Non-Markovianity, entropy production, and Jarzynski equality
J. Martins, L. Defaveri, D. O. Soares-Pinto, S. M. D. Queirós, W. A. M. Morgado
NNon-Markovianity, entropy production, andJarzynski equality
Jackes Martins
Departamento de F´ısica, Pontif´ıcia Universidade Cat´olica, 22452-970, Rio de Janeiro,Brazil
Lucianno Defaveri
Departamento de F´ısica, Pontif´ıcia Universidade Cat´olica, 22452-970, Rio de Janeiro,Brazil
Diogo O. Soares-Pinto
Instituto de F´ısica de S˜ao Carlos, Universidade de S˜ao Paulo CP 369, 13560-970, S˜aoCarlos, S˜ao Paulo, Brazil
S´ılvio M. Duarte Queir´os
Centro Brasileiro de Pesquisas F´ısicas, Rua Dr Xavier Sigaud, 150, 22290-180 Rio deJaneiro – RJ, Brazil, and National Institute of Science and Technology for ComplexSystems
Welles A.M. Morgado
Departamento de F´ısica, Pontif´ıcia Universidade Cat´olica, 22452-970, Rio de Janeiro,Brazil, National Institute of Science and Technology for Complex SystemsE-mail: [email protected]
Abstract.
We explore the role a non-Markovian memory kernel plays on informationexchange and entropy production in the context of a external work protocol. TheJarzynski Equality is shown to hold for both the harmonic and the non-harmonicmodels. We observe the memory function acts as an information pump, recovering partof the information lost to the thermal reservoir as a consequence of the non-equilibriumwork protocol. The pumping action occurs for both the harmonic and non-harmoniccases. Unexpectedly, we found that the harmonic model does not produce entropy,regardless of the work protocol. The presence of even a small amount of non-linearityrecovers the more normal entropy producing behavior, for out-of-equilibrium protocols. a r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p on-Markovianity, entropy production, and Jarzynski equality
1. Introduction
Fluctuation theorems, for entropy and work, provide us with an important, andsometimes unexpected, insight into the inner workings of non-equilibrium drivensystems [1, 2, 3, 4, 5, 6, 7, 8]. In the case of work, fluctuation theorems state thedissipated work obeys an exact, and restrictive, relation [4, 9, 10]. These systems – atfirst in an equilibrium state – are then perturbed in a reproducible way, by means of atime protocol on some interaction variable. The validity of the Jarzynski relation hasbeen shown in the case of a non-Markovian description of the system [11], characterizedby a memory function responsible for the dissipation of energy [12, 13]. As we shallsee, the memory function may act as a information backflow channel between systemand thermal reservoir. Understanding the flow of information is important and mayhave many applications. Herein, we shall approach the question by studying, in detail,a non-linear non-Markovian Brownian particle model under the action of an externalprotocol.We approach the problem of information flow via the study of the interaction of asmall system with a thermal bath. The system is also coupled to an external systemby means of an external variable (piston position) that changes in time according to apredefined protocol. The present work has similarities to an older model that was solvedexactly [14], where the Jarzynski equality [15] was verified for the harmonically boundparticle under the action of white noise. Despite its somewhat simplicity, Langevinsystems are quite useful and reproduce behavior found in more complex systems, suchas Fluctuation Theorems [6].In the present work, we generalize the friction coefficient to include a memorykernel, leading to a non-Markovian behavior. We also generalize the potential to a non-harmonic interaction. The memory kernel expresses exchange of information throughtime, which is at the core of non-Markovian behavior. Whereas harmonic systems arepossible to treat mathematically, they present weaknesses in terms of their oversimplifiedphysics. For instance, harmonic potentials do not couple well with higher order noisecumulants [16, 17] and efficient, but purely harmonic, machines cannot be build [18, 19].We shall use the Shannon out-of-equilibrium entropy in order to define theinstantaneous entropy for the system [20, 21, 22]. The out-of-equilibrium entropy is auseful quantity for understanding the interplay of the information flow and the heatand work exchanges for the system. The piston (work) protocol acts as a controlknob, allowing us to tune the exchanges of entropy and work with the environment.The entropic budget, production and exchanges between system and reservoir, showvery intriguing properties such as: for the purely harmonic model the piston protocoldoes not produce entropy; there is constant backflow of information between systemand reservoir. By moving away from equilibrium limitations we hope to be able tounderstand the dynamics of information flow and control in these simple systems.Indeed, one of the points we are most interested in is the relationship between theJarzynski equality, which can be seen as a restriction upon the non-equilibrium behavior on-Markovianity, entropy production, and Jarzynski equality W d = W − ∆ F : (cid:68) e − β W d (cid:69) = 1 . Since the result above is protocol independent, we can choose a protocol thattypically generates very far from equilibrium states (such as compressing a gas almostinstantaneously), yielding very large values for the dissipated work. Typically β (cid:104) W d (cid:105) (cid:29) ⇒ < e − β W d (cid:28) . With very high probability, every time the experiment is repeated we get e − βW d (cid:28) W d < ⇒ e − β W d >
1, and take the average of theexponential back to 1. We shall call these rare event states as Free-Lunches (FL) [24].We shall illustrate below a possible free-lunch for a far from equilibrium protocol. Weshould keep in mind that a FL is the outcome of the initial condition and the externalprotocol applied to it.Other entropic effects can be studied in the context of the Jarzynski equality. Inparticular, the non-Markovian Brownian model allows for the exchange of informationalong time due to the presence of the memory kernel. The excitation of slowhydrodynamic modes can generate retarded kernel functions that allow for the Brownianparticle in the present to interact with its own state in the past. An interesting questionarises: can it recover (partially at least) the information it lost, via dissipation, to thethermal bath earlier? We shall attempt to shed light on this topic. Also, we shall tryand understand some of the roles played by the non-linearities on the production andtransfer of entropy for these systems.In section II, we define the non-Markovian model its procedure and illustrate withan example of a so called “free-lunch”. In section III, we solve the harmonic non-Markovian model exactly. In section IV, we study the properties of the entropy flux forthe model. We describe the entropy oscillations and show that a harmonic model doesnot produce entropy. In section V, we exhibit the numerical results for the non-linearnon-Markovian model. In section VI, we briefly discuss the results herein.
