = 0 . ± . (cid:104) d (cid:105) , i.e. q ( t ) = d ( t ) − (cid:104) d (cid:105) . c) was obtained from experimental data of Refs. [68, 69] and d) wasdetermined from molecular dynamics simulations in Ref. [39]. Further details can be found in Appendix D. “Transient aging”in C t a ( τ ) arises whenever there is a region ( t a , τ ) where Υ( t a , τ ) >
0. In the case of PGK (panel d) t rel is not reached withinthe simulation time, which renders the system virtually eternally time asymmetric and “forever aging” [39, 102]. ral units for the Rouse polymer and single file grow withthe number of particles as ∝ N (see e.g. Fig. D2 in theAppendix D8); for any duration of an observation onemay find a N that makes DTA appear as everlasting).One appreciates that Υ( t a , τ ) truly quantifies the de-gree of broken time-translation invariance and not cor-relations with the value of the observable at t a . This isalso the reason why Υ( t a , τ ) decays to zero on a time-scale shorter than C t a ( τ ). C t a ( τ ) starts at 1 and decaysto zero as a result of “forgetting the initial condition”.Because the probability density of being found at a givenpoint always depends trivially on t a (cid:54) = 0 (see Eq. (5)) irre- spective of whether time-translation invariance in Eq. (2)is broken or satisfied, C t a ( τ ) displays non-stationaritymanifested in a t a -dependence even for time-translationsymmetric dynamics. Conversely, Υ( t a , τ ) is constructedto not be affected by such spurious non-stationarity. In-stead, it reflects how far the latent degrees of freedom aredisplaced from equilibrium at time t a . In other words,Υ( t a , τ ) compares the probability densities of the actualdynamics with those of fictitious dynamics that have thesame probability density at time t a but in which at time t a the latent degrees of freedom are quenched to equilib-rium (see Eq. (B25)). Figure 4.
Attenuation and disappearance of dynamical time asymmetry upon approaching stationary initialconditions.
Gradual vanishing of the time asymmetry index Υ( t a , τ ) when the initial distribution of the projected coordinate p ( q ) is sampled from a distribution being closer and closer to the density of the invariant measure, p inv ( q ) for: a) Rousemodel of a polymer chain (the parameters are the same as in Fig. 3); The initial end-to-end distance is sampled from intervals(from left to right): q =9.85, q ∈ [9 − q ∈ [7 −
13] and q ∈ [4 − (cid:104) d (cid:105) , that is, d ( t ) = q ( t ) + (cid:104) d (cid:105) ) are sampled (from left to right)from the following intervals: q ∈ [1 ,
3] nm, q ∈ [ − ,
7] nm, q ∈ [ − ,
10] and q ∈ [ − ,
12] nm, respectively. When the initialcondition is sampled from a distribution closer to the invariant measure, DTA vanishes confirming the claims of our theory.
One can look at Υ( t a , τ ) in two ways; as a function of τ at fixed t a and as a function of t a at fixed τ . Whilethe former intuitively reflects how the relaxation of theobservable to equilibrium depends on the instantaneous(“initial”) state of the latent degrees of freedom at time t a , the latter measures how the correlation of the value ofthe observable at two times separated by τ changes due tothe relaxation of the latent degrees of freedom to equi-librium. The time asymmetry index therefore providesaccess to the dynamics of hidden degrees of freedom cou-pled to the observable through an analysis of time-seriesderived from measurements on the observable.A verification that a breaking of time-translation in-variance occurs whenever the distribution of initial con-ditions sampled by the experiment has not converged tothe equilibrium distribution follows from inspection ofΥ( t a , τ ) evolving from an ensemble of initial conditionsbeing closer and closer to an equilibrium distribution,i.e. Ω → Ξ (see Fig. 4 for the Rouse chain and DNA-hairpin). Indeed, Υ( t a , τ ) progressively vanishes whenthe initial condition becomes sampled from a distributionapproaching the invariant measure, p ( q ) → p inv ( q ).In the Appendix C we prove that this is a general effect(Theorem 1), independent of any details of the dynamics. DISCUSSION
Non-stationary behavior of physical observables is tra-ditionally considered as being important in systems withglassy, aging dynamics, such as polymer, spin or colloidalglasses, that attain glassy properties upon a quench in anexternal parameter [26–31]. During, for example, a tem-perature quench, the system (e.g. a supercooled liquid ora set of spins) at some point cannot keep pace with rapidchanges in the bath, and is pushed out of equilibrium [14].After the quench at t = 0 the observable is thus (at leastweakly) non-stationary – it is sampled from and averagedover a non-equilibrium ensemble, i.e. p ( q ) (cid:54) = p inv ( q ).The absence of such an obvious quench rendered the ori-gin of non-stationary, apparent aging behavior in biolog-ical macromolecules somewhat mysterious [36, 37, 39–43]. However, in biological systems the observable canbecome quenched implicitly, e.g. by the ’locking in’ ofa protein’s configuration by a chaperone [74], the config-urational requirements for enzymatic catalysis [41–43],or simply by the under-sampling of equilibrium such asin single-molecule experiments and particle-based com-puter simulations [39], such that p ( q ) (cid:54) = p inv ( q ). Inan experiment one can check for non-stationarity of ini-tial conditions, e.g. by inspecting whether histograms ofthe observable (also referred to as the “occupation timefraction” or “empirical density”) at t = 0 and at all latertimes coincide [103].Here, we highlight a more general and wide-spread as-pect of out-of-equilibrium dynamics of physical observ-ables – dynamical time asymmetry. The requirements forDTA to occur are much weaker than for aging, and it ismanifested in a very broad variety of experimental situa-tions, and in particular, one may also expect aging phys-ical observables probed in many experiments to displayDTA. Even measurements on polymer, spin and colloidalglasses have built-in underlying projections. For exam-ple, in tensile creep experiments in polymeric glasses themotion in a (cold) polymer is projected onto a local, effec-tively one-dimensional flow [26]. In supercooled liquidsand colloidal glasses the dynamics is typically projectedonto local particle displacements, pair correlation func-tions and structure factors [30, 31, 33, 34]. In bulk ex-periments with spin glasses and supercooled liquids onemeasures quantities such as the average single-spin auto-correlation function [21, 104] , magnetization, conduc-tance or the dielectric constant, which correspond to pro-jections of many-particle dynamics onto a scalar param-eter [29, 32, 49]. In biological macromolecules the pro-jection may correspond to [37, 39] or depend on [41–43]some internal distance within the macromolecule. Theseprojections lead to non-Markovian observables evolvingfrom non-stationary initial conditions which are in turnexpected to show DTA. In fact we can appreciate thatthe physical origin of DTA in both, ’traditional’ glassysystems [26–31] and biological matter [36, 37, 39–43],is qualitatively the same and simply results from non-stationary initial conditions of non-Markovian observ-ables (see Observation 2 in the Appendix C). In mostof these aforementioned systems the dynamics is also ag-ing [26–31, 36, 37, 39].It is important to realize that it is not possible to inferfrom a finite measurement whether the observed processis genuinely non-ergodic (i.e. a result of some true lo-calization phenomenon in phase space) or whether theobservation is made on an ergodic system but on a time-scale shorter the relaxation time [85] (note that a compar-ison of the dynamics of PGK in Fig. 3d with a transientshorter than the relaxation time in any of the remainingexamples in Fig. 3a-c shows no qualitative difference). Atheoretical description of both scenarios on time-scalesshorter than the relaxation time is in fact identical (fordetails see [85] as well as [24] in the context of glasses).Although sporting characteristics commonly associ-ated with aging, DTA and aging are not quite the samething. DTA does not require the relaxation to take placeon time-scales that grow indefinitely with the age of thesystem t a beyond the largest times accessible within anexperiment or simulation, nor does it impose require-ments on the precise form of the dependence on t a . It islikely to be a ubiquitous phenomenon that is frequentlyobserved in measurements of projected observables. Inturn, aging does not imply a broken time-translation in-variance according to Eq. (2).Note, however, that many paradigmatic models of ag-ing dynamics (e.g. continuous-time random walks withdiverging mean waiting times and fractional diffusion [45, 83, 84]) display a (strongly) broken time-translationinvariance. Furthermore, most experimental observa-tions of aging dynamics monitor projected observables,e.g. magnetization, single-spin auto-correlation functionsaveraged over the sample and potentially also over disor-der [26–31, 36, 37, 39]. The dynamics of these observablesis thus almost surely non-Markovian [85] and expected todisplay DTA.The observation of Υ( t a , τ ) > t a and τ implies that the dynamics of the observable q ( t )fundamentally changes in the course of time as a results ofthe relaxation of hidden DOF, and does not reflect corre-lations with the value of the observable at zero time q (0).That is, the effective equations of motion for q ( t ) trulychange in time. In biological systems and in particularenzymes and other protein nanomachines non-stationaryeffects are thought to influence function, e.g. memoryeffects in catalysis [41–43]. This is particularly impor-tant because some larger proteins potentially never relaxwithin their life-times, i.e. before they become degraded(note that relaxation corresponds to attaining the spon-taneous unfolding-refolding equilibrium). This rendersthe dynamically time asymmetric regime virtually ’for-ever lasting’ and implies that the system is aging [39].As proteins are produced in the cell in an ensemble offolded configurations under the surveillance of chaper-ones [74], our theory implies that DTA during function[41–43] should arise naturally and generically due to thememory of a protein’s preparation.We expect DTA to be particularly pronounced inmeasurements on systems with entropy-dominated, tem-porally heterogeneous collective conformational dynam-ics involving (transient) local structure-formation wherethe background DOF evolve on the same time-scaleas the observable [38], and we suggest the breakingof time-translation invariance to be closely related tothe phenomenological notion of “dynamical disorder” inbiomolecular dynamics [41–43, 71].Our results have some intriguing implications. First,a quench in an external parameter and the mere under-sampling of equilibrium distributions give rise to quali-tatively equivalent manifestations (but potentially witha largely different magnitude and duration) of DTA assoon as the observable follows a non-Markovian evolution(see Appendix C, Observation 2). This has importantpractical consequences in fields such as single-moleculespectroscopy and computer simulations of soft and bi-ological matter, which often suffer from sampling con-straints. Second, broken time-translation invariance is’in the eye of the beholder’, insofar as its degree dependson the specific observable; there should exist a (poten-tially less) reduced coordinate, not necessarily accessibleto experiment (e.g. when we follow all degrees of free-dom), according to which the same system will exhibitvirtually time-translation invariant dynamics. However,auto-correlation functions will show a t a -dependence foressentially any non-stationary initial condition in any sys-tem.0A broken time-translation invariance was shown to belinked to a form of entropy embodied in a time asymmetryindex that is a measure of the instantaneous thermody-namic displacement of latent, hidden degrees of freedomfrom their stationary state. The time asymmetry indexmay therefore be used to probe systematically the time-scale of dynamics of hidden, slowly relaxing degrees offreedom relative to the time-scale of the evolution of theobservable. In particular, it may be useful as a prac-tical tool to discriminate between situations where thehidden degrees of freedom evolve through a sequence oflocal equilibria that would yield small values of the timeasymmetry index Υ from those cases where their evolu-tion is transient and slow on the time-scale of the ob-servable thus implying a significant Υ. For example, Υmay potentially provide additional insight into the domi-nant folding mechanism of a protein from single-moleculeforce-spectroscopy data [105], in particular about themuch debated heterogeneity of folding trajectories andits functional relevance [106, 107].The present theory ties dynamical time asymmetry ina general setting to both the non-stationary prepara-tion of an observable and its non-Markovian time evo-lution. Thereby it connects aspects of the better knownphenomenology of aging of projected observables withthe broken time-translation invariance observed in recentmeasurements on in soft and biological materials on acommon footing. Moreover, dynamical time asymmetryis suggested to be a ubiquitous phenomenon in biologicaland materials systems. ACKNOWLEDGMENTS
We thank Krishna Neupane and Michael T. Woodsidefor providing unlimited access to their DNA-hairpin dataand Peter Sollich for clarifying discussion about physi-cal aging and critical reading of the manuscript. Thefinancial support from the
Deutsche Forschungsgemein-schaft (DFG) through the
Emmy Noether Program ”GO2762/1-1” (to AG) and from the
Department of Eenergy through the grant DOE BER FWP ERKP752 (to JCS)are gratefully acknowledged.
APPENDIX
In this Appendix we present the main theorems needed for the article with the corresponding proofs.We treat the problem in a general setting, that is, not assuming that the full system is initially preparedin equilibrium. Further included are analytical results with details of calculations for the Rouse polymerand single file diffusion, all details of the numerical analyses of the DNA-hairpin and protein PGK dataand further supporting results.
CONTENTS
A. Definitions, notation and preliminaries 1B. Dynamics of the projected lower-dimensional observable 31. Spectral theory of projected dynamics 32. Three-point dynamics and breaking of time-translation invariance 5C. Main theorems with proofs 6D. Physical models, experimental and simulation data 101. Fictitious dynamical time asymmetry in a time-translation invariant system: the Brownian particle in a box102. Rouse polymer model 113. Single file diffusion 154. Analysis of experimental and simulation data 17a. DNA-hairpin 18b. Yeast 3-phosphoglycerate kinase (PGK) 20References 23
Appendix A: Definitions, notation and preliminaries
We consider a stable conservative mechanical system in a continuous domain Ω ∈ R d that is at least weakly coupledto a thermal bath with Gaussian statistics with the longest correlation time τ b being much shorter than that of thesystem, τ s (i.e. τ b (cid:28) τ s ) such that the bath can be considered as representing stationary white noise on the time-scaleof the system’s dynamics [73]. The thermal bath is either external or the result of integrating out an additionalsubset of internal degrees of freedom that relaxes much faster than the system. At any time t the state of the systemis specified by a d -dimensional state (column) vector x t ∈ R d , whose entries are generalized coordinates x t,i . Notethat the dynamics in soft matter and biological systems is typically strongly overdamped which we also assume here.The extension to underdamped systems is conceptually straightforward (since we consider microscopically reversibledynamics) [108], but since a broken time-translation invariance in soft and biological matter is not tied to momenta,we omit these for convenience. We are strictly interested in the evolution of x t for t (cid:29) τ b . It is well known that undercertain technical conditions imposed on the dynamics of the bath [73], which we will not further detail here but arestrictly granted for the physical systems relevant to the discussion, x t evolves according to the Itˆo equation d x t = F ( x t ) dt + σ d W t (A1)where W t is a d -dimensional vector of independent Wiener processes whose increments have a Gaussian distributionwith zero mean and variance dt , i.e. E [ dW t,i dW t (cid:48) ,j ] = δ ij δ ( t − t (cid:48) ) dt , E [ · ] denotes the expectation over the ensemble ofWiener increments and where σ is a d × d symmetric noise matrix. If momentum coordinates were included σ wouldbe positive semi-definite with zeros in the sector of position variables and non-zero terms proportional to the frictionconstant γ in the momentum sector, and is strictly positive definite with terms ∝ γ − for over-damped dynamics (i.e.for γ (cid:29)
1) [108]). We focus on microscopically reversible dynamics, that is, we consider d -dimensional Markoviandiffusion with a d × d symmetric positive-definite diffusion matrix D = σσ T / M = D /k B T (with β − ≡ k B T being the thermal energy) in a drift field F ( x ), such that M − F ( x ) = −∇ ϕ ( x ) is a gradient flow. Thedrift field F ( x ) : R d → R d , is either nominally confining (in this case Ω is open) or is accompanied by correspondingreflecting boundary conditions at ∂ Ω (in this case Ω is closed) thus guaranteeing the existence of an invariant measureand hence ergodicity [73, 108].On the level of probability measures in phase space the dynamics is governed by the (forward) Fokker-Planckoperator ˆ L : V → V , where V is a complete normed linear vector space with elements f ∈ C ( R d ). In particular,ˆ L = ∇ · D ∇ − ∇ · F ( x ) . (A2) F ( x ) is assumed to be sufficiently confining, i.e. lim x →∞ P ( x , t ) = 0 , ∀ t sufficiently fast to assure that ˆ L correspondsto a coercive and densely defined operator on V with a pure point spectrum [109–111]. ˆ L propagates probabilitymeasures µ t ( x ) in time, which will throughout be assumed to possess well-behaved probability density functions P ( x , t ), i.e. dµ t ( x ) = P ( x , t ) d x . The nullspace of ˆ L (i.e. the solution of ˆ L P eq ( x ) = 0) is the equilibrium (Maxwell-)Boltzmann-Gibbs distribution, P eq ( x ) = Q − e − βϕ ( x ) , with partition function Q = (cid:82) Ω d x e − βϕ ( x ) . We define the(forward) propagator ˆ U ( t ) = e ˆ L t that is the generator of a semi-group ˆ U ( t + t (cid:48) ) = ˆ U ( t ) ˆ U ( t (cid:48) ). The formal solution ofthe Fokker-Planck equation ( ∂ t − ˆ L ) P ( x , t ) = 0 is thereby given as P ( x , t ) = ˆ U ( t ) P ( x , x t will be denoted by (cid:104)·(cid:105) and in the case of a physical observable B ( x t ) is given by (cid:104)B ( x t ) (cid:105) ≡ (cid:90) B ( x ) dµ t ( x ) ≡ (cid:90) Ω B ( x ) P ( x , t ) d x ≡ (cid:90) Ω B ( x ) ˆ U ( t ) P ( x , d x (A3)Part of the analysis will involve the use of spectral theory in Hilbert space, for which it is convenient to introducethe bra-ket notation; the ’ket’ | g (cid:105) represents a vector in V written in position basis as g ( x ) ≡ (cid:104) x | g (cid:105) , and the ’bra’ (cid:104) h | as the integral (cid:82) d x h † ( x ). The scalar product is defined with the Lebesgue integral (cid:104) h | g (cid:105) = (cid:82) d x h † ( x ) g ( x ). Inthis notation we have the following evolution equation for the probability density function starting from an initialcondition | p (cid:105) : | p t (cid:105) = e ˆ L t | p (cid:105) . Since the process is ergodic we have lim t →∞ e ˆ L t | p (cid:105) = | eq (cid:105) , where (cid:104) x | eq (cid:105) = P eq ( x ). Wealso define the (typically non-normalizable) ’flat’ state | – (cid:105) , such that (cid:104) x | – (cid:105) = 1 and (cid:104) – | p t (cid:105) = 1. Hence, ∂ t (cid:104) – | p t (cid:105) = 0and (cid:104) – | ˆ L = 0.Whereas ˆ L by itself is not self-adjoint, it is orthogonally equivalent to a self-adjoint operator, i.e. the operatorˆ L s = e βϕ ( x ) / ˆ L e − βϕ ( x ) / is self-adjoint, and, moreover the operator e βϕ ( x ) ˆ L is self-adjoint (for a proof see [108]).Because any self-adjoint operator in Hilbert space is diagonalizable, ˆ L is diagonalizable as well, but with a separateset of left and right bi-orthonormal eigenvectors (cid:104) ψ Lk | and | ψ Rk (cid:105) , respectively. That is, ˆ L| ψ Rk (cid:105) = − λ k | ψ Rk (cid:105) and (cid:104) ψ Lk | ˆ L = − λ k (cid:104) ψ Lk | with real eigenvalues λ k ≥ λ = 0, | ψ R (cid:105) = | eq (cid:105) , (cid:104) ψ L | = (cid:104) – | ,and (cid:104) ψ Lk | ψ Rl (cid:105) = δ kl . Moreover, since e βϕ ( x ) ˆ L is self-adjoint it follows that that | ψ Lk (cid:105) = e βϕ ( x ) | ψ Rk (cid:105) . The resolution ofidentity is given by = (cid:80) k | ψ Rk (cid:105)(cid:104) ψ Lk | and the propagator by ˆ U ( t ) = (cid:80) k | ψ Rk (cid:105)(cid:104) ψ Lk | e − λ k t .The Markovian Green’s function of the process x t corresponds to the conditional probability density function for alocalized initial condition (cid:104) x | p (cid:105) = δ ( x − x ) and is defined as Q ( x , t | x ,
0) = (cid:104) x | ˆ U ( t ) | x (cid:105) , such that the probabilitydensity starting from a general initial condition | p (cid:105) becomes P ( x , t, p ) = (cid:104) x | ˆ U ( t ) | p (cid:105) ≡ (cid:82) d x p ( x ) Q ( x , t | x , Q ( x , t | x ,
0) = (cid:88) k ψ Rk ( x ) ψ Lk ( x )e − λ k t , (A4)where the semi-group property means that Q ( x , τ | x ,
0) = Q ( x , t + τ | x , t ) is independent of t as is easily verified via (cid:90) Ω d x (cid:48) Q ( x , t | x (cid:48) , t (cid:48) ) Q ( x (cid:48) , t (cid:48) | x ,
0) = (cid:88) k,l ψ Rk ( x ) (cid:104) ψ Lk | ψ Rl (cid:105) ψ Ll ( x )e − λ k ( t − t (cid:48) ) − λ l t (cid:48) ≡ Q ( x , t | x , , (A5)where we have used that (cid:104) ψ Lk | ψ Rl (cid:105) = δ k,l .In the presence of a time-scale separation giving rise to local equilibrium the system’s dynamics may be coarse-grained further into a discrete-state Markov jump master equation (see e.g. [112, 113]). In this case the configurationspace would be discrete and d − dimensional, ˆ L would be replaced by a d × d symmetric stochastic matrix M , andthe Fokker-Planck equation by the master equation ddt Q = M Q . Since this situation corresponds to an approximate,lower-resolution dynamics of the system that is mathematically simpler and the mapping between the Fokker-Planckequation and Markov-state jump dynamics is well-known [103, 108, 112] and does not introduce any further conceptualchanges (the complete spectral-theoretic approach in particular remains unchanged), we will without any loss ofgenerality focus on the continuous scenario. Appendix B: Dynamics of the projected lower-dimensional observable