2. The non-Markovian model
We shall study the Jarzynski equality by means of a massive Brownian particle underthe action of external driving force and by a combination of harmonic and quartic on-Markovianity, entropy production, and Jarzynski equality m ˙ v ( t ) = − (cid:90) t dt (cid:48) φ ( t − t (cid:48) ) v ( t (cid:48) ) − k x ( t ) − k [ x ( t ) − L ( t )] − k x ( t ) + ξ ( t ) , (1)˙ x ( t ) = v ( t ) . (2)The colored Gaussian noise, ξ ( t ), induces a memory kernel, φ ( t − t (cid:48) ), consequentlyturning the dynamics intrinsically non-Markovian. The mechanical behavior induced bysuch kernel is quite unusual and instructive. We study it carefuly in appendix AppendixA. Such noise is characterized by (cid:104) ξ ( t ) (cid:105) = 0 , (3) (cid:104) ξ ( t ) ξ ( t (cid:48) ) (cid:105) = γ Tτ e − | t − t (cid:48)| τ , (4) φ ( t − t (cid:48) ) = (cid:104) ξ ( t ) ξ ( t (cid:48) ) (cid:105) T . (5)The external driving force protocol is given by [14] L ( t ) = L (cid:16) − e − tλ (cid:17) . (6)Although it can be shown that Jarzynski equality ( J E ) is valid for non-Markoviandynamics [11], even when considering models more general than those described bygeneralized Langevin equations, there are many facets of this problem that deserve acareful look, and the present approach is helpful. Thus, to verify
J E for such dynamics,we will follow a series of steps described below: • The system is initially at equilibrium with a reservoir at temperature T . The initialconditions ( x , v ) are consequently given by the Boltzmann-Gibbs distribution(with L (0) = 0); • At t = 0 + , an external force is applied, causing a displacement described by theprotocol L ( t ) = L (cid:16) − e − t/λ (cid:17) and realizing external work W over the system upto t = τ (both the form of L ( t ) and the finite value of τ were chosen without loss ofgenerality and can be generalized to more complicated forms of L ( t ) and τ → ∞ ); • The process above is repeated many times, measuring the work W each run, andthe non-equilibrium average (cid:68) e − β W (cid:69) is computed; • The equilibrium free-energies are computed for cases F (0) ≡ F ( L (0)) and F ( τ ) ≡ F ( L ( τ )), yielding ∆ F = F ( τ ) − F (0).Consequently, J E reads (cid:104) exp {− β W }(cid:105) = exp {− β ∆ F } . (7)The JE is an important constraint that our simulation results must obey. We arespecially interested in the change of the instantaneous entropy [21, 20] during the processdue to the non-Markovian nature of the dissipation. In the next section we exploit theexactly solvable case of k = 0. Before that we look at a special case where a seeminglyviolation of the 2 nd law of Thermodynamics occur. on-Markovianity, entropy production, and Jarzynski equality We can construct a simple case of a rare event (a.k.a. a free-lunch - FL) of the Jarzynskiprotocol, similar to the one in reference [25], by studying the expansion of a systemformed by an ideal gas ( N particles) bound by thermally conducting walls, always incontact with a thermal reservoir at temperature T = β − , and a thermally insulatedpiston, controlled by some given external force protocol.If the gas, initially (at t = 0) is considered to be in an equilibrium macro-state M at volume V and temperature T , we can chose a particular micro-state µ = (cid:110) r N , p N (cid:111) ∈ M as our initial condition. Next, the piston is moved fast, atvelocity v piston (faster than any of the gas particles, hence no gas-piston contact duringthe present expansion), from its initial position to a final position, at time t = τ ,such that V ( τ ) = V = 100 V . It should be clear that no work is done by the pistonupon the gas in such case. After a long time interval such that τ (cid:29) (cid:29) τ , thesystem thermalizes at volume V . Thus, when t = τ + τ , the system is at a micro-state µ f = (cid:110) r Nt = τ + τ , p Nt = τ + τ (cid:111) = (cid:110) r ∗ N , p ∗ N (cid:111) ∈ M f , where M f is an equilibrium macro-stateat at volume V and temperature T . We now define the two phase protocol to be usedfor the J E as: • Start with the gas at thermal equilibrium (100 V , T ), at the macro-state M f described earlier; • Phase 1: from t = 0 up to t = τ , the piston is kept at its initial position; • Phase 2: from t = τ up to t = τ + τ , the piston is moved very fast with theopposite velocity as above ( − v piston ), such that the final volume is now V .Obviously, the system does not receive any work during phase 1 but, in general,an enormous amount of work shall be done upon it during phase 2, in order tocompress the system from 100 V → V . Thus, the typical dissipated work W d = W ext − ( F ( V ) − F (100 V )) would be quite large.However, let us chose the initial microscopic state of the gas, at the beginning ofthe protocol, as the thermally equilibrated micro-state µ (cid:48) i = µ f ≡ (cid:110) r ∗ N , − p ∗ N (cid:111) , themicro-state obtained from the final state above by inverting all the molecular velocities,which is in the same macro-state M f (two micro-states, with all their particle momentainverted, are both equilibrium micro-states whenever one of them is, due to the detailedbalance property [26]). Consequently, the gas is going to clearly reverse its trajectoryand spontaneously evolve towards the final state µ (cid:48) f = µ ≡ (cid:110) r N , − p N (cid:111) . During phase 1of the protocol, the external work shall be null, and during phase 2 the gas will have nocontact with the piston, as both are retracing back their trajectories of the expansion.So, W ext = 0 in phase 2, given that the initial equilibrium state µ (cid:48) i = µ f has been chosen.The final state, µ (cid:48) f = µ is a thermally equilibrated system at ( T, V ). In this case, thedissipated work can be easily shown to be W ( F L ) d = 0 − ∆ F = T ∆ S = − N T ln (cid:18) VV (cid:19) = − N T ln 100 , on-Markovianity, entropy production, and Jarzynski equality Figure 1.
In (a), (b) and (c), we work out the steps that will be useful for constructingthe protocol for an ideal gas example of a rare event (a free lunch). The velocities ofthe particles are schematically represented by green arrows in these pictures. In (a), at t = 0, the N-particle ideal gas is at an equilibrium microstate µ at temperature T andvolume V . We then move the piston very fast, faster than any particle of the gas, insuch a way that at time t = τ (cid:28) V , which isdepicted in (b). We then let the system thermalize for a very long time τ (cid:29)
1, whereit reaches a microstate µ f , an equilibrium configuration, at t = τ + τ , shown in (c).In (d), (e) and (f) the protocol, amd the rare state, are defined (now the velocities arerepresented in red). basically the piston protocol chosen for the problem correspondsto the inverse of the piston motion in (a)-(c). at time t (cid:48) = 0, we start from a givenequilibrium microstate at temperature T and volume 100 V , represented in (d). Sincethe N-particle ideal gas is going to eventually be violently compressed (100 V → V )the dissipated work shall be huge for any “normal” microstate. However, if we pick asour initial microstate µ f , the final microstate of (c) (with the velocities inverted), welet it evolve up to t (cid:48) = τ , represented in (e). We then compress the piston very rapidlyso that at time t (cid:48) = τ + τ the piston and system reach the equilibrium configurationrepresented by (f). on-Markovianity, entropy production, and Jarzynski equality e − βW ( FL ) d = e N ln 100 = 100 N (cid:29) , which is a very large contribution to the average of e − βW , in the J E . Micro-states suchas µ (cid:48) i are exceedingly rare, and their contribution to the instantaneous averages maybe quite small, but their effect on the subsequent entropy behavior can be huge dueto their effect of spontaneous entropy decreasing which is a very large contribution tothe average of e − βW in the J E , and ultimately guarantees the validity of the 2nd lawof Thermodynamics in its Free Energy version. Micro-states such as µ (cid:48) i are exceedinglyrare, and their contribution to the instantaneous averages may be quite small, but theireffect on the subsequent entropy behavior can be huge due to their effect of spontaneousentropy decreasing.The mechanical properties of the non-Markovian dissipation are important enoughto the understanding of the entropy behavior. In Appendix Appendix A we study thedamping of the initial velocity of the Brownian particle for the low and high dissipationregimes.
3. Non-Markovian linear model
The dynamics of the system is given by the generalized Langevin equation defined before,when k = 0, where the potential energy for the system is given by U ( x ) = k x + k x − L ( t )] . (8)Consequently, the total energy of the system is given by the Hamiltonian H = mv k x k x − L ( t )) . (9)The process starts with the system at equilibrium with a thermal reservoir attemperature T , where the initial conditions ( x , v ) are Boltzmann-Gibbs distributed.Then, at t = 0, an external force is applied causing the displacement of the piston givenby L ( t ) = L (cid:16) − e − t/λ (cid:17) , and also doing work on the system, up to the instant theprotocol stops, t = τ . Both the form of L ( t ) and the finite value of τ where chosenwithout loss of generality. It is important to notice that the time-scale λ can be set toany positive value, with λ → ∞ corresponding to a reversible thermodynamic process. In order to obtain the non-equilibrium steady-state (NESS) probability distribution ,we can start the system at any initial condition ( x , v ) at t , and then obtain theinstantaneous distribution at time t f [27, 14]. Taking t f → ∞ it is possible to obtainthe NESS probability distribution. Despite the fact that the dissipation term induces on-Markovianity, entropy production, and Jarzynski equality p S ( x, v, L ), given the pistonposition L , is still in the Boltzmann-Gibbs form p S ( x, v, L ) = (cid:113) ( k + k ) m π T e − mv T − ( k k T (cid:16) x − k Lk k (cid:17) . (10)Consequently, the initial equilibrium distribution p ( x , v ) = p S ( x , v , L = 0) reads p ( x , v ) = p S ( x , v , L = 0)= (cid:113) ( k + k ) m π T e − mv T − ( k k x T . (11)From the stationary distribution p S ( x, v, L ) we can obtain the equilibriumHelmholtz free energy F ( T, L ) directly since the equilibrium entropy is given by S ( T, L ) = − (cid:90) dx dv p S ( x, v, L ) ln p S ( x, v, L ) , and the internal energy is U ( T, L ) = (cid:90) dx dv p S ( x, v, L ) H ( x, v ) . Thus, combining these results to find the free energy ( F = U − T S ) we obtain F ( T, L ) = k k k + k L T ln (cid:113) ( k + k ) m π T , which is the same as in Ref. [14], where it was derived for the Markovian harmonicmodel. Then, the free-energy difference is just∆ F = F ( L f , T ) − F (0 , T ) = k k k + k L f (cid:18) k + 1 k (cid:19) − L f . (12) We need to construct the generating function (cid:68) exp {− i u W θ }} (cid:69) for the work function W θ , the external work done upon the system from t = 0 to t = θ , where F is theaverage over the initial conditions and (cid:104) F (cid:105) is the average over the noise, both for anygiven function F . The cumulant generating function (CGF) is then ln (cid:68) exp {− i u W θ }} (cid:69) .Hence, the Jarzynski equality can be obtained through the analytic continuation, i.e., u = − iT , of the CGF G ( u ) ≡ ln (cid:68) exp {− i u W θ }} (cid:69) = ∞ (cid:88) n =1 ( − i u ) n n ! (cid:68) W nθ (cid:69) c . (13)Due to the linearity of the model, the distribution is forcibly Gaussian and an exactsolution ensues. Only the average and the variance of W θ will be non-zero. Observethat the Jarzynski equality should occur at u = − iT , sinceexp (cid:26) G (cid:18) − iT (cid:19)(cid:27) = (cid:42) exp (cid:26) − W θ T (cid:27)(cid:43) , which, due to the J E , we must have G (cid:16) − iT (cid:17) ≡ − ∆ FT . on-Markovianity, entropy production, and Jarzynski equality The work expression in Eq. (13) corresponds to the external work done upon the system,given by the L ( t ) protocol. It reads W θ = − k (cid:90) θ dt dLdt ( x ( t ) − L ( t ))= ∆ U − k (cid:90) θ dt ∂ L ( t ) ∂ t x ( t ) , ≡ ∆ U + I θ , (14)with ∆ U = k L /
2. It is the coupling of x ( t ) and ∂ L ( t ) ∂ t that will give rise to theirreversible work loss. Thus, lets rewrite the integrals in the form I θ = − k (cid:90) θ dt ∂ L ( t ) ∂ t x ( t )= − k (cid:90) θ dt ∂ L ( t ) ∂ t (cid:90) ∞ dt δ ( t − t ) x ( t )= − k L λ (cid:90) θ dt e − tλ (cid:90) ∞ dt (cid:90) ∞−∞ dq π e ( iq + (cid:15) )( t − t ) x ( t )= k L λ (cid:90) ∞−∞ dq π (cid:18) e − [ λ − ( iq + (cid:15) ) ] θ − (cid:19) λ − ( iq + (cid:15) ) ˜ x ( iq + (cid:15) ) , where ˜ x ( iq + (cid:15) ) corresponds to the Laplace-Fourier Transform of the position.The cumulants of W θ will be given by (cid:104) W θ (cid:105) c = ∆ U + (cid:104) I θ (cid:105) c , (15) (cid:68) W θ (cid:69) c = (cid:68) I θ (cid:69) c , (16) (cid:68) W n ≥ θ (cid:69) c = 0 . (17)Consequently, the CGF in Eq. (13) can be easily derived as G ( u ) ≡ ln (cid:68) exp {− i u W θ }} (cid:69) = − iu ∆ U +ln (cid:68) exp {− i u I θ }} (cid:69) = − iu ∆ U − i u (cid:68) I θ (cid:69) c − u (cid:68) I θ (cid:69) c . (18)The calculations, for both first and second order cumulants, are rather cumbersome butstraightforward. These cumulants are obtained exactly for the linear harmonic model.In the following, three important quantities are κ (real), and κ = κ ∗ (complex). Theyare the zeroes of the factor R ( s ), for the harmonic case, defined as˜ x ( s ) = ˜ ξ ( s ) R ( s ) . We also write κ = κ R + i κ I . The actual expressions are very long and cumbersome,but can be worked out directly without much problem. We study these expressionsand their consequence in appendix Appendix B. An interesting phase diagram arisescorresponding to κ I being real or complex (in such case all the κ , , are real). This willhave important consequences for the time behavior of the information entropy of thesystem. on-Markovianity, entropy production, and Jarzynski equality I θ below. (cid:68) I θ (cid:69) c In order to calculate the cumulant of a dynamicalfunction, say F , we first take the noise average (cid:104) F (cid:105) , then we take the average over theinitial conditions F . The only first cumulant contribution is (cid:68) I θ (cid:69) c = (cid:42) k L λ (cid:90) ∞−∞ dq π ˜ x ( iq + (cid:15) ) (cid:18) e − [ λ − ( iq + (cid:15) ) ] θ − (cid:19) λ − ( iq + (cid:15) ) (cid:43) c = 12 k L (cid:16) e − θλ − (cid:17) ( − τ + λ ) λ m τ (1 + κ λ ) (1 + κ λ ) (1 + κ λ ) − k L (cid:16) e − θλ − (cid:17) m τ κ κ κ − k L (cid:18) e ( κ λ − θλ − (cid:19) (1 + τ κ ) m τ κ ( κ − κ ) ( κ − κ ) ( κ λ − k L (cid:18) e ( κ λ − θλ − (cid:19) (1 + τ κ ) m τ κ ( κ − κ ) ( κ − κ ) ( κ λ − − k L (cid:18) e ( κ λ − θλ − (cid:19) (1 + τ κ ) m τ κ ( κ − κ ) ( κ − κ ) − ( κ λ − (cid:68) I θ (cid:69) c Like the first order cumulant, we break thecalculation into the following parts: (cid:68) I θ (cid:69) c = A + A + A + A + A , where the expressions for the A i terms can be found in Appendix Appendix C.The verification of J E comes from the calculation G (cid:18) u = − iT (cid:19) = ln (cid:42) exp (cid:26) − W θ T (cid:27)(cid:43) = − ∆ UT − (cid:68) I θ (cid:69) c T + A + A + A + A + A T . The lengthy expressions above can be easily simplified yielding ⇒ G (cid:18) u = − iT (cid:19) = k k k + k L θ − ∆ FT , showing that the Jarzynski equality holds exactly for the non-Markovian linear case.