In order to describe the dynamics of the r -dimensional projected observable q = Γ ( x ) : R d → R r with r < d and q lying in some orthogonal system in Euclidean space q ∈ Ξ( R r ) ⊂ Ω( R d ), we define the operator ˆ P x ( Γ ; q ), such that,when applied to some function Z ( x ) ∈ V , ˆ P x ( Γ ; q ) gives (see [85])ˆ P x ( Γ ; q ) Z ( x ) ≡ (cid:90) Ω d x δ ( Γ ( x ) − q ) Z ( x ) , (B1)where δ ( y ) is to be understood in the distributional sense. We can now define the (in general) non-Markovian two-point conditional probability density of projected dynamics starting from q ∈ Ξ , where the subdomain Ξ is notnecessarily simply connected, with the extended operator ˆ P x ( Γ ; q ∈ Ξ ) = (cid:82) Ξ d q ˆ P x ( Γ ; q ) in terms of the single-pointand joint two-point density P p ( q ∈ Ξ ) and P p ( q , t, q ∈ Ξ ), respectively, as G p ( q , t | q ∈ Ξ ) = P p ( q , t, q ∈ Ξ ) P p ( q ∈ Ξ ) ≡ ˆ P x (Γ; q ) ˆ P x ( Γ ; q ∈ Ξ ) Q ( x , t | x , p ( x )ˆ P x ( Γ ; q ∈ Ξ ) p ( x ) (B2)with the convention that P p ( q , t, q ) and G p ( q , t | q ) stand for Ξ corresponding to a single point q . The full systemis said to have a stationary preparation if and only if p ( x ) = P eq ( x ), whereas the projected observable is said tohave a stationary preparation if and only if Ξ = Ξ. Note that lim t →∞ P p ( q , t, q ∈ Ξ ) = P eq ( q ) (cid:82) Ξ d q P p ( q ),where we have defined P eq ( q ) ≡ ˆ P x (Γ; q ) P eq ( x ) as well as P p ( q ) ≡ ˆ P x ( Γ ; q ) p ( x ). In turn it follows thatlim t →∞ G p ( q , t | q ∈ Ξ ) = P eq ( q ). Eq. (B2) demonstrates that the entire time evolution of projected dynamicsstarting from a fixed condition q depends on the initial preparation of the full system p ( x ) as denoted by thesubscript, which is the first signature of the non-stationary nature of projected dynamics. In addition, the dynamicsdescribed by Eq. (B2) is, except for quite exotic projections Γ ( x ), non-Markovian (see [85]).We can now define averages and two-point correlation functions of q ( t ). The n -th moment of the position averagedover an ensemble of all projected non-Markovian evolutions prepared in the point q while the full system at t = 0 isprepared in the state p ( x ) is given by (cid:104) q ( t ) n (cid:105) Ξ p ≡ (cid:90) Ξ d qq n G p ( q , t | q ∈ Ξ ) , (cid:104) q n (cid:105) Ξ p ≡ (cid:90) Ξ d q q n P p ( q ) (B3)where we are here only interested in n = 1 ,
2, whereas the most general tensorial two-point (0 , t ) (non-aging) correlation(i.e. covariance) matrix is defined as C Ξ ( t ; p ) ≡ (cid:104) q ( t ) ⊗ q (0) (cid:105) Ξ p − (cid:104) q ( t ) (cid:105) Ξ p ⊗ (cid:104) q (cid:105) Ξ p = (cid:90) Ξ d q (cid:90) Ξ d q ( q ⊗ q ) P p ( q , t, q ) − (cid:104) q ( t ) (cid:105) Ξ p ⊗ (cid:104) q (cid:105) Ξ p , (B4)such that lim t →∞ C ( t ; p ) = 0 , ∀ p , where from the scalar version is in turn obtained by taking the trace C Ξ ( t ; p ) ≡ (cid:104) q ( t ) · q (0) (cid:105) Ξ p − (cid:104) q ( t ) (cid:105) Ξ p · (cid:104) q (cid:105) Ξ p = Tr C Ξ ( t ; p ) (B5)with the convention C Ξ ( t ; P eq ) = (cid:104) q ( t ) · q (0) (cid:105) Ξ eq − (cid:104) q ( t ) (cid:105) Ξ eq · (cid:104) q (cid:105) Ξ eq ≡ C Ξ ( t ). We can equivalently define the time-dependent variance of q ( t ) with q (0) = q ∈ Ξ as σ ( t ; p ) ≡ (cid:104) q ( t ) (cid:105) Ξ p − ( (cid:104) q ( t ) (cid:105) Ξ p ) (B6)
1. Spectral theory of projected dynamics
We now use spectral theory of the Markovian Green’s function in Eq. (A4) to analyze the general properties of thenon-Markovian time evolution of the projected lower-dimensional observable q ( t ). As the initial preparation of thefull system p ( x ) was found to determine the point-to-point propagation of the probability density of q , we begin byexpanding the initial condition of the full system p ( x ) in the eigenbasis of ˆ L , i.e. p ( x ) = (cid:80) l | ψ Rl (cid:105) (cid:10) ψ Ll | p (cid:11) . Theonly assumptions made for p ( x ) are that it is normalized, Lebesgue integrable (such that (cid:10) ψ Ll | p (cid:11) exists) and locallysufficiently compact to assure that the projection at time t = 0 does not project onto an empty set of the observable q . By further introducing the elements of the following infinite-dimensional matricesΨ kl ( q ) = (cid:104) ψ Lk | δ ( Γ ( x ) − q ) | ψ Rl (cid:105) , Ψ kl (Ξ ) = (cid:90) Ξ d q (cid:104) ψ Lk | δ ( Γ ( x ) − q ) | ψ Rl (cid:105) (B7)where lim Ξ → q Ψ kl (Ξ ) = Ψ kl ( q ), we can express P p ( q , t | q ∈ Ξ ) in Eq. (B2) as P p ( q , t, q ∈ Ξ ) = (cid:88) k e − λ k t Ψ k ( q ) (cid:88) l Ψ kl (Ξ ) (cid:10) ψ Ll | p (cid:11) (B8)and since the preparation of the projected observable is P p ( q ∈ Ξ ) = (cid:80) l Ψ l (Ξ ) (cid:10) ψ Ll | p (cid:11) , the conditional non-Markovian two-point density as G p ( q , t | q ∈ Ξ ) = (cid:80) k e − λ k t Ψ k ( q ) (cid:80) l Ψ kl (Ξ ) (cid:10) ψ Ll | p (cid:11)(cid:80) l Ψ l (Ξ ) (cid:10) ψ Ll | p (cid:11) . (B9)For a stationary preparation of the full system, i.e. p ( x ) = P eq ( x ), we have that (cid:10) ψ Ll | P eq (cid:11) = δ l, and hence P eq ( q ∈ Ξ ) = Ψ (Ξ ) as well as G eq ( q , t | q ∈ Ξ ) = P eq ( q , t, q ∈ Ξ ) P eq ( q ∈ Ξ ) = (cid:80) k Ψ k ( q )Ψ k (Ξ )e − λ k t Ψ (Ξ ) . (B10)As a result (cid:104) q ( t ) (cid:105) Ξ p = (cid:88) k e − λ k t (cid:18)(cid:90) Ξ d qq Ψ k ( q ) (cid:19) (cid:80) l Ψ kl (Ξ ) (cid:10) ψ Ll | p (cid:11)(cid:80) l Ψ l (Ξ ) (cid:10) ψ Ll | p (cid:11) (cid:104) q ( t ) (cid:105) Ξ eq = (cid:88) k e − λ k t (cid:18)(cid:90) Ξ d qq Ψ k ( q ) (cid:19) Ψ k (Ξ )Ψ (Ξ ) . (B11)Furthermore, we find thatlim Ξ → Ξ Ψ kl (Ξ ) = (cid:90) Ξ d q (cid:104) ψ Lk | δ ( Γ ( x ) − q ) | ψ Rl (cid:105) = (cid:104) ψ Lk | (cid:90) Ξ d q δ ( Γ ( x ) − q ) | ψ Rl (cid:105) = (cid:104) ψ Lk | ψ Rl (cid:105) = δ k,l , (B12)where the order of integration can be exchanged since the delta function in the distributional sense is smooth (i.e. thelimit to a ’true’ delta-function is taken after the integrals) and the domain of the q integration Ξ by definition includesall mappings q = Γ ( x ) such that (cid:82) Ξ d q δ ( Γ ( x ) − q ) = 1. As a result lim Ξ → Ξ G eq ( q , t | q ∈ Ξ ) = Ψ ( q ) = P eq ( q ) , ∀ t .Using these spectral-theoretic results it follows immediately that the elements of the general tensorial second momentmatrix read (cid:104) ( q ( t ) ⊗ q (0)) ij (cid:105) Ξ p = (cid:88) k e − λ k t (cid:18)(cid:90) Ξ i dq i q i Ψ k ( q i ) (cid:19) (cid:88) l (cid:10) ψ Ll | p (cid:11) (cid:32)(cid:90) Ξ ,j dq ,j q ,j Ψ kl ( q ,j ) (cid:33) (cid:104) ( q ( t ) ⊗ q (0)) ij (cid:105) Ξ eq = (cid:88) k e − λ k t (cid:18)(cid:90) Ξ i dq i q i Ψ k ( q i ) (cid:19) (cid:32)(cid:90) Ξ j, dq ,j q ,j Ψ k ( q ,j ) (cid:33) , (B13)which, once plugged into Eq. (B4) together with Eq. (B11) and the right member of Eq. (B3), yield the tensorialcorrelation (or covariance) matrix C ( t ; p ). The case treated in the main text, that is, when the projected coordinateis one-dimensional and the full-system’s preparation is stationary, follows trivially by appropriate simplification ofEq. (B1) and insertion into Eq. (B10), which leads to C ( t ; eq) ≡ (cid:104) q ( t ) q (0) (cid:105) Ξ eq − (cid:104) q ( t ) (cid:105) Ξ eq (cid:104) q (0) (cid:105) Ξ eq = (cid:88) k e − λ k t (cid:18)(cid:90) Ξ dqq Ψ k ( q ) (cid:19)(cid:18)(cid:90) Ξ dq q Ψ k ( q ) − Ψ k (Ξ )Ψ (Ξ ) (cid:90) Ξ dq q Ψ ( q ) (cid:19) . (B14)As we now show in the following section dynamical time asymmetry (i.e. broken time-translation invariance) isinherently tied to non-Markovian three-point probability density functions of the projected observable.
2. Three-point dynamics and breaking of time-translation invariance
In order to describe dynamical time asymmetry we introduce two times, the so-called “aging” (or “waiting”) time, t a , and the observation time window τ = t − t a . More precisely, we consider, as in the previous section, that the fullsystem was prepared at t = 0 in a general (not necessarily stationary) state p ( x ), whereby the choice of time originis dictated by the initiation of an experiment or the onset of a phenomenon. The actual observation starts at somelater (aging) time t a ≥ t and hence has a duration τ = t − t a . An example of anon-stationary preparation of a full system would be a temperature quench of a system equilibrated at some differenttemperature. We assume, as before, that only the lower-dimensional observable q ( t ) is observed for all times t ≥ C Ξ t a ( τ ; p ) ≡ C Ξ t a ( τ ; p ) C Ξ t a (0; p ) = (cid:104) q ( τ + t a ) ⊗ q ( t a ) (cid:105) Ξ p − (cid:104) q ( τ + t a ) (cid:105) Ξ p ⊗ (cid:104) q ( t a ) (cid:105) Ξ p (cid:104) q ( t a ) ⊗ q ( t a ) (cid:105) Ξ p − (cid:104) q ( t a ) (cid:105) Ξ p ⊗ (cid:104) q ( t a ) (cid:105) Ξ p (B15)such that ˆ C t a ( τ ; p ) ≡ Tr ˆ C t a ( τ ; p ) and for the one-dimensional coordinate starting from a system prepared in astationary state that is studied in the main paperˆ C Ξ t a ( τ, eq) ≡ ˆ C Ξ t a ( τ ) ≡ C Ξ t a ( τ ) C Ξ t a (0) = (cid:104) q ( τ + t a ) q ( t a ) (cid:105) Ξ − (cid:104) q ( τ + t a ) (cid:105) Ξ (cid:104) q ( t a ) (cid:105) Ξ (cid:104) q ( t a ) (cid:105) Ξ − ( (cid:104) q ( t a ) (cid:105) Ξ ) . (B16)From the definitions of aging observables in Eqs. (B15-B16) it follows that these are inherently tied to three-pointprobability density functions at times 0 , t a , and t a + τ . The full system’s dynamics, corresponding to a Hamiltoniandynamics coupled to a Markovian heat bath, is Markovian and time-translation invariant. The three-point jointdensity therefore reads P p full ( x , t a + τ, x (cid:48) , t a , x ) = Q ( x , t a + τ | x (cid:48) , t a ) Q ( x (cid:48) , t | x , p ( x ) . (B17)Using the definitions from the previous section and introducing the shorthand notation ˆ P x , x (cid:48) , x ( Γ ; q , q (cid:48) , q ∈ Ξ ) ≡ ˆ P x ( Γ ; q ) ˆ P x (cid:48) ( Γ ; q (cid:48) ) ˆ P x ( Γ ; q ∈ Ξ ) the three-point joint density is defined as P p ( q , t a + τ, q (cid:48) , t a , q ∈ Ξ ) ≡ ˆ P x , x (cid:48) , x ( Γ ; q , q (cid:48) , q ∈ Ξ ) P p full ( x , t a + τ, x (cid:48) , t a , x )= (cid:88) k,l e − λ k τ − λ l t a Ψ k ( q )Ψ kl ( q (cid:48) ) (cid:88) m Ψ lm (Ξ ) (cid:104) ψ Lm | p (cid:105) . (B18)Under the milder (as far as the non-stationarity of q ( t ) is concerned) assumption that the full system at t = 0 is inequilibrium, that is p ( x ) = P eq ( x ) as we have assumed in the main text, (cid:104) ψ Lm | P eq (cid:105) = δ m, and Eq. (B18) simplifiesto P eq ( q , t a + τ, q (cid:48) , t a , q ∈ Ξ ) ≡ ˆ P x , x (cid:48) , x ( Γ ; q , q (cid:48) , q ∈ Ξ ) P P eq full ( x , t a + τ, x (cid:48) , t a , x )= (cid:88) k,l e − λ k τ − λ l t a Ψ k ( q )Ψ kl ( q (cid:48) )Ψ l (Ξ ) . (B19)The corresponding three-point conditional probability densities are in turn defined by G p ( q , t a + τ, q (cid:48) , t a | q ∈ Ξ ) ≡ P p ( q , t a + τ, q (cid:48) , t a , q ∈ Ξ ) P p ( q ∈ Ξ )= (cid:80) k,l e − λ k τ − λ l t a Ψ k ( q )Ψ kl ( q (cid:48) ) (cid:80) m Ψ lm (Ξ ) (cid:104) ψ Lm | p (cid:105) (cid:80) l Ψ l (Ξ ) (cid:10) ψ Ll | p (cid:11) , (B20) G eq ( q , t a + τ, q (cid:48) , t a | q ∈ Ξ ) ≡ P eq ( q , t a + τ, q (cid:48) , t a , q ∈ Ξ ) P eq ( q ∈ Ξ )= (cid:80) k,l e − λ k τ − λ l t a Ψ k ( q )Ψ kl ( q (cid:48) )Ψ l (Ξ )Ψ (Ξ ) . (B21)A broken time-translation invariance is, however, most explicitly visible by means of what we will refer to as thetwo-point conditioned Green’s function:˜ G p ( q , t a + τ | q (cid:48) , t a , q ∈ Ξ ) ≡ P p ( q , t a + τ, q (cid:48) , t a , q ∈ Ξ ) P p ( q , t , q ∈ Ξ )= (cid:80) k,l e − λ k τ − λ l t a Ψ k ( q )Ψ kl ( q (cid:48) ) (cid:80) m Ψ lm (Ξ ) (cid:104) ψ Lm | p (cid:105) (cid:80) k e − λ k t a Ψ k ( q (cid:48) ) (cid:80) l Ψ kl (Ξ ) (cid:10) ψ Ll | p (cid:11) , (B22)˜ G eq ( q , t a + τ | q (cid:48) , t a , q ∈ Ξ ) ≡ P eq ( q , t a + τ, q (cid:48) , t a , q ∈ Ξ ) P eq ( q , t a , q ∈ Ξ )= (cid:80) k,l e − λ k τ − λ l t a Ψ k ( q )Ψ kl ( q (cid:48) )Ψ l (Ξ ) (cid:80) k e − λ k t a Ψ k ( q (cid:48) )Ψ k (Ξ ) . (B23)By means of Eqs. (B20) and (B21) we can now determine aging expectation values entering Eq. (B15) and Eq. (B16),which, for a general matrix element (cid:104) q i ( τ + t a ) q j ( t a ) (cid:105) read (cid:104) q i ( τ + t a ) q j ( t a ) (cid:105) Ξ p = (cid:90) Ξ j dq i (cid:90) Ξ j dq j q i q j G p ( q i , t a + τ, q j , t a | q ∈ Ξ ) (cid:104) q i ( τ + t a ) q j ( t a ) (cid:105) Ξ eq = (cid:90) Ξ i dq i (cid:90) Ξ j dq j q i q j G eq ( q i , t a + τ, q j , t a | q ∈ Ξ ) (B24)The dynamics of the projected observable q ( t ) is typically referred to as aging if correlation functions likeˆ C t a ( τ ; p ) , ˆ C t a ( τ ; p ) and/or C t a ( τ ) defined in Eqs. (B4-B5) depend on t a . However, the observables in Eq.