4. Entropy and the Jarzynski Equality: analytical results
We shall exploit the properties of the entropy change of the linear model during theaction of the protocol. The Gaussian property, coupled with the linearity of the modelreveals some surprising consequences as those shown in the following. on-Markovianity, entropy production, and Jarzynski equality In the context of applying information theory to the analysis of non-equilibrium systems,the use of the so called out-of-equilibrium notions for entropy and free-energies becomesquite useful [21, 20]. For instance, it allows to calculate the maximum work extractablefrom a system undergoing a non-equilibrium process. Under this perspective, in thepresent case, lets start assuming that the system is always in contact with a thermal bathat constant temperature T . Being the non-equilibrium informational entropy defined as( k B = 1) S sys ( t ) = − (cid:90) dx dv ρ ( x, v, t ) ln ρ ( x, v, t ) , (19)where the internal energy is U ( t ) = (cid:90) dx dv ρ ( x, v, t ) H ( x, v ) , (20)and the out-of-equilibrium free-energy reads F ( t ) = U ( t ) − T S ( t ) . (21)For the reservoir, the corresponding change of entropy is given by by the negativeof the amount of heat flowing towards the system, j Q , since the equilibrium reservoirdoes not produce entropy. We have∆ S R ( t ) = − T (cid:90) t ds j Q ( s ) . (22)The heat flow expression can be easily obtained as [28] j Q ( s ) = ξ ( s ) v ( s ) − (cid:90) s dt (cid:48) φ ( s − t (cid:48) ) v ( t (cid:48) ) v ( s ) , (23)yielding ∆ S R = 1 T (cid:90) t ds (cid:18)(cid:90) s dt (cid:48) φ ( s − t (cid:48) ) v ( t (cid:48) ) v ( s ) − ξ ( s ) v ( s ) (cid:19) . (24)The change in the system entropy can be found from∆ S sys ( t ) = − (cid:90) dx dv ( ρ ( x, v, t ) ln ρ ( x, v, t ) − ρ ( x, v,
0) ln ρ ( x, v, . (25)Consequently, the total entropy change, i.e., for system and thermal reservoir, is thesum of the therms in Eqs. (24) and (25):∆ S tot ( t ) = ∆ S sys ( t ) + ∆ S R ( t ) . (26)Next we are going to obtain the exact results for the linear case scenario. At the beginning of the protocol ( t = 0), the system is in thermal equilibriumat temperature T with the reservoir and, thus, the initial probability distribution p ( x , v ) ≡ p ( x , v , t = 0) is Gaussian distributed, due to the harmonic nature ofthe elastic interactions and of the quadratic form for the kinetic energy. Hence p ( x , v ) = (cid:113) ( k + k ) m π T e − mv T − ( k k x T . on-Markovianity, entropy production, and Jarzynski equality k = 0) of Eq.1, m ˙ v ( t ) = − (cid:90) t dt (cid:48) φ ( t − t (cid:48) ) v ( t (cid:48) ) − k x ( t ) − k [ x ( t ) − L ( t )] + ξ ( t ) . (27)In appendix Appendix A, we study the mechanical consequences of the non-Markovianmemory kernel, which tell us that the competition between the memory time-scale andthe dissipation time-scale can lead to oscillations of velocity in time. Hence, we mightexpect that the statistical consequence of the mechanical oscillations would be entropyoscillations over time.Due to the linearity of the stochastic equations of motion, we can use the Green’sfunction approach [14]. The particular solution (which has the noise function ξ ( t ) andthe protocol for L ( t ) as the source terms) and the homogeneous one (which depends onthe initial quantities x and v ) are combined below: x ( t ) = (cid:90) t dt (cid:48) g ( t − t (cid:48) ) (cid:104) ξ ( t (cid:48) ) + k L ( t (cid:48) ) (cid:105) + x f ( t ) + m v g ( t ) , (28) v ( t ) = (cid:90) t dt (cid:48) ˙ g ( t − t (cid:48) ) (cid:104) ξ ( t (cid:48) ) + k L ( t (cid:48) ) (cid:105) + x ˙ f ( t ) + m v ˙ g ( t ) , (29)where the Green function g ( t ) and the auxiliary function f ( t ) are given by g ( t ) = lim (cid:15) → (cid:90) ∞−∞ dq π e ( iq + (cid:15) ) t R ( iq + (cid:15) ) , f ( t ) = lim (cid:15) → (cid:90) ∞−∞ dq π m ( iq + (cid:15) ) + ˜ φ ( iq + (cid:15) ) R ( iq + (cid:15) ) e ( iq + (cid:15) ) t , (30)with R ( s ) = ms + ˜ φ ( s ) s + k + k , ˜ φ ( s ) being the Laplace transform of the dampingkernel φ ( t ), and the former can be rewritten as R ( s ) = ms + γs τ s + k + k = mτ s + ms + ( γ + τ ( k + k )) s + k + k τ s = m ( s − κ )( s − κ )( s − κ )1 + τ s , (31)where the numerator is in a more compact form using its roots κ , κ and κ . Thevalues of κ will depend on the system parameters, a more detailed analysis of thepossible results can be found in Appendix Appendix B.The next step is to solve for the cumulants of the instantaneous distribution, namelythe averages and variances, of x ( t ) and v ( t ). Since (cid:104) ξ ( t ) (cid:105) = 0, the averages for theposition and the velocity of the particle can be calculate from µ x ( t ) = (cid:104) x ( t ) (cid:105) = (cid:90) t dt (cid:48) g ( t − t (cid:48) ) k L ( t (cid:48) ) + (cid:104) x (cid:105) f ( t ) + (cid:104) v (cid:105) m, g ( t ) (32) µ v ( t ) = (cid:104) v ( t ) (cid:105) = (cid:90) t dt (cid:48) ˙ g ( t − t (cid:48) ) k L ( t (cid:48) ) + (cid:104) x (cid:105) ˙ f ( t ) + (cid:104) v (cid:105) m ˙ g ( t ) , (33)where, due to the symmetrical nature of the initial conditions, (cid:104) x (cid:105) = (cid:104) v (cid:105) = 0. Theonly surviving contribution to the averages is a (deterministic) term corresponding tothe deterministic integral of L ( t (cid:48) ) . That contribution is associated to the changes forthe equilibrium averages of x due to the moving of the so called piston, i.e., the freeextremity of the spring linking the Brownian particle to the external system. on-Markovianity, entropy production, and Jarzynski equality − . . . . .
81 0 10 20 30 40 50 (a) σ t σ xx σ vv σ xv − . − . − . − . . . . . . . . (b) µ t µ x µ v Figure 2.
We compare the numerical results (points) with the theoretical predictions(solid lines) for the σ -functions in (a) and for the µ -functions in (b). For the numericalintegrations we used k = k = 2, γ = 0 . m = τ = T = λ = L = 1, with x = v = 0, the figure represents the process of thermalization of the system. Notethat the frequency of observed oscillations in panel b) is identical for µ x and µ v , withit’s value being determined by the imaginary root κ I (cid:39) .