(B24) onlycapture linear correlations in systems with broken time-translation invariance, and moreover display a t a -dependenceeven in Markovian systems which are time-translation invariant but evolve from a non-stationary initial condition(see Lemma 2 below). These correlation functions are therefore by no means conclusive indicators of broken time-translation invariance. We therefore propose the time asymmetry index , Υ – a new, conclusive (albeit not unique)indicator of broken time-translation invariance, which we define asΥ Ξ ( t a , τ ) ≡ ˆ D q , q (cid:48) [ G p ( q , τ + t a , q (cid:48) , t a | q ∈ Ξ ) || G p ( q , τ | q (cid:48) ) G p ( q (cid:48) , t a | q ∈ Ξ )] , (B25)where we have introduced the Kullback-Leibler divergence (or relative entropy)ˆ D y , y [ p || q ] ≡ (cid:90) (cid:90) supp p d y d y p ( y , y ) ln p ( y , y ) q ( y , y ) , (B26)which has the property ˆ D y , y [ p || q ] ≥ p ( y , y ) is equal to q ( y , y ) almosteverywhere [114]. The rationale behind this choice is that it is defined to measure exactly the existence and degree ofbroken time-translation invariance and we will use this property in the following section to assert the necessary andsufficient conditions for the emergence of dynamical time asymmetry. We are now in a position to prove the centralclaims in the manuscript. Appendix C: Main theorems with proofs
Definition 1.
Time-translation invariance [76, 115]. The dynamics of the observable q ( t ) resulting from the projectiondefined in Eq. (B1) of the full Markovian dynamics x t evolving according to Eq. (A1) is said to relax to equilibriumin a time-translation invariant manner (i.e. stationary) if and only if the two-point conditioned Green’s function inEqs. (B22-B23) does not depend on t a , that is ˜ G p ( q , t a + τ | q (cid:48) , t a , q ∈ Ξ ) = ˜ G p ( q , t (cid:48) + τ | q (cid:48) , t (cid:48) , q ∈ Ξ ) , ∀ τ, t (cid:48) > . Definition 2.
Dynamical time asymmetry. The dynamics of the projected observable q ( t ) is said to be dynamicallytime asymmetric if its relaxation to equilibrium is not time-translation invariant. Definition 3.
Trivial non-stationarity. The dynamics of the projected observable q ( t ) is said to be trivially non-stationary if the relaxation is time-translation invariant but evolves from a non-equilibrium initial condition of the full system, p ( x ) (cid:54) = P eq ( x ) . Theorem 1.
The dynamics of the observable q ( t ) resulting from the projection defined in Eq. (B1) of the full Marko-vian dynamics x t evolving according to Eq. (A1) is time-translation invariant if and only if at least one of the followingis true:(1) the projected dynamics q ( t ) is Markovian(2) the full system and projected observable are both prepared in and sampled from equilibrium, that is p ( x ) = P eq ( x ) , Ξ → Ξ such that lim Ξ → Ξ P eq ( q ∈ Ξ ) → .If either of these two assumptions is true Υ Ξ ( t a , τ ) = 0 , ∀ t a , τ > .Proof. We first prove sufficiency. If the projection ˆ P is such that above holds then G p ( q , τ + t a , q (cid:48) , t a | q ∈ Ξ ) = G p ( q , τ | q (cid:48) ) G p ( q (cid:48) , t a | q ∈ Ξ ) for any τ, t a , q , q (cid:48) , q , Ξ and p ( x ), such that the logarithmic term in Eq. (B26) isidentically zero everywhere and hence Υ Ξ ( t a , τ ) = 0 , ∀ t a , τ . Conversely, if is true then due to Eq. (B12) we have G eq ( q , t | q ∈ Ξ) = Ψ ( q ) = P eq ( q ) , ∀ t and according to Eq. (B20) also G p ( q , t a + τ, q (cid:48) , t a | q ∈ Ξ) = G p ( q , τ | q (cid:48) ),such that the logarithmic term in Eq. (B26) is again identically zero everywhere and hence Υ Ξ ( t a , τ ) = 0 , ∀ t a , τ .This proves sufficiency.To prove necessity we first recall that ˆ D y , y [ p || q ] = 0 if and only if p ( y , y ) is equal to q ( y , y ) almost everywhere[114]. In addition, as a result of Eq. (B7) and irrespective of the projection (as long as it does not project onto anempty set) the time evolution of G p ( q , τ + t a , q (cid:48) , t a | q ∈ Ξ ) in Eq. (B20) and Eq. (B21) as well as G p ( q , τ | q (cid:48) ) and G p ( q (cid:48) , t a | q ∈ Ξ ) in Eq. (B9) is smooth and continuous ∀ t a > , τ >
0. Moreover, Ψ k ( q ) (cid:54) = Ψ lk ( q ) for l (cid:54) = 0 exceptfor potentially on a set of q with zero measure because of Eq. (B7) and since (cid:104) ψ L | and (cid:104) ψ Ll | are linearly independent.Therefore, because G p ( q , τ + t a , q (cid:48) , t a | q ∈ Ξ ) ≥ , ∀ τ, t a , q , q (cid:48) , q , Ξ and p ( x ) the Kullback-Leibler divergencein Eq. (B26) cannot not be zero almost everywhere except if either one or both of the statements or above aretrue. This completes the proof of necessity. Corollary 1.1.
The dynamics of the projected observable q ( t ) displays a dynamical time asymmetry as soon as theprojection ˆ P renders it non-Markovian and it is initially not prepared in, and averaged over, an equilibrium initialcondition, i.e. P p ( q ∈ Ξ ) (cid:54) = P eq ( q ∈ Ξ) = 1 . If this is true then Υ Ξ ( t a , τ ) > at least on a dense set of t a and τ with non-zero measure.Proof. The proof follows immediately from a straightforward extension of the proof of Theorem 1.
Lemma 2.
Aging correlation functions like ˆ C t a ( τ ; p ) , ˆ C t a ( τ ; p ) and/or C t a ( τ ) defined in Eqs. (B4-B5) are not con-clusive indicators of the dynamical time asymmetry because they cannot discriminate between trivial non-stationarityand broken time-translation invariance, that is, they can display a dependence on t a even if the relaxation to equilib-rium is time-translation invariant.Proof. A simple example suffices to prove this claim. Consider that the observable y t is evolving according to Markovdynamics (for conditions imposed on ˆ P for this to occur please see [85]) with Green’s function Q ( y , t | y ). It is notdifficult to show that the relaxation to equilibrium is time-translation invariant. Namely, consider, the probabilitydensity of y at a time τ + t a given that at time t a the system was found in a point q (cid:48) whereby it evolved there froman initial probability density p ( y ):˜ G p ( y , τ + t a | y (cid:48) , t a , y ) ≡ (cid:82) Ξ d y Q ( y , τ + t a | y (cid:48) , t a ) Q ( y (cid:48) , t a | y ) p ( y ) (cid:82) Ξ d y Q ( y (cid:48) , t a | y ) p ( y )= Q ( y , τ + t a | y (cid:48) , t a ) = Q ( y , τ | y (cid:48) ) , (C1)where we allow (redundantly) and under-sampling of p by setting Ξ (cid:54) = Ξ. Clearly, and expectedly, the relaxationprocess y ( t a ) → y ( t a + τ ) is time-translation invariant – it depends only on y (cid:48) = y ( t a ) but does not depend on howthis state was reached. Analogously to Eq. (B20) we also define the three-point joint probability density of y evolvingfrom an initial probability density p ( y ) G p ( y , τ + t a , y (cid:48) , t a | Ξ ) ≡ (cid:82) Ξ d y Q ( y , τ + t a | y (cid:48) , t a ) Q ( y (cid:48) , t a | y ) p ( y ) (cid:82) Ξ d y p ( y )= Q ( y , τ | y (cid:48) ) Q ( y (cid:48) , t a | Ξ ) , (C2)where the propagation from y (cid:48) = q ( t a ) → q ( t a + τ ) only depends on q ( t a ) but not on how this state was reached.It follows immediately that the time asymmetry index Υ for this process is identically zero (See Theorem 1 andCorollary 1.1). Nevertheless, because G p ( y , τ + t a , y (cid:48) , t a | Ξ ) in Eq. (C2) depends on the probability that the systemis found at time t a in q (cid:48) (but not on how it got there) the aging correlation function obtained from Eq. (C2) woulddisplay a dependence on t a as long as p ( y ) (cid:54) = P eq and Ξ (cid:54) = Ξ, which implies non-stationarity in a trivial sense(i.e. this would equally well be the case even simple Brownian diffusion of a particle in a box, which is manifestlytime-translation invariant).Conversely, it is also possible that the relaxation is indeed not translationally invariant according to Definition 1but a dependence on t a only arises in the evolution of higher order (even or odd) correlation functions and is notvisible in first order correlations in Eqs. (B15-B16). Observation 1.
Characteristic scaling of aging correlation functions. An interesting and very common observationin the existing literature on glassy, aging dynamics is an irreducible structure C t a ( τ ; p ) = f ( τ + t a , t a ) (see e.g.[46, 47, 116, 117]) frequently accompanied by a characteristic power-law scaling of aging autocorrelation functions[39, 46, 47, 117]. A particularly striking observation is the frequently observed so-called ’full aging’ regime where theobservation time window τ becomes much longer than the aging time t a , τ (cid:29) t a , and the following simple scalingemerges C t a ( τ ; p ) ∝ t a /τ [20, 49, 96, 97]. Below we explain the emergence of irreducible structure as well as power-law-decaying aging autocorrelation functions incl. the full aging within the context of our spectral-theoretic approach. Theorem 3.