1. On panel a) the frequencyof oscillations is a combinaton of κ I and 2 κ I (with the latter being the dominant), as39 is proportional to square terms of g ( t ) and f ( t ). Since the variances are defined by σ xx ( t ) = (cid:104) x ( t ) x ( t ) (cid:105) − µ x ( t ) , (34) σ vv ( t ) = (cid:104) v ( t ) v ( t ) (cid:105) − µ v ( t ) , (35) σ xv ( t ) = (cid:104) x ( t ) v ( t ) (cid:105) − µ x ( t ) µ v ( t ) , (36)it is straightforward to show that the second moments contributions due to the protocolswill be identically canceled by the averages products. This remains true regardless theprotocol we use. Indeed, after a little algebra, we can write σ xx ( t ) = (cid:90) t dt dt g ( t − t ) g ( t − t ) (cid:104) ξ ( t ) ξ ( t ) (cid:105) + (cid:104) x (cid:105) f ( t ) + (cid:104) v (cid:105) m g ( t ) , (37) σ vv ( t ) = (cid:90) t dt dt ˙ g ( t − t ) ˙ g ( t − t ) (cid:104) ξ ( t ) ξ ( t ) (cid:105) + (cid:104) x (cid:105) ˙ f ( t ) + (cid:104) v (cid:105) m ˙ g ( t ) , (38) σ xv ( t ) = (cid:90) t dt dt g ( t − t ) ˙ g ( t − t ) (cid:104) ξ ( t ) ξ ( t ) (cid:105) + (cid:104) x v (cid:105) m { f ( t ) ˙ g ( t ) + ˙ f ( t ) g ( t ) } ++ (cid:104) x (cid:105) ˙ f ( t ) f ( t ) + m (cid:104) v (cid:105) ˙ g ( t ) g ( t ) , (39)where we can see that the variances σ xx , σ xx , and σ xx do not depend on L or λ . InFig. 2, we can note the time-evolution of the variances for a system coupled to a thermalbath, given that the initial conditions are fixed at x = v = 0. This has interestingconsequences with respect to the evolution of the total entropy, as we shall see in thefollowing.The instantaneous probability distribution for the system, in terms of the averagesand variances above, is given by p ( x, v, t ) = 12 π (cid:113) σ xx σ vv − σ xv exp (cid:40) − σ vv ( x − µ x ) + σ xx ( v − µ v ) − σ xv ( x − µ x )( v − µ v ) σ xx σ vv − σ xv (cid:41) . (40) on-Markovianity, entropy production, and Jarzynski equality µ x,y .The exact form for the entropy can be obtained after the substitution of Eq. (40)into the Shannon form as in Eq. (19) S ( t ) = 12 ln (cid:16) σ xx ( t ) σ vv ( t ) − σ xv ( t ) (cid:17) + ln 2 πe, (41)which is completely independent of the protocol variables. In fact, for more generalprotocols still keeping the system harmonic and Gaussian, the elimination of the protocolrelated deterministic terms in the equations defining the variances related to the protocolalso occur, which makes the variance independent of the protocol. Hence, the entropyobtained in Eq. (41) is independent of the protocol variables ∂S∂L = ∂S∂λ = 0 . (42)Observe that σ xx ( t ) σ vv ( t ) − σ xv ( t ) is always a positive quantity. That guarantees theentropy S ( t ) is well defined. In the following we exploit this rather unexpected resultfor the non-Markovian harmonic model.It is important to highlight that if one starts with the system thermalized withthe same temperature as that of the thermal reservoir, the variances shall keep theirequilibrium values. the only effect of the protocol is to displace the averages of theBrownian variables.The non-usual entropy behavior that we observe is mostly encoded in the transientbehavior of σ xv ( t ), which is zero at equilibrium. Correlations of velocity and position areintimately linked to the slow hydrodynamic modes, that build up in a fluid perturbedby the motion of a Brownian particle, giving rise to the dissipative memory functionunderlying the present non-Markovian model. In Fig. 3, it is possible to see the evolutiontowards equilibrium of the entropy for such a system. It is highly non-trivial, asthe entropy exchange rate fluctuates very strongly (see inset of figure 3). In all thesimulations of the linear system, the set of variables used were k = k = 2 , γ = 0 . , τ = m = 1 . We stress out that in these simulations the pulling protocol is irrelevant.
In order for a system to produce entropy it is necessary to take that system to a non-equilibrium state, i.e., to realize non-quasi-static processes on it. Take the presentpulling protocol model, unless the pulling rate is quasi-static ( λ → L ( t = 0) = 0, the system stays inthat equilibrium state and no entropy is produced. A fast pulling protocol might be anecessary condition for the production of entropy. But is it a sufficient one?Let us start by clarifying the locus of entropy production in the present model.The triad system of interest (the Brownian particle and springs), external system,and thermal reservoir (see Sekimoto’s book [29]) characterizes well the work and heat on-Markovianity, entropy production, and Jarzynski equality . .
52 0 5 10 15 2000 . . . . . S ( t ) t T T f = 0 . T T f = 0 . T T f = 0 . T T f = 0 . d S / d t t Figure 3.
Entropy evolution for a Brownian particle system with initial fixedconditions thermalized at various initial temperatures while the inset shows the entropychange rate. Even tough the change rate displays oscillatory behaviour, since both g ( t ) and f ( t ) oscillate, the entropy is still a monotonically function, increasing towardsequilibrium as dS/dt ≥ γ = 0 . k = k = 2and T f = m = λ = L = τ = 1. exchanged by the system. We assume that the external system is a pure work reservoir,injecting no entropy into the system of interest. We assume it does work into thesystem of interest in an ordered way, with no increase of its entropy. The thermalreservoir may well exchange entropy with the system of interest, due to the exchangedheat, but it will not produce entropy itself, since we assume it to be in a state ofequilibrium itself. Indeed, this is in sharp contrast with athermal reservoirs, such asPoisson reservoirs [30, 31], which continuously produce entropy to preserve its non-equilibrium athermal state Hence, the only possible source for any produced entropylies with the system of interest itself.Consequently, the total entropy of the system and the thermal reservoir togethershall vary by the amount produced in the system itself. The total entropy budget shallfollow ∆ S total ( t ) = ∆ S ( t ) + ∆ S R ( t ) = Π S ( s ) , (43)where S total corresponds to the total entropy variation of system, external system andreservoir, and Π S ( t ) is the total entropy produced in the system during the interval(0 , t ).Since only the pulling protocol L ( t ) would be capable of taking the system awayfrom equilibrium [14], no entropy will be generated in the system as time goes on. From on-Markovianity, entropy production, and Jarzynski equality ∂S∂L = ∂S∂λ = 0 , we deduce that the result is the same if we take λ → ∞ , which is the quasi-staticprotocol. Hence, the system is always at equilibrium, without any entropy production,yielding Π S ( t ≥
0) = 0 . (44)In fact, in order for the entropy production to be non-zero during the protocol, thepresence of a non-harmonic potential is essential. In the following we shall study thecase of a (small) non-linear term in the potential. The essential effect of the presence of a memory kernel is to feed the present withinformation from the past, hence the non-Markovian property of the model. A possibleoutcome of a non-Markovian model might be information backflow, which is defined as dS/dt <
0. We shall see that it will be the case for the present model. However, thenon-Markovian model not always leads to information backflow.Another interesting effect shall be called “entropy oscillations”, where we canidentify the traces of a typical oscillatory behavior for the time-evolution of the entropy S ( t ). For the present model, oscillations are present for the regimes of high-and low-dissipation, being absent for an intermediate range of the dissipation intensity.We have analyzed a few scenarios, shown in Fig. 4, where the system startsthermalized at a temperature T , which can be chosen arbitrarily. In Fig. 4, we observethat, for high enough values for the coefficient of dissipation γ , the entropy change ratebecomes negative, showing oscillations for a range of time much longer than the memorytime-scale τ = 1 . Interestingly, the memory function acts as an information pump, recoveringpartially some of the information lost to the reservoir due to the information backflow( dS/dt < κ I = 2 . κ i . The spectral analysisof the entropy as a function of time ferrets out the oscillating behavior of the variancesquite clearly. In appendix Appendix B we study the behavior of the κ ’s.The presence of oscillations due to κ I (cid:54) = 0 does not guarantee information backflow.There are other factors that contribute to whether the entropy will decrease or not. Inorder to understand this point, let us take a look at Fig. 6. For instance, if τ = 0 thenthe noise becomes Gaussian white and the entropy will not decrease regardless of κ I (cid:54) = 0,for the pulling protocol studied here. Information backflow is a direct manifestation ofthe coupling of the non-Markovian memory kernel with a high dissipation regime. on-Markovianity, entropy production, and Jarzynski equality . . . . . . − . . . . . S ( t ) t γ = 0 . γ = 1 γ = 5 d S / d t t Figure 4.
We show the evolution of entropy for different values of the dampingconstant γ , while the inset shows the entropy change rate. Unlike in Fig. 3, thevalues of γ are high enough so that the system displays information backflow, thechange rate now becomes negative and the entropy is no longer a monotonicallyincreasing function. The parameters are the initial temperature T = 0 . T f = 1 . k = k = 2 and m = τ = λ = L = 1. The phase diagrams in figure B2 shows the ranges of parameters that favoroscillations in the entropy (in the sense of peaks on the spectrum, such as those infigure 5). However, in the high dissipation range, on the right in the diagrams offigure B2, information backflow might as well happen. In order to check for it, we testedthe existence of the backflow for several values below and above τ c , as shown in figure 6.We observe that: for τ = 0 no backflow is present, as expected; for τ > τ c = 0 .
05 weobserve several instances of backflow. However, for τ = 0 . τ , observing thebackflow is not guaranteed.In the next section we include a small amount of non-linearity in our model via aweak quartic potential k x / k T /k (cid:28)
5. Numerical results for the non-linear model
In our simulations, we have run a series of runs of the process for the range of parametersgiven by m = 1 . , k = 2 . , k = 2 . , k = 0 . , L = 1 . , τ = 1 . , T = 1 . , γ =0 . , λ = 1 . . The number of runs of the protocol driven process is 2 × . The resultspresented in the following correspond to averages over these simulations. The initialstates are sampled over with the thermalized equilibrium distribution. on-Markovianity, entropy production, and Jarzynski equality . . . . . . . . .
91 0 2 4 6 8 10 | ∆ ˜ S ( ω ) | / | ∆ ˜ S ( ) | ω γ = 5 Figure 5.
The spectrum for ∆ S = S ( t ) − S ( ∞ ) of figure 4 for γ = 5. The large valueat ω = 0 occurs because the entropy approaches the limit from beneath, incurring alarge area. Subsequent peaks are related to multiples of the natural frequency drivingthe Green function g ( t ), which for the parameters is κ I ≈ . − . . . . . . d S / d t t τ = 0 τ = 0 . τ = 0 . τ = 0 . τ = 0 . τ = 1 . Figure 6.
The results above represent the time derivative of the entropy for τ = 0 , . , . , . , . , .
0. The other variables are γ = 5 , m = 1. In this case τ c = m/ (4 γ ) = 0 . . Observe that for τ = 0 we have dS/dt >
0, hence no backflow ofentropy. on-Markovianity, entropy production, and Jarzynski equality t (cid:1) J d i s , J i n j Figure 7.
Injected (dashed red line) and dissipated (continuous blue line) heat duringthe protocol.
We now generalize the harmonic model to a non-harmonic one by including a weakquartic potential term k x / k T /k (cid:28)
1, so the quartic potentialcan be considered as a small energetic correction for the harmonic potential energy.We can split the total heat exchanged with the reservoir up to time t , Q ( t ), intothe injected J inj ( t ) and dissipated parts J diss ( t ).m as defined below. Q ( t ) = ∆ J ( t ) , = J inj ( t ) + J diss ( t ) , (45)where J inj ( t ) = (cid:90) t ds ξ ( s ) v ( s ) , (46) J diss ( t ) = − (cid:90) t ds (cid:90) s dt (cid:48) φ ( t − t (cid:48) ) v ( t (cid:48) ) v ( t ) . (47)In Fig. 7 we show the results from the simulations.In Fig. 8, we learn the total heat exchanged ∆ J ( t ) tends to saturate around anegative value. This is due to the fact that part of the work, done upon the systemduring the protocol, becomes heat and is transferred to the reservoir.We can also obtain the probability distribution for the injected heat p ( J inj ) for theduration of the protocol. This is shown in Fig. 9. It shows exponential tails and clearly on-Markovianity, entropy production, and Jarzynski equality (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) t (cid:2) J Figure 8.
Total heat absorbed by the system. Observe that it tends to a negativevalue since it expresses the dissipated work done by the external system during theprotocol. (cid:1) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) J inj P (cid:1) J i n j (cid:2) Figure 9.
The probability distribution for the injected heat J inj for the completeprotocol exhibits a markedly exponential behavior at the tails. It suggests that itsatisfies a fluctuation theorem of sorts. on-Markovianity, entropy production, and Jarzynski equality J inj P (cid:1) J i n j (cid:2) (cid:3) P (cid:1) (cid:1) J i n j (cid:2) Figure 10.
The fluctuations off the injected heat for the whole protocol obey afluctuation theorem form. The red line is not a data adjustment: it is the theoreticalcurve obtained in reference [32] for a Markovian model, hence the slight non-conformityof the data points. The actual adjustment would have a higher angular coefficient, asa consequence of the non-Markovian memory kernel. suggests that a relations of the fluctuation theorem form [32]ln p ( J inj ) p ( − J inj ) = 2 µ J inj σ J inj J inj , (48)shall hold. Actually this is verified in Fig. 10 to a very good degree.That kind of behavior is already well known [32, 17] where the distributions areobtained for systems in contact with thermal and athermal heat baths. The action ofthe pulling protocol in the system will be felt as an equivalent thermodynamic worktransfer, as we see in the following. The work probability has an symmetric form around a non-zero average (see Fig. 11,where it is clear that (cid:104) W ext (cid:105) > W side), since the externalwork is done by stretching the spring ( L (0) = 0 → L ( t ) > p ( W ) = 1 (cid:113) πσ W exp (cid:34) ( W − µ W ) σ W (cid:35) (49)The same dependence has been found for a similar model [14] (on difference was k = 0).A fluctuation relation can be extracted on the formln p ( W ) p ( − W ) = 2 µ W σ W W, (50) on-Markovianity, entropy production, and Jarzynski equality (cid:1) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) W P (cid:1) W (cid:2) Figure 11.
Probability distribution for the work W done upon the system by theexternal system. The form of the work fits a Gaussian distribution, already found forsimilar models [14] W P (cid:1) W (cid:2) (cid:3) P (cid:1) (cid:1) W (cid:2) Figure 12.
Fluctuation relation obtained for the work transfer from the externalsystem. Observe that the relation above upholds the Gaussian character of the workdistribution. which agrees with the Gaussian character of the work distribution, and satisfiesCrooks [5] and Jarzynski [4] relations. on-Markovianity, entropy production, and Jarzynski equality t S s y s Figure 13.
Variation of the entropy as a function of time for the case of non-linearsystem under a Jarzynski pulling protocol.
In accordance with the oscillatory behavior for the superposition of injected anddissipated heat (see Fig. 8), we analysed the behavior of the entropy of the system, S sys ( t ), which shows oscillations and information backflow. The oscillatory dependenceinduced us to perform a spectral analysis which indicates a peak close to ω = 3 as inFig 14. Comparing the non-linear simulation shown in Fig. 13 with the linear modelwhere all parameters are the same, except for k = 0 was done. This is shown infigure 3. Interestingly, we noticed that the linear non-Markovian model does not presentinformation backflow while the non-linear model does.
6. Concluding Remarks
In the present work, we studied the energetics, and the entropic, aspects of non-Markovian massive models subjected to external pulling protocols, obeying the Jarzynskiequality (JE). The importance of rare events for the non-equilibrium dynamics of asystem Cannot be downplayed. The verification of the JE only occurs thanks to theserare events, as can be readily calculated for a few cases, such as the one presented herein.More specifically, we work out two models, a linear (harmonic) one and a (slightly)non-linear (an-harmonic) one. The linear model allows for exact analytical treatments,while we exploit the non-linear case numerically.The non-Markovian harmonic model can be solved exactly, and we can obtainexact probability distribution functions for its dynamic variables. Harmonic models canexhibit quite singular behavior in the context of small classical system thermodynamic on-Markovianity, entropy production, and Jarzynski equality Ω " Figure 14.