A representation result. Let ϕ be a positive real number smaller than 1, < ϕ < , and the functions g ( t ) : R + → R and g ( τ, t ) : R + × R + → R be smooth for t > and t, τ > , respectively. A matrix element of agingcorrelation functions, C t a ,i,j ( τ ; p ) or C t a ( τ ; p ) , defined in Eqs. (B15-B16) has the following irreducible structure inthe form of a stationary contribution g ( t ) and a non-stationary contribution g ( τ, t ) : C t a ,i,j ( τ ; p ) = (1 − ϕ ) g ( τ ) + ϕg ( τ, t a ) . Proof.
We recall the definition of aging expectation values in Eq. (B24) and simply split the double sum in Eqs. (B20)and (B21) into two parts (cid:80) k ≥ (cid:80) l ≥ → (cid:80) k ≥ l =0 + (cid:80) k ≥ (cid:80) l ≥ . The second term in the numerator of Eqs. (B15-B16)is nominally non-stationary (i.e. depends on both τ = t − t a and t a ). Collecting terms we obtain the representationstated in the theorem. Definition 4.
Self-similar dynamics. Let us write a general non-aging correlation function in Eq. (B4) as C ij ( t ; p ) = (cid:88) k> w ijk (Ξ ; p )e − λ k t . The dynamics of the projected observable q ( t ) is said to be (transiently) self-similar on a time-scale < t (cid:46) λ − k min (see e.g. [118]) if a time-scale change t → tδ does not change the relaxation beyond a renormalization of the weights, w ijk → ˜ w ijk ( δ ) . That is if λ k and w k are not independent, such that ∃ k min ∈ Z + and constants ( τ , α, δ, y ) ∈ R + with < δ < and y < such that for k ∈ Z + > k min we have λ k = ( δ k τ ) − and w k = δ − αk y . Then we have, on thetime-scale < τ (cid:28) t (cid:46) λ − k min C ij ( t ; p ) = (cid:88) Self-similar time asymmetric dynamics. Let us write the non-normalized aging correlation function,i.e. the numerator in Eqs. (B15-B16), compactly as C t a ,ij ( t ; p ) = (cid:88) k> (cid:88) l> (cid:16) w ijkl (Ξ ; p )e − λ k ( t − t a ) − λ l t a − ˜ w ijkl (Ξ ; p )e − λ k t − λ l t a (cid:17) . (C4) We now extend the idea of self-similar scaling in Definition 4 to aging correlations. Let C , C , C ∈ R + , C ∈ R , λ k = ( δ k τ ) − , w kl = δ − αk δ − αl y , ˜ w kl = δ − αk δ − αl y for suitably chosen < δ , δ , δ < , y < , τ = t − t a and ( τ , α, k min ) as in Definition 4. Then we have, for < τ (cid:28) t, τ, t a (cid:46) λ − k min asymptotically C t a ,ij ( t ; p ) (cid:39) C + y Γ( α ) (cid:34) C ln δ − (cid:18) ττ (cid:19) − α − C ln δ − (cid:18) tτ (cid:19) − α (cid:35) + y Γ( α )ln δ − (cid:18) t a τ (cid:19) − α (cid:34) C + Γ( α )ln δ − (cid:18) ττ (cid:19) − α − Γ( α )ln δ − (cid:18) tτ (cid:19) − α (cid:35) + O ( κ max ) (C5) where κ max ≡ α − max (cid:32) C δ − β min ln δ − , C δ − β min ln δ − , C δ − β min ln δ − , Γ( α )( δ δ ) − β min ln δ ln δ , Γ( α )( δ δ ) − β min ln δ ln δ (cid:33) and β min = αk min . Ona “good” scale of t a , i.e. where y Γ( α ) C ( τ /t a ) α / ln δ − (cid:39) const ≡ B is effectively constant (that is, varies slowly)with respect to ( t a /t ) α then C t a ,ij ( t ; p ) (cid:39) C + B (cid:20) C ln δ C ln δ (cid:18) t a τ (cid:19) α − C ln δ C ln δ (cid:18) t a τ + t a (cid:19) α (cid:21) + O (cid:18)(cid:20) τ t a τ (cid:21) α (cid:19) (C6) Moreover, when τ (cid:29) t a we have the the “anomalous full aging” scaling C t a ,ij ( t ; p ) (cid:39) C + B (cid:34) C ln δ − C ln δ C ln δ (cid:18) t a τ (cid:19) α + α C ln δ C ln δ (cid:18) t a τ (cid:19) α +1 (cid:35) + O (cid:32)(cid:20) t a τ (cid:21) α +2 (cid:33) = B + B (cid:18) t a τ (cid:19) ν + O (cid:32)(cid:20) t a τ (cid:21) α +2 (cid:33) (C7) where ν = α if C ln δ (cid:54) = C ln δ and ν = α + 1 otherwise.Proof. To prove the proposition we split each of the double sums as (cid:88) k> (cid:88) l> = (cid:88) Note that the scaling-from in Definition 4 arises, for example, when the observable corresponds toan internal distance within a single macromolecule (such e.g as the Rouse chain) [92] or within individual proteinmolecules [93, 94], as well as in diffusion on fractal objects [95]. Proposition 2. Logarithmic relaxation and ’full aging’. Let C t a ,ij ( t ; p ) be written as in Eq. (C4) in Proposition 1and let α = 0 , δ = δ and C = C = C ∈ R + (i.e. /f self-similar scaling with logarithmic relaxation [20, 49, 96];in fact a simple change of integration variable x → δ − x in Eqs. (C3)-(C5) with α = 0 shows that this case ismathematically equivalent to the analysis in [49, 96]). Then we have, for < τ (cid:28) t, τ, t a (cid:46) λ − k min asymptotically C t a ,ij ( t ; p ) (cid:39) C + C y ln δ − ln (cid:18) τ + t a τ (cid:19) + y ln δ − ln (cid:18) λ k min t a (cid:19)(cid:20) C + ln (cid:18) τ + t a τ (cid:19)(cid:21) + O ( κ ) (C8) where κ = max ( λ k min t, λ k min τ, λ k min t a ) . Moreover, in te limit τ (cid:29) t a we find C t a ,ij ( t ; p ) (cid:39) C + y ln δ − (cid:18) C + ln δ − ln δ − ln (cid:18) λ k min t a (cid:19)(cid:19) t a τ + C y ln δ − ln (cid:18) λ k min t a (cid:19) + O ( κ ) , (C9) such that when λ k min t a = O (1) we recover the so-called ’full aging’ scaling [49, 96, 97] C t a ,ij ( t ; p ) (cid:39) C + C t a τ (C10)0 Proof. The proof of the proposition is straightforward and follows from noticing that in the limit x (cid:28) , x ) = − ln( x ) + γ + O ( x ). Plugging into the expression in Proposition 1 we find, upon elementary manipulations,the result in Proposition 2. The ’full aging’ scaling is further obtained by Taylor expanding the logarithm to firstorder. Remark 2.1. The representation of C t a ,i,j ( τ ; p ) given in Theorem 3 is indeed frequently observed in experiments onglassy systems [117], while self-similar aging dynamics with a power law scaling as in Proposition 1 has been observedboth in glassy systems [47, 116, 117] as well as in individual protein molecules [39] and, in a similar form, emergesin the case of phenomenological so-called continuous time random walk models with diverging waiting times [119].Notably, in the specific case of /f self-similar dynamics our analysis also recovers the well-known, yet puzzling, “fullaging” limiting scaling of C t a ,i,j ( τ ; p ) (see Eq. (C10) [20, 49, 96, 97], and in particular the presence of the logarithmiccorrection in t a to the full aging scaling in Eq. (C9) may potentially explain the observation that a perfect f ( t a /τ ) collapse is only observed for a specific “good” range values of the aging time t a [97]. Moreover, the power-law agingin Eq. (C7) for t (cid:29) t a agrees with the “renewal aging” in fractional dynamics [45] (since z α is the leading order termof the expansion of the incomplete Beta function, B ( z, α, − α ) , as z → ). These specific results, as wall as others,therefore emerge as special cases of the framework presented in this work. Observation 2. With respect to dynamical time asymmetry, the under-sampling of equilibrium is equivalent to atemperature quench. Let P T eq ( x ) be the equilibrium probability density function of the full system at a temperature T prior to a quench in temperature, that is different from the ambient temperature T , i.e. T > T . Expanding P T eq ( x ) inthe eigenbasis of ˆ L (at the ambient temperature) we find P T eq ( x ) = (cid:80) l | ψ Rl (cid:105) (cid:10) ψ Ll | P T eq (cid:11) , where lim T → T (cid:10) ψ Ll | P T eq (cid:11) = δ l .Let us further assume that the observable q is fully sampled from P T eq ( x ) (like in the case of a supercooled liquid), i.e. Ξ = Ξ , such that according to Eq. (B12) we have Ψ kl (Ξ) = δ kl . G P T eq ( q , t | q ∈ Ξ) = (cid:88) k e − λ k t Ψ k ( q ) (cid:10) ψ Lk | P T eq (cid:11) G P T eq ( q , t a + τ, q (cid:48) , t a | q ∈ Ξ) = (cid:88) k,l e − λ k τ − λ l t a Ψ k ( q )Ψ kl ( q (cid:48) ) (cid:104) ψ Ll | P T eq (cid:105) . since Ψ kk (Ξ) = (cid:104) – | P T eq (cid:11) = 1 . According to Theorem 1 and Corollary 1.1 a temperature quench gives rise to brokentime-translation invariance as longs as the projection renders the dynamics non-Markovian. Now consider an systemprepared in equilibrium p ( x ) = P eq ( x ) but with the projected observable undersampled from said equilibrium, i.e. fora domain q ∈ Ξ ⊂ Ξ such that P eq ( q ∈ Ξ ) (cid:54) = 1 . Then (see Eq. (B10) and Eq. (B21)) G eq ( q , t | q ∈ Ξ ) = (cid:88) k e − λ k t Ψ k ( q ) Ψ k (Ξ )Ψ (Ξ ) G eq ( q , t a + τ, q (cid:48) , t a | q ∈ Ξ ) = (cid:88) k,l e − λ k τ − λ l t a Ψ k ( q )Ψ kl ( q (cid:48) ) Ψ l (Ξ )Ψ (Ξ ) , which has a broken time-translation invariance as long as the projection renders the dynamics non-Markovian accordingto Theorem 1 and Corollary 1.1. Clearly, the only difference between the two non-Markovian time evolutions is in thefactor ψ Ll | P T eq (cid:105) versus Ψ l (Ξ ) / Ψ (Ξ ) , which demonstrates that the effect of temperature quench and under-samplingof equilibrium are indeed (qualitatively) virtually indistinguishable as stated in the observation. Appendix D: Physical models, experimental and simulation data1. Fictitious dynamical time asymmetry in a time-translation invariant system: the Brownian particle in abox Consider the propagator (i.e. the probability density) of a Browninan particle with diffusion coefficient D L , G ( x, t | x , 0) (without loss of generality we express length in units of L and time in units of L /D such that x → x/L and t → tD/L ) with x ∈ [0 , ∂ t G ( x, t | x , 0) = ∂ x G ( x, t | x , , ∂ x G | x =0 = ∂ x G | x =1 = 0 , (D1)1with initial condition G ( x, | x ) = δ ( x − x ). The spectral expansion of the Green’s function of the problem reads G ( x, t | x , 0) = ∞ (cid:88) k =0 ψ k ( x ) ψ k ( x )e − λ k t , ψ k ( x ) = (cid:112) − δ ,k cos( kπx ) , λ k = k π , (D2)where we note that the problem is self-adjoint and hence ψ Rk ( x ) = ψ Lk ( x ) = ψ k ( x ). Let us define I k ≡ − δ k + (1 − δ k ) √ kπ ) − /k π J k,l ≡ − δ k δ l + I k δ l + I l δ k + δ kl / kπ + 2(1 − δ kl )( k + l )[cos( kπ ) cos( lπ ) − / [( k − l ) π ] . (D3)Then we have (cid:104) x ( t ) (cid:105) = (cid:90) dxG ( x, t | x , x = ∞ (cid:88) k =0 I k ψ k ( x )e − λ k t (cid:104) x ( τ + t a ) x ( t a ) (cid:105) = (cid:90) dx (cid:90) dx G ( x, τ + t a | x , t a ) G ( x , t a | x , xx = ∞ (cid:88) k =0 I k e − λ k ( τ + t a ) ∞ (cid:88) l =0 J k,l ψ l ( x )e − λ l t a , (D4)which enter the definition of the aging correlation function C t a ,x ( τ ) = (cid:104) x ( τ + t a ) x ( t a ) (cid:105) − (cid:104) x ( τ + t a ) (cid:105)(cid:104) x ( t a ) (cid:105)(cid:104) x ( t a ) x ( t a ) (cid:105) − (cid:104) x ( t a ) (cid:105) . (D5)When the initial distribution is not a point but is sampled from a flat distribution between a and b (as in the examplein the main text), i.e. uniformly from a domain Ω = [ a, b ], we instead define L k ( b, a ) = δ k + (1 − δ k ) √ kπb ) − sin( kπa )] /kπ ( b − a ) (D6)and then (cid:104) x ( t ) (cid:105) Ω = (cid:90) dx (cid:90) Ω dx G ( x, t | x , P ( x ) x = ∞ (cid:88) k =0 I k L k ( b, a )e − λ k t (cid:104) x ( τ + t a ) x ( t a ) (cid:105) Ω = (cid:90) dx (cid:90) dx (cid:90) Ω dx G ( x, τ + t a | x , t a ) G ( x , t a | x , P ( x ) xx = ∞ (cid:88) k =0 I k e − λ k ( τ + t a ) ∞ (cid:88) l =0 J k,l L l ( b, a )e − λ l t a . (D7)Once inserted in Eq. (D5) Eq. (D7) deliver the aging autocorrelation function shown in Fig. 2 in the main textthat displays fictitious dynamical time asymmetry (i.e. trivial dependence on t a ). In the meantime, the relaxationdynamics is time-translation invariant according to Definition 1 as a result of Theorem 1 (see also Eq. (2) in the maintext), since it is Markovian and thus satisfies the Chapman-Kolmogorov semi-group property (see Lemma 2). Thefictitious dynamical time asymmetry is thus a result of trivial non-stationarity in Definition 3. 