We analyze the spectrum of the entropy variations. The peaks at,approximately, 3, 6, 9, 12 and 15 are clearly visible. behavior. For instance, such models lead to ballistic heat conduction not obeying FourierLaw; or that strictly harmonic potentials (with time invariant spring hardness) cannotbe used to build machines with positive efficiencies.We have first demonstrated, exactly, the JE for a class of protocols that are infact quite general. We also have studied the mechanical effects of the non-Markovianmemory kernel. Its highly unusual properties can be appreciated by focusing in thebehavior of the instantaneous entropy.The evolution of the Shannon informational entropy for the harmonic system can bestudied exactly, since the initial state corresponds is described by a Gaussian equilibriumdistribution. The Gaussian character of the noise, and the linearity between variables on-Markovianity, entropy production, and Jarzynski equality T , starting at an equilibrium state at temperature T , would always be at equilibrium,the entropy would not vary. Two interesting cases may happen: firstly, if the system isinitially at equilibrium, at the same temperature of the reservoir, it stays at equilibrium,regardless of the protocol. The only action of the protocol is to change the averageposition of the Brownian particle during the process; Secondly, if the system starts inequilibrium, at a temperature which is distinct from the reservoir’s, the system willreach a non-equilibrium state where all the entropy variation is due to flux to and frothe reservoir, since no entropy is produced by the action of the external work protocolno matter how apparently far from equilibrium are its actions! The effect above is onemore strange consequence of the harmonic type of potentials.The presence of the non-Markovian memory kernel may induce actual oscillationson the entropy. Akin to the velocity oscillations, the entropy oscillations are due tothe fact that the memory kernel time-scale τ (cid:54) = 0 . It disappears as τ → . Thus, forthe appropriate range of parameters, the memory kernel acts as an information pump,recovering it (partially) from the thermal bath. In fact, this constitutes strong evidencethat the presence of the memory kernel indicates the formation of structures (which canstore information) in the bath, such as slow hydrodynamic modes in Brownian-Fluidmodels. Taking a more realist approach to the problem, we studied a non-harmonicmodel, where we introduced a small quartic potential as a perturbation term. Similarlyto earlier models, the injected heat yields a fluctuation relation in the form of anasymmetric large deviation function. The work transmitted from the external systemobeys the Crooks relation.The presence of the non-linear terms somehow restores “normality” to the evolutionof the total entropy and its production during the protocol. In this case we observethat the entropy production rate, by the system, is non-zero. Concerning the entropyoscillations, they are persistent since their cause is that τ (cid:54) = 0 and distinct harmonicscan be detected by spectral analysis. The principal components are the same as theharmonic case if k /k (cid:28) Acknowledgements
W.A.M.M. would like to thank the Brazilian agency CNPq. D.O.S.P. acknowledgesthe Brazilian funding agencies CNPq (Grants No. 307028/2019-4), FAPESP (GrantNo. 2017/03727-0) and the Brazilian National Institute of Science and Technology ofQuantum Information (INCT/IQ). This study was financed in part by Coordena¸c˜ao deAperfei¸coamento de Pessoal de N´ıvel Superior - Brasil (CAPES) - Finance Code 001. [1] G. Gallavotti, and E. G. D. Cohen, 1995
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Physical Review E , 022110.[33] J. Bouchaud, and R. Cont 1998, The European Physical Journal B , 543. Appendix A. A mechanical view of the non-Markovian kernel
As an illustration of the behavior induced by the memory kernel, let us study the velocityattenuation when the interaction potentials and the energy injection are turned off.Thus, starting from the simplified equation of motion, the non-Markovian dissipationdynamics reads m ˙ v + (cid:90) t φ ( t − t (cid:48) ) v ( t (cid:48) ) dt (cid:48) = m ˙ v + (cid:90) t γτ e − ( t − t (cid:48) ) /τ v ( t (cid:48) ) dt (cid:48) = 0 , (A.1)with initial condition v (cid:54) = 0. To simplify the problem, let us define the inverse of thedissipation time-scale τ − diss = Γ = γ/m and let us re-scale time by τ (effectively making on-Markovianity, entropy production, and Jarzynski equality τ = 1) so that the equation of motion may be written in the far simpler form:˙ v + Γ (cid:90) t e t − t (cid:48) v ( t (cid:48) ) dt (cid:48) = 0 . (A.2)To solve this equation we will employ the Laplace transform, we obtain s ˜ v ( s ) − v + Γ ˜ v ( s )1 + s = 0 −→ ˜ v ( s ) = (1 + s ) s + s + Γ v . (A.3)The inverse can be calculated by using a version of Mielin integration (rotated by π /2in the complex plane) as v ( t ) = v (cid:90) ∞−∞ dq π iq + (cid:15) ( iq + (cid:15) ) + ( iq + (cid:15) ) + Γ e iqt . (A.4)The integration is evaluated using the residue theorem, so we are interested in the polesfrom the roots of the denominator: q ± = i ± (cid:115) Γ − . (A.5)We can see that, depending on the value of Γ, we may have three distinct regimes(depicted in Fig. A1): v ( t ) v = e − t/ cosh (cid:16) t (cid:113) − Γ (cid:17) + sinh (cid:16) t √ − Γ (cid:17) √ − (Γ < ) ,e − t/ (cid:16) t (cid:17) (Γ = ) ,e − t/ cos (cid:16) t (cid:113) Γ − (cid:17) + sin (cid:16) t √ Γ − (cid:17) √ − (Γ > ) . (A.6)For a similar calculation, see reference [33].Remarkably, only for sufficiently large values of Γ we obtain oscillations , e.g. τ diss τ < . Expressing this result in the original variables, the critical damping is4 τ γ c m = 1 . (A.7)For the high dissipation regime, for the situations when the velocity of the particlereaches zero, the memory function brings back information from the past motion of theparticle accelerating it back to non-zero velocities (of opposite sign, of course). Appendix B. Behaviour of κ i In order to understand the nature of the Green function g ( t ) and the auxiliary function f ( t ) defined in section 4.2, we must study the nature of the roots of the numerator of R ( s ), which we will refer as κ i , that is mτ ( s − κ )( s − κ )( s − κ ) = mτ s + ms + (Γ + τ ν ) s + ν , (B.1) on-Markovianity, entropy production, and Jarzynski equality − . − . . . . .
81 0 5 10 15 20 v / v t/τ γ = γ c γ < γ c γ > γ c Figure A1.
We highlight the behavious of the three regimes. where we have used Γ = γ/m and ν = ( k + k ) /m . We can use the discriminant ofthe numerator of R ( s ) that we shall refer as ∆ and is calculated as∆ = Γ − τ − ν + 4Γ τ (5 − τ ) ν − τ (2 + 3Γ τ ) ν − τ ν . (B.2)Note that if we make τ = 0 the discriminant becomes ∆ = Γ − ν , which is the usualresult for a system with white noise.It is possible to obtain the values of κ ’s exactly by solving the third degreepolynomial. The expressions become: κ = − τ − − τ − τ + 2 / (cid:16) − τ − τ ν + √− τ ∆ (cid:17) / / (cid:16) − τ − τ ν + √− τ ∆ (cid:17) / κ = − τ − τ − τ − ( − / (cid:16) − τ − τ ν + √− τ ∆ (cid:17) / / (cid:16) − τ + 18 τ ν − √− τ ∆ (cid:17) / (B.3) κ = − τ − ( − / − Γ τ − τ + ( − / (2) / (cid:16) − τ − τ ν + √− τ ∆ (cid:17) / / (cid:16) − τ − τ ν + √− τ ∆ (cid:17) / , where to simplify the answer we used ∆ as the discriminant. Note that the dependencebetween the variables (Γ, ν and τ ) is highly nontrivial, and from the solutions is not veryclear the regimes one could obtain. We demonstrate some typical values for a couple ofexamples in Fig. B1.Despite the complexity, some general properties can still be extracted. The realcomponent of the κ ’s will always be negative, indicating that the solutions will always on-Markovianity, entropy production, and Jarzynski equality − − . − . − . − . − . − . − . − . − .
10 0 0 . . (a) R e ( κ ) τ ντ Γ τ = 0 . τ = 0 . τ = 0 .
50 00 . . . . . . . . . (b) κ I τ ντ Γ τ = 0 . τ = 0 . τ = 0 . Figure B1.
We show the real part of the roots κ in panel (a) and the imaginary partof the roots κ (labeled κ I ) in panel (b) as a function of ν for different values of Γ (allscaled by τ ). It is possible to note that the real part is always negative and that κ I grows with ν for sufficiently large values. . . . . . . . . . . . ν τ Γ τ (a)with oscilationswithout oscilationsBA . . . . . . ν τ Γ /ν with oscilationswithout oscilations(b) Figure B2.