2. Rouse polymer model The Rouse polymer chain [120, 121] is a flexible macromolecule consisting of harmonic springs of zero rest-length.The potential energy of the macromolecule with N +1 point-like units (here referred to as ’beads’) with a configuration R ∈ R N +1) ≡ { R i } , where R i ∈ R is given by U ( R ) = (cid:80) Ni =1 3 βb | R i +1 − R i | , where b is the so-called Kuhn lengthdescribing the size of a chain segment (i.e. the characteristic distance between two beads) and will for convenience (andwithout any loss of generality) here be set to b = √ 3. The dynamics is assumed to evolve according to overdampeddiffusion (with all beads having a equal diffusion coefficient D ) in a heat bath with zero mean Gaussian white noise,i.e. according to the system of coupled Itˆo equations d R t = − βD MR t dt + √ Dd ˆ W t , (D8)2where ˆ W t denotes for a 3( N + 1)-dimensional vector of independent Wiener processes whose increments have aGaussian distribution with zero mean and variance dt : E [ d ˆ W t,i d ˆ W t (cid:48) ,j ] = δ ij δ ( t − t (cid:48) ) dt . E [ · ] denotes the expectationover the ensemble of Wiener increments. The interaction matrix M is the 3( N + 1) × N + 1) tridiagonal Rousesuper-matrix whose elements are M ij (where denotes the 3 × N + 1) × ( N + 1) matrix M has elements M ii = (2 − δ i − δ iN ) and M ii +1 = M ii − = − 1. On the level of a probability density function theItˆo process Eq. (D8) corresponds to the N -body Fokker-Planck equation, which, introducing the operator ∇ ≡ {∇ i } reads ∂ t P ( R , t ) = D (cid:16) ∇ T ∇ + β ∇ T MR (cid:17) P ( R , t ) , (D9)which has the structure of Eq. (A2) and can be decoupled as follows. We first rotate the coordinate system to normalcoordinates Q ∈ R N +1) = { Q i } , ∀ i ∈ [0 , N ] [122], i.e. R i = SQ i ( Q i ∈ R ) and ∇ i = S ∇ Q i with the ( N +1) × ( N +1)orthogonal matrix S , S − = S T , which diagonalizes the Rouse matrix, Λ = S T MS , where Λ ik = 4 sin (cid:18) kπ N + 1) (cid:19) δ ik ≡ λ k δ ik , S ik = (cid:114) N + 1 cos (cid:18) (2 i − kπ N + 1) (cid:19) , ∀ k > , (D10)and S i = ( N + 1) − / , ∀ i (which is not required; see footnote). Introducing the 3( N + 1) × N + 1) super-matrix S ,whose elements are S ik , the transformation to normal coordinates is found to decouple the Fokker-Planck equationEq. (D9): ∂ t P ( Q , t ) = D (cid:16) ∇ T S T S ∇ + β ∇ T S T MSQ (cid:17) P ( Q , t )= D N (cid:88) i =1 (cid:2) ∂ Q i + βλ i ∂ Q i Q i (cid:3) P ( Q , t ) , (D11)whose structure implies that the solution factorizes P ( Q , t ) = (cid:81) Ni =1 P ( Q i , t ) and P ( Q i , t ) is simply the well-knownsolution of a 3-dimensional Ornstein-Uhlenbeck process. In particular, the density of the invariant measure andGreen’s function read P eq ( Q ) = N (cid:89) i =1 (cid:18) λ i π (cid:19) / e − λ i Q i / (D12) Q ( Q , t | Q (cid:48) , t (cid:48) ) = N (cid:89) i =1 (cid:18) λ i π (1 − e − λ i ( t − t (cid:48) ) ) (cid:19) / exp (cid:32) − λ i ( Q i − Q (cid:48) i e − λ i ( t − t (cid:48) ) ) − e − λ i ( t − t (cid:48) ) ) (cid:33) . (D13)The Gaussian structure of the solution will permit explicit results not requiring a spectral decomposition of theFokker-Planck operator (which, however, is well-known [120]).We are here interested in the dynamics of the end-to-end distance of the polymer, q ( t ) ≡ | q ( t ) | = | R N +1 ( t ) − R ( t ) | ,which would be typically probed in a single-molecule FRET or optical tweezers experiment. To make minimalassumptions we assume a stationary initial preparation of the full system, i.e. P ( R ) = P eq ( R ). Since q ( t ) at anyinstance depends on all other degrees of freedom R k ( t ) , ∀ k ∈ [2 , N ] its dynamics is strongly non-Markovian. In normalcoordinates this corresponds to q ( t ) = N (cid:88) i =1 |A i Q i | = N (cid:88) i =1 (cid:114) N + 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:20) cos (cid:18) (2 N − kπ N + 1) (cid:19) − cos (cid:18) kπ N + 1) (cid:19)(cid:21) Q i (cid:12)(cid:12)(cid:12)(cid:12) , (D14)having defined A i in Eq. (D14), such that introducing d Q ≡ (cid:81) Ni =1 d Q i the projection operator Eq. (B1) can be shownto correspond to ˆ P Q ( Γ ; q ) = q (cid:90) π dϕ (cid:90) π dθ (cid:90) Ω d Q δ (cid:32) N (cid:88) i =1 A i Q i − q ( q, ϕ, θ ) (cid:33) , (D15)and the non-Markovian conditional two-point probability density is calculated according to Eq. (B2) and leads, upona lengthy but straightforward computation via a Fourier transform FT { q → v } , i.e. ˆ P Q ( Γ ; q ) f ( Q ) FT −−→ ˜ f ( v ) FT − −−−→ f ( q ) (cid:82) dϕ (cid:82) dθ −−−−→ f ( q ), to G eq ( q, t | q ) = 1 √ πγ φ ( t ) − (cid:112) − φ ( t ) qq exp (cid:18) − q + q γ (1 − φ ( t ) ) + q γ (cid:19) sinh (cid:18) qq φ ( t )2 γ (1 − φ ( t ) ) (cid:19) , (D16)where we have defined φ ( t ) = N (cid:88) i =1 A i λ i e − λ i t , γ = φ (0) . (D17)The (non-aging) autocorrelation function in Eq. (B5), C ( t ) = (cid:104) q ( t ) q (0) (cid:105) − (cid:104) q ( t ) (cid:105)(cid:104) q (0) (cid:105) , can in turn be shown to begiven by C ( t ) = 4 γπ (cid:34) (cid:18) φ ( t ) + 2 (cid:19) arctan (cid:32) φ ( t ) (cid:112) − φ ( t ) (cid:33)(cid:35) − γπ , (D18)which decays to zero as t → ∞ . We now address the three-point conditional probability density Eq. (B21) and agingautocorrelation function Eq. (B16), which are much more challenging. As such a complex calculation has, to the bestof our knowledge, not been performed before for any stochastic system, we here present a more detailed derivation.We start with Eq. (B17), plug in Eqs. (D12) and use Eq. (D15) to first calculate the three-point joint density ofthe vectorial counterpart, i.e. P eq ( q , t a + τ, q (cid:48) , t a , q ). We now perform a triple Fourier transform˜ P eq ( u , t a + τ, v , t a , w ) ≡ (cid:90) d q e − i u · q (cid:90) d q (cid:48) e − i v · q (cid:48) (cid:90) d q e − i w · q P eq ( q , t a + τ, q (cid:48) , t a , q ) (D19)and carry out all integrations over Q , Q (cid:48) and Q (a total of 3 N integrals each) and introduce the short-hand notation S t = φ ( t ) to find˜ P eq ( u , t a + τ, v , t a , w ) = 1(2 π ) exp (cid:0) − γ ( w − v − u ) − S t a w T v − S t u T w − S τ u T v (cid:1) . (D20)We now invert back all three Fourier transforms and introduce auxiliary functions X τ,t a ≡ γ − γ ( S t a − S t − S τ ) +2 S t a S t S τ as well as Y τ,t a ≡ γS t − S t a S τ and Z τ,t a ≡ γS τ − S t S t a (keeping in mind that t = τ + t a ) to find P eq ( q , t, q (cid:48) , t a , q ) = (4 π X τ,t a ) − / × exp (cid:18) − ( γ X τ,t a + Y τ,t a ) q + ( γ X τ,t a + Z τ,t a ) q (cid:48) + ( γ − S t a ) q γ − S t a ) X τ,t a (cid:19) × exp (cid:18) − ( S t a X τ,t a + Y τ,t a Z τ,t a )2( γ − S t a ) X τ,t a q (cid:48) T q + Y τ,t a X τ,t a q T q + Z τ,t a X τ,t a q T q (cid:48) (cid:19) . (D21)Before we perform the angular integrations, ( qq (cid:48) q ) (cid:82) π dϕ (cid:82) π dθ (cid:82) π dϕ (cid:48) (cid:82) π dθ (cid:48) (cid:82) π dϕ (cid:82) π dθ , we introduce the final setof auxiliary functions (i.e. the third in the hierarchy of our notation):Λ τ,t a ≡ S t a X τ,t a − Y τ,t a Z τ,t a γ − S t a ) X τ,t a , Λ τ,t a ≡ Y τ,t a X τ,t a , Λ τ,t a ≡ Z τ,t a X τ,t a , Ω Λ τ,ta (cid:18) q q (cid:48) q a b c (cid:19) ≡ erfi a Λ τ,t a Λ τ,t a q (cid:48) + b Λ τ,t a | Λ τ,t a q + c Λ τ,t a q | (cid:113) τ,t a Λ τ,t a Λ τ,t a , (D22)4which, after a long and laborious computation leads to the exact result P eq ( q, τ + t a , q (cid:48) , t a , q ) = qq (cid:48) q π (cid:18) γ − S t a [ S t a X τ,t a − Y τ,t a Z τ,t a ] Y τ,t a Z τ,t a (cid:19) / × exp − S t a ( γ − S t a ) q S t a X τ,t a − Y τ,t a Z τ,t a ) − ( γ + S t a Z τ,ta Y τ,ta ) q (cid:48) + ( γ + S t a Y τ,ta Z τ,ta ) q γ − S t a ) × (cid:40) Ω Λ τ,ta (cid:18) q q (cid:48) q − + − (cid:19) − Ω Λ τ,ta (cid:18) q q (cid:48) q + + − (cid:19) + Ω Λ τ,ta (cid:18) q q (cid:48) q + − + (cid:19) + Ω Λ τ,ta (cid:18) q q (cid:48) q + + + (cid:19)(cid:41) . (D23)The conditional three-point density is in turn obtained from Eq. (D23) by G eq ( q, τ + t a , q (cid:48) , t a | q ) = P eq ( q, τ + t a , q (cid:48) , t a , q ) /P eq ( q ) . (D24)Having obtained all quantities required for the computation of the aging correlation function and the time asymmetryindex Υ, the remaining integrals C t a ( τ ) = (cid:90) ∞ dq (cid:90) ∞ dq (cid:48) qq (cid:48) G eq ( q, τ + t a , q (cid:48) , t a | q ) − (cid:90) ∞ dqqG eq ( q, τ + t a | q ) (cid:90) ∞ dqqG eq ( q, t a | q ) (D25)as well as Υ q ( t a , τ ) = (cid:90) ∞ dq (cid:90) ∞ dq (cid:48) G eq ( q, τ + t a , q (cid:48) , t a | q ) log (cid:18) G eq ( q, τ + t a , q (cid:48) , t a | q ) G eq ( q, τ | q (cid:48) ) G eq ( q (cid:48) , t a | q ) (cid:19) (D26)are performed using an adaptive Gauss-Kronrod routine [123]. The results for a Rouse chain with N = 50 beads arepresented in Fig. D1 and the corresponding time asymmetry index Υ q ( t a , τ ) in Fig. 3a in the manuscript. . . . . . . . . . 09 0 5 10 15 20 25 30 qP eq ( q ) 00 . . . . . . . . . qG ( q, t | q = 9 . t = 0 . G ( q, t = 0 . , q , t a = 0 . | q = 9 . q q . . . . . . G ( q, t = 200 , q , t a = 100 | q = 9 . q q . . . . . Figure D1. The top left panel shows the density of the invariant measure P eq ( q ), while the top right panel depicts the conditionaltwo-point density G eq ( q, t | q ) in Eq. (D16) for q = 9 . 85 and three different t . The bottom panels show the conditional three-point probability density function G eq ( q, τ + t a , q (cid:48) , t a | q ) in Eq. (D24) for two combinations of t = τ + t a and t a . P eq ( q ) (Fig. D1, top left) is concentrated in theregime 0 < q < 30 with a maximum at q peak = 9 . ≈ 10. The evolution of the conditional two-point conditionalprobability density for an ensemble of trajectories starting at the typical distance q peak , G eq ( q, t | q peak ) (Fig. D1, topright) evolves smoothly towards P eq ( q ) with a relaxation time t rel = λ − ≈ . 4. Notably, the corresponding three-point density G eq ( q, t, q (cid:48) , t a | q peak ) (Fig. D1, bottom) shows strong long-time correlations in the evolution of q ( t ), e.g.even for aging times t a = 100 (which are already of the order of, but still smaller than, t rel ) the value of q at time t = 2 t a is strongly correlated with its value q (cid:48) at t a (Fig. D1, bottom right). This long-lasting correlations, which arethe result of the projection of the full 3( N + 1)-dimensional dynamics of the polymer onto a single distance coordinate q ( t ), are responsible for the dynamical time asymmetry.Note that the relaxation time scales quadratically with the length of the chain, in.e. t rel ∝ N and therefore thedynamical time asymmetry extends, for long polymers N (cid:38) over many orders in time. However, for such longchains the computation or Υ at short t a , τ becomes numerically unstable. In Fig. D2 we demonstrate the quadraticgrowth of relaxation time-scales displaying dynamical time asymmetry with increasing N . a) τ t a . . . b) τ . . . c) τ . . . . Figure D2. Υ( t a , τ ) for a) N = 10, b) N = 10 , and c) N = 10 portraying a growing time-scale of dynamical time asymmetry. 3. Single file diffusion The single file model refers to the overdamped Brownian motion of a system of N particles with hard core exclusioninteractions, which for simplicity (and because the finite-size scenario is obtained by a simple re-scaling of space) weassume to be point-like and confined to an interval of unit length L = 1 [101, 124, 125]. We express length in units of L and time in units of τ = D/L , where D corresponds to the diffusion coefficient which is assumed to be equal for allparticles. The state of the system is completely described with the vector of particle positions x = ( x , . . . , x N ). Weare interested in tagged-particle dynamics and therefore our projected observable corresponds to the position of the i -th particle, q ( t ) = x i ( t ). The full system’s dynamics is driven solely by entropic driving forces because the potentialenergy is strictly zero. In turn, the free energy landscape (i.e. the potential of the mean force acting on the taggedparticle) corresponds to the entropic landscape, whereas the potential energy hypersurface is perfectly flat.The Fokker-Planck equation for the Green’s function with initial condition Q ( x , t = 0 | x ) describing the dynamicsof N reads ( ∂ t − (cid:88) i ∂ x i ) Q ( x , t | x ) = 0 , Q ( x , t = 0 | x ) = N (cid:89) i =1 δ ( x i, − x i ) , (D27)which is solved under N − x i +1 → x i ( ∂ x ,i +1 − ∂ x ,i ) Q ( x , t | x ) = 0 , ∀ i. (D28)The system is exactly solvable with the coordinate Bethe ansatz, which yields explicit results for the spectral expansionof Q ( x , t | x ) [101, 125], i.e. Q ( x , t | x ) = (cid:80) k ψ R k ( x ) ψ L k ( x )e − λ k t according to Eq. (A4), where we introduced the N -tuple k = ( k , . . . , k N ) , k i ∈ N , ∀ i . Expressions for the eigenfunctions ψ R k ( x ) , ψ L k ( x ) are given in [101, 125] and theeignevalues corresponding to λ k = (cid:80) Ni =1 k i π . Note that the relaxation time t rel = 1 /λ once re-scaled to natural6units in terms of the collision time (i.e. t col = ( L/N ) /D scales as ∝ N .The projection operator is turn defined by Eq. (B1) with δ ( x i − q ), which according to Eq. (B7) yields, upon sometedious algebra, G eq ( q, t | q ∈ Ξ ) in Eq. (B10) and G eq ( q, t, q (cid:48) , t a | q ∈ Ξ ) in Eq. (B21) with matrix elementsΨ kl ( x ) = m l ! N L ! N R ! (cid:88) { n i } T j ( x ) j − (cid:89) i =1 L i ( x ) N (cid:89) i = j +1 R i ( x ) (D29)where m l ! = (cid:81) i m k i is the multiplicity of the eigenstate with m k i corresponds to the number of times a particularvalue of k appears in the tuple and N L , N R are the number of particles to the left and right from the tagged particle,respectively. The sum (cid:80) { n i } is over all permutations of the elements of the N -tuple k . For the equilibrium densitywe find P eq ( q ) = N ! N L ! N R ! q N L (1 − q ) N R . In Eq. (D29) we have defined the auxiliary functions T j ( x ) = λ k = λ l = 0 √ λ k/l πx ) λ k = 0 or λ l = 02 cos( λ k πx ) cos( λ l πx ) otherwise (D30)(D31) L j ( x ) = x λ k = λ l = 0 √ λ k/l πx ) λ k/l π λ k = 0 or λ l = 0 x + sin(2 λ k πx )2 λ k πx λ k = λ l λ k cos( λ l πx ) sin( λ k πx ) − λ l cos( λ k πx ) sin( λ l πx ) π ( λ k − λ l ) otherwise (D32) R j ( x ) = − x λ k = λ l = 0 −√ λ k/l πx ) λ k/l π λ k = 0 or λ l = 01 − x − sin(2 λ k πx )2 λ k πx λ k = λ l − λ k cos( λ l πx ) sin( λ k πx ) + λ l cos( λ k πx ) sin( λ l πx ) π ( λ k − λ l ) otherwise (D33)The autocorrelation functions C ( t ) and C t a ( τ ) can now be calculated using Eqs. (B14) and (B16), respectively, wheretrivially (cid:104) q ( t ) (cid:105) = ( N l + 1) / ( N + 1) and (cid:104) q (0) (cid:105) = ( N L + 2)( N L + 1) / [( N + 2)( N + 1)]. As it was impossible to carryout this final step analytically, we carried out the integrals in Eqs. (B14) and (B16) depicted in Fig. 3b in the maintext numerically according to the trapezoidal rule.The computation of the time asymmetry index Υ in Eq. (B25), which was as well performed using the trapezoidalrule on a grid of 100 points, is extremely challenging even for moderate values of N . By repeating the integrationusing a smaller grid of 50 points we double-checked that the integration routine converged to a sufficient degree. Theresults for a single file of N = 5 particles tagging the third particle are presented in Fig. 3b in the main text and inFig. D3.The density of the invariant measure of the tagged central particle (Fig. D3, top left) peaks in the center of the unitbox, q peak = 0 . 5, and decays towards the borders due to the entropic repulsion with the neighbors. The evolution ofthe conditional two-point conditional probability density for an ensemble of trajectories starting at the typical distance q peak , G eq ( q, t | q peak ) (Fig. D3, top right) evolves smoothly towards P eq ( q ) with a relaxation time t rel = λ − ≈ . G eq ( q, t, q (cid:48) , t a | q peak ) (Fig. D3, bottom) shows strong long-time correlations in the evolution of q ( t ), e.g. even for agingtimes t a = 0 . 125 (which are already of the order of, but still smaller than, t rel ) the value of q at time t = 2 t a is stronglycorrelated with its value q (cid:48) at t a (Fig. D1, bottom right). This long-lasting correlation reflects prolonged entropicbottlenecks (i.e. ’traffic-jams’), which require collective rearrangements of many particles and thus decorrelate slowly,giving rise to strong memory effects and dynamical time asymmetry (see Fig. 3b in the main text) as soon as q isinitially not sampled from P eq ( q ).7 . . . . . . . . 82 0 0 . . . . . . . . . qP eq ( q ) 012345 0 0 . . . . . . . . . qG ( q, t | q = 0 . t = 0 . . . G ( q, t = 0 . , q , t a = 0 . | q = 0 . . 25 0 . . 75 1 q . . . q G ( q, t = 0 . , q , t a = 0 . | q = 0 . . 25 0 . . 75 1 q . . . q Figure D3. Single file. The top left panel depicts the density of the equilibrium measure P eq ( q ) of the third particle in a singlefile of five particles, q ( t ) = x ( t ). The top right panel shows a two-point conditional probability density function G ( q, t | q )for different values of t , where q = 0 . P eq ( q )). The bottom panels depict the three-point density G ( q, t, q (cid:48) , t a | q ) at different τ and t a evolving from the same initial condition. To produce G ( q, t, q (cid:48) , t a | q ) the spectral expansionEq. (A4) was truncated at maximum Bethe eigenvalue 225 π . 4. Analysis of experimental and simulation data We now consider the time series of a low-dimensional projected observable sampled at discrete time steps that isfrequently encountered in the analysis of experimental data. Therefore we first translate all definitions in Sec. B todiscrete time series and explain in detail how to carry out the complete analysis of aging dynamics for such systems.To ease the application of these new concepts we also provide a C++ routine TSymmetryFinder that will be madeavailable on GitHub.We consider a discrete time series of length N – in our case a one-dimensional physical observable q ( t i ) – sampledat constant time intervals with spacing t i +1 − t i = ∆ t, ∀ i . We pre-process the data by evaluating the mean value overthe time series q = N − (cid:80) Ni =1 q ( t i ) and then center the data by subtracting the mean value, q ( t i ) → q ( t i ) − q . In thefirst stage we determine the (non-aging) autocorrelation function C ( τ = n τ ∆ t ) = 1( N − n τ )∆ t N − n τ (cid:88) i =1 q ( t i + n τ ) q ( t i )∆ t, n τ (cid:28) N (D34)and determine the relaxation time as t r : min t i [ C ( t i ) / C (0) < (cid:15) ], where we choose (cid:15) = 0 . 05. All data are henceforthanalyzed such that n max ≡ n τ max ≈ n r = t r / (∆ t ) in order to assure sufficient sampling when evaluating slidingaverages.Next we determine the equilibrium probability density function and two-point conditional probability density(Eq. (B2)) as a histogram taken over the data. We introduce bins B i centered at q i with a width δq and definethe characteristic function of a bin B i [ q ( t i )] = 1 if q i − δq/ ≤ q ( t i ) < q i + δq/ Ξ [ q ]be the indicator function of the initial condition. Let us further define n l ( n τ ) = max i q ( t i ) ∈ Ξ : n l + n τ ≤ N . The8equilibrium probability density function is then determined as P eq ( q i ) = ( N δq ) − N (cid:88) i =1 B i [ q ( t i )] (D35)and the 2-point conditional probability density as G ( q i , n τ | q ∈ Ξ ) = (cid:80) n l ( n τ ) i =1 B i [ q ( t i + n τ )] Ξ [ q ( t i )] δq (cid:80) n l ( n τ ) i =1 Ξ [ q ( t i )] , (D36) G ( q i , n τ | q j ) = (cid:80) n l ( n τ ) i =1 B i [ q ( t i + n τ )] B j [ q ( t i )] δq (cid:80) n l ( n τ ) i =1 Ξ [ q ( t i )] , (D37)The three-point conditional density Eq. (B21) is defined for n τ (cid:48) ≤ n τ analogously as G ( q i , n τ , q j , n τ (cid:48) | q ∈ Ξ ) = (cid:80) n l ( n τ ) i =1 B i [ q ( t i + n τ )] B j [ q ( t i + n τ (cid:48) )] Ξ [ q ( t i )] δq (cid:80) n l ( n τ ) i =1 Ξ [ q ( t i )] . (D38)Introducing δn = n τ − n a the time-asymmetry index is in turn determined as a double sumΥ Ξ ( t a , τ ) = δq (cid:88) i,j G ( q i , δn + n a , q j , n a | q ∈ Ξ ) log G ( q i , δn + n a , q j , n a | q ∈ Ξ ) G ( q i , δn + n a | q j ) G ( q i , n a | q ∈ Ξ ) , (D39)whereas the normalized aging correlation function (Eq. (B16) is determined according toˆ C t a ( τ ) = (cid:104) q ( t δn + n a ) q ( t n a ) (cid:105) − (cid:104) q ( t δn + n a ) (cid:105)(cid:104) q ( t n a ) (cid:105)(cid:104) q ( t n a ) (cid:105) − (cid:104) q ( t n a ) (cid:105) , (D40) (cid:104) q ( t δn + n a ) q ( t n a ) (cid:105) ≡ (cid:88) i,j q i q j G ( q i , δn + n a , q j , n a | q ∈ Ξ ) , (D41) (cid:104) q ( t i ) (cid:105) ≡ (cid:88) i,j q i G ( q i , n i | q ∈ Ξ ) , (D42)where τ = δn ∆ t and t a = n a ∆ t and we note that by construction (i.e. due to the centering of data) (cid:104) q ( t i ) (cid:105) = 0 , ∀ i .100 bins in q ( t δn + n a ) and 100 bins in q ( t n a ) were used for each combination of τ and t a to determine G ( q i , δn + n a , q j , n a | q ∈ Ξ ) , G ( q i , δn + n a | q j ) and G ( q i , n a | q ∈ Ξ ) and in turn Υ Ξ ( t a , τ ). a. DNA-hairpin Dual optical tweezers data of the DNA hairpin were kindly provided by the Woodside group [69] in the form of aconstant trap measurements of the DNA hairpin 30R50T4, sampled at 400 kHz, for trap stiffness 0 . pN/nm in oneoptical trap and 1 . pN/nm in the other. The time series was 2 . · ms long. The normalized aging correlationfunction C t a ( τ ) and dynamical time asymmetry index Υ are depicted in Figs. 3c and 4b in the manuscript. Here, weadditionally present in Fig. D4, for illustrative purposes and for the sake of completeness, the density of the invariant(equilibrium) measure P eq ( q ) and exemplary two-point conditional probability G ( q, t | q ∈ Ξ ) and the three-pointconditional density G ( q, τ + t a , q (cid:48) , t a | q ∈ Ξ ), respectively, for various t . The histograms in the relative deviations q ( t i ) = q raw ( t i ) − q were determined by binning the interval from -25 nm to +15 nm into 100 bins and q = 3 . 47 nm.These probability density functions are shown in Fig. D4.The density of the invariant measure of the extension of the hairpin is bimodal, reflecting the existence of twolong-lived conformational states (Fig. D5, top left). The evolution of the conditional two-point conditional probabilitydensity for an ensemble of trajectories starting at the typical distance q peak , G eq ( q, t | q peak ) (Fig. D5, top right) evolvessmoothly towards P eq ( q ) with a relaxation time t rel = λ − ≈ ms , and nicely depicts the onset of conformationaltransitions (see red line).Another striking feature of hairpin dynamics is seen in the corresponding three-point density, G eq ( q, t, q (cid:48) , t a | q peak )(Fig. D5, bottom), which depicts, alongside the linear correlations along and near the diagonal q = q (cid:48) that werealso present in the Rouse polymer and tagged-particle diffusion in a single-file, prominent non-linear correlations (seeoff-diagonal peaks). This readily reveals that temporal correlations in the motion persist beyond the time-scale of9 . . . . . . . − − − − − qP eq ( q ) 00 . . . . . . . . . − − − − − qG ( q, t | q = 0 . t = 0 . . G ( q, t = 0 . , q , t a = 0 . | q = 0 . − − − − q − − − − q . . . . . . G ( q, t = 200 , q , t a = 100 | q = 0 . − − − − q − − − − q . . . . . . . . . Figure D4. DNA Hairpin. In the top left panel depicts the density of the equilibrium measure P eq ( q ) of the centered timeseries q ( t i ) = q raw ( t i ) − q , while the top right shows a two-point conditional probability density function G ( q, t | q ∈ Ξ ) fordifferent values of t , where Ξ = [0 . − . , . . 2] nm. The bottom panels depict the three-point density G ( q, t, q (cid:48) , t a | q ∈ Ξ )at different τ and t a evolving from the same initial condition. q = 0 . ± . . 01 0 . τ [ms]0 . . t a [ m s ] . . . . . . . . . q = − . ± . . 01 0 . τ [ms]0 . . t a [ m s ] . . . . . . . . . . . Figure D5. DNA Hairpin, second example. The time asymmetry index for the DNA hairpin data as in Fig. D4 but withthe initial condition Ξ = [ − − . , − . 2] nm. conformational transitions, that is, the hairpin relaxation dynamics post transition remembers the configurations priorto the transition even on time-scales of (cid:38) 200 ms, which reflects a very long range of broken Markovianity. However,a comparison with the corresponding time asymmetry index in Fig. 3c in the main text shows that at aging times t a = 100 ms the dynamics is already time-translation invariant. This is a nice and clear practical demonstration ofthe important conceptual difference between the notion of relaxation with a broken time-translation invariance and0memory effects in time-translation invariant relaxation of a low-dimensional physical observable.In order to demonstrate the robustness of these observations with respect to specific the initial condition q = 1(0)(as long as p ( q ) (cid:54) = P eq ( q ) that is) we also present in Fig. D5 the results for a different set of initial conditions. Theresults in Fig. D5 show qualitatively the same features and are fully consistent with the statements in the manuscript.Finally, we asses the statistical uncertainty of determining Υ from the experimental time-series. We do so byperforming the analysis on an ensemble of trajectories obtained by randomly removing 10 (from the total of 50, i.e.20% of the data) trajectories and averaging over 20 repetitions of a data-set created in this manner. We quantify thestatistical error by determining the local standard deviation of Υ, i.e. σ Υ = (cid:113) N − i (cid:80) N i i =1 Υ i − ( N − i (cid:80) N i i =1 Υ i ) where N i =20. The results are shown in Fig. D6 and depict a local error that is smaller than 1%. − − τ [ ms ]10 − − t a [ m s ] . . . t a , τ ) 10 − − τ [ ms ]10 − − t a [ m s ] . . . . σ Υ Figure D6. Statistical error in the DNA-hairpin analysis. a) average Υ( τ, t a ) determined from an ensemble with forcedunder-sampling (i.e. by omiting 20% of the data); b) the standard deviation of the local Υ( τ, t a ). b. Yeast 3-phosphoglycerate kinase (PGK) Atomistic Molecular Dynamics (MD) simulation of yeast PGK were carried out by Hu et al. [39], starting fromthe PDB structure 3PGK with a duration of 1 . · ps. The observable q ( t i ) refers here to the distance betweenthe center of mass of the N-terminal domain (residues 1-185) and the center of mass of C-terminal domain (residues200-389). In Fig. D7 we depict the density of the invariant (equilibrium) measure P eq ( q ) and exemplary two-pointconditional probability G ( q, t | q ∈ Ξ ) and the three-point conditional density G ( q, τ + t a , q (cid:48) , t a | q ∈ Ξ ), respectively,for various t . The histograms in the relative deviations q ( t i ) = q raw ( t i ) − q were determined by binning the intervalfrom -0.2 nm to +0.3 nm into 100 bins and q = 0 . 67 nm.The density of the invariant measure P eq ( q ) (Fig. D7, top left) is unimodal and effects of a poorer statistics are readilydiscernable through the roughness of the curve. The evolution of the conditional two-point conditional probabilitydensity for an ensemble of trajectories starting at the typical distance q peak , G eq ( q, t | q peak ) is shown in Fig. D7 (topright panel) and reveals that the the dynamics along q is strongly localized (i.e. G eq ( q, t | q peak barely changes between t = 100 ps and t = 3000). Note that PGK did not relax within the duration of the trajectory. The correspondingthree-point density, G eq ( q, t, q (cid:48) , t a | q peak ) (Fig. D7, bottom), shows that the observable almost does not relax at allwithin 6 × ps (compare left and right panel). As in the case of the Rouse polymer G eq ( q, t, q (cid:48) , t a | q peak ) shows strongand long-lasting correlations between positions, and comparison with the dynamical time asymmetry index in Fig. 3din the main text reveals that the dynamics has a strongly time-translation invariance. These findings corroborate theoriginal analysis of Hu et al. [39] who observed aging effects.Similar to the DNA hairpin we also present in Fig. D8 the results for the dynamical time asymmetry index fora different set of initial conditions and two different choices of the observable q ( t ) for PGK, which demonstrate therobustness of the results.PGK obviously does not relax within the duration of the trajectory and more generally it is conceivable that larger,complex proteins do not relax at all during their life-time [39], which makes them virtually ’forever aging’ [126],which may have important consequences for their biological function. Such aging effects on function were observabedin single-enzyme turnover statistics [41–43] and have so-far been rationalized only with ad-hoc phenomenologicalmodels [42, 43]. The present theoretical framework provides a unifying mechanistic understanding dynamics with1 − . − . . . . qP eq ( q ) 024681012 − . − . . . . qG ( q, t | q = 0 . t = 5 . G ( q, t = 11 . , q , t a = 5 . | q = 0 . − . . . q − . . . q G ( q, t = 6 ∗ , q , t a = 3 ∗ | q = 0 . − . . . q − . . . q Figure D7. PGK. The top left panel depicts the density of the equilibrium measure P eq ( q ) of the centered time series q ( t i ) = q raw ( t i ) − q , while the top right shows a two-point conditional probability density function G ( q, t | q ∈ Ξ ) for differentvalues of t , where Ξ = [0 . − . , . 01 + 0 . G ( q, t, q (cid:48) , t a | q ∈ Ξ )at different τ and t a evolving from the same initial condition. broken time-translation invariance in soft and biological matter and will pave the way for deeper and more systematicinvestigations of the potential biological relevance of memory and dynamical time asymmetry for enzymatic catalysis.2 q = 0 . ± . , res : 12 − τ [ps]10100 t a [ p s ] . . . . . . . . . . . q = − . ± . , res : 12 − τ [ps]10100 t a [ p s ] . . . . . . . . . . q = 0 . ± . , res : CM − CM 10 100 1000 τ [ps]101001000 t a [ p s ] . . . . . . . . . q = − . ± . , res : CM − CM 10 100 1000 τ [ps]101001000 t a [ p s ] . . . . . . . Figure D8. PGK, second example. Top: dynamical time asymmetry index for yeast PGK when q ( t ) corresponds to distancebetween the center of masses of the N- and C- terminal domains (respectively residues 1-185 and 200-389) for a pair differentinitial conditions. 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