We highlight range of values for the system parameters so that oscillationsmay be observed. In each panel we use a different set of dimensionless parameters, for(a) we use ντ and Γ τ and for (b) we use ντ and γ/ν . The two marked points will bestudied in the next figure. Note that in panel (a), by making ν = 0 (no external force)we recover the result from equation (A.7), that is Γ C > γ C τ /m = 1 /
4, so that thesystem displays oscillations. And in panel (b), by making τ = 0 (removing the bathmemory) we recover γ/m > ν which defines the underdamped of a harmonic oscillator.It is also important to note that for a sufficiently large value of τ , the discriminant willalways be negative and the system will display oscillations. approach a limit, and never diverge. Since all the coefficients of the polynomial arereal and positive, the sign of the discriminant will determine the nature of the roots.We are interested in differentiating the cases where all roots are real (∆ ≥ < τ as the timescale and using thedimensionless parameters ντ and Γ τ or choosing 1 /ν as the timescale and using thedimensionless parameters ντ and Γ /ν . Both cases are displayed in Fig. B2. on-Markovianity, entropy production, and Jarzynski equality − . . . . .
54 0 10 20 30 40 50 g , f t/τ gf Figure B3.
We highlight the behaviour of the Green function g ( t ) and the auxiliaryfunction f ( t ) for the points A and B highlighted in Fig. B2. For a positive discriminant(the blue lines) the functions only decay while for a negative discriminant (the red lines)the functions display oscillatory behaviour. We can write the Green function, with a positive discriminant, as g ( t ) = κ τ − mτ ( κ − κ )( κ − κ ) e − κ t + κ τ − mτ ( κ − κ )( κ − κ ) e − κ t + κ τ − mτ ( κ − κ )( κ − κ ) e − κ t , (B.4)and for a negative discriminant ( κ , = κ R + ± i κ I ) we have oscillations with frequencyIm( κ ) = Im( κ ) = κ I g ( t ) = (1 − κ τ ) e − κ t mτ ( κ I + ( κ − κ R ) ) + ( κ τ − e − κ R t mτ ( κ I + ( κ − κ R ) ) cos( κ I t ) + ( κ − κ R − κ κ R τ + ( κ I + κ R ) τ ) e − κ R t mτ ( κ I + ( κ − κ R ) ) sin( κ I t ) , (B.5)and the auxiliary function for positive discriminant is f ( t ) = κ τ − κ + Γ mτ ( κ − κ )( κ − κ ) e − κ t + κ τ − κ + Γ mτ ( κ − κ )( κ − κ ) e − κ t + κ τ − κ + Γ mτ ( κ − κ )( κ − κ ) e − κ t , (B.6)and for negative discriminant f ( t ) = κ τ − κ + Γ mτ ( κ − κ )( κ − κ ) e − κ t + (Γ + κ + ( κ I + κ R ( κ T − κ ) τ )) e − κ R t mτ ( κ I + ( κ − κ R ) ) cos( κ I t ) ++ (Γ + κ + ( κ I + κ R ( κ R − κ )) τ ) e − κ R t mτ ( κ I + ( κ − κ R ) ) sin( κ I t ) . (B.7)In Fig. B3 we display the behaviour of both regimes. on-Markovianity, entropy production, and Jarzynski equality Appendix C. Second order terms
Here we present the expressions for the second order terms: A = (cid:18) e − θλ − ( κ λ − θλ + 1 (cid:19) T γ L k m τ κ ( κ λ −
1) ( κ − κ ) ( κ − κ ) − (cid:18) e − θλ − ( κ λ − θλ + 1 (cid:19) T γ L k m τ κ ( κ λ −
1) ( κ − κ ) ( κ − κ )+ (cid:18) e − θλ − ( κ λ − θλ + 1 (cid:19) T γ L k m τ κ ( κ λ −
1) ( κ − κ ) ( κ − κ ) − (cid:16) e − θλ − (cid:17) T γ L k λ m τ ( κ λ −
1) ( κ λ −
1) ( κ λ − (cid:18) e ( κ λ − θλ − (cid:19) T γ L k (1 + τ κ ) m τ ( κ λ − ( κ − κ ) ( κ − κ ) κ + (cid:18) e ( κ λ − θλ − (cid:19) T γ L k (1 + τ κ ) m τ ( κ λ − ( κ − κ ) ( κ − κ ) κ + (cid:18) e ( κ λ − θλ − (cid:19) T γ L k (1 + τ κ ) m τ ( κ λ − ( κ − κ ) ( κ − κ ) κ − (cid:18) e ( κ λ − θλ − (cid:19) (cid:18) e ( κ λ − θλ − (cid:19) T γ L k (2 + τ κ + τ κ ) m τ ( κ λ −
1) ( κ λ −
1) ( κ − κ ) ( κ − κ ) ( κ − κ ) ( κ + κ )+ 2 (cid:18) e ( κ λ − θλ − (cid:19) (cid:18) e ( κ λ − θλ − (cid:19) T γ L k (2 + τ κ + τ κ ) m τ ( κ λ −
1) ( κ λ −
1) ( κ − κ ) ( κ − κ ) ( κ − κ ) ( κ + κ ) − (cid:18) e ( κ λ − θλ − (cid:19) (cid:18) e ( κ λ − θλ − (cid:19) T γ L k (2 + τ κ + τ κ ) m τ ( κ λ −
1) ( κ λ −
1) ( κ − κ ) ( κ − κ ) ( κ − κ ) ( κ + κ ) A = (cid:16) e − θλ − (cid:17) T L k ( k + k ) A + A = 2 (cid:18) e ( κ λ − θλ − (cid:19) (cid:16) e − θλ − (cid:17) T L k (1 + τ κ ) m τ ( κ λ −
1) ( κ − κ ) ( κ − κ ) κ − (cid:18) e ( κ λ − θλ − (cid:19) (cid:16) e − θλ − (cid:17) T L k (1 + τ κ ) mτ ( κ λ −
1) ( κ − κ ) ( κ − κ ) κ + 2 (cid:18) e ( κ λ − θλ − (cid:19) (cid:16) e − θλ − (cid:17) T L k (1 + τ κ ) mτ ( κ λ −
1) ( κ − κ ) ( κ − κ ) κ + 2 (cid:16) e − θλ − (cid:17) T L k m τ κ κ κ A = 2 (cid:18) e ( κ λ − θλ − (cid:19) (cid:16) e − θλ − (cid:17) T L k (1 + τ κ ) ( k + k ) m τ ( κ λ − κ ( κ − κ ) ( κ − κ ) κ κ − (cid:18) e ( κ λ − θλ − (cid:19) (cid:16) e − θλ − (cid:17) T L k (1 + τ κ ) ( k + k ) m τ ( κ λ − κ ( κ − κ ) ( κ − κ ) κ κ on-Markovianity, entropy production, and Jarzynski equality
32+ 2 (cid:18) e ( κ λ − θλ − (cid:19) (cid:16) e − θλ − (cid:17) T L k (1 + τ κ ) ( k + k ) m τ ( κ λ − κ ( κ − κ ) ( κ − κ ) κ κ + (cid:16) e − θλ − (cid:17) T L k ( k + k ) m τ κ κ κ + (cid:18) e ( κ λ − θλ − (cid:19) T L k (1 + τ κ ) ( m κ + k + k ) m τ ( κ λ − κ ( κ − κ ) ( κ − κ ) + (cid:18) e ( κ λ − θλ − (cid:19) T L k (1 + τ κ ) ( m κ + k + k ) m τ ( κ λ − κ ( κ − κ ) ( κ − κ ) + (cid:18) e ( κ λ − θλ − (cid:19) T L k (1 + τ κ ) ( m κ + k + k ) m τ ( κ λ − κ ( κ − κ ) ( κ − κ ) − (cid:18) e ( κ λ − θλ − (cid:19) (cid:18) e ( κ λ − θλ − (cid:19) T L k (1 + τ κ ) (1 + τ κ ) ( m κ κ + k + k ) m τ ( κ λ − κ κ ( κ λ −
1) ( κ − κ ) ( κ − κ ) ( κ − κ )+ 2 (cid:18) e ( κ λ − θλ − (cid:19) (cid:18) e ( κ λ − θλ − (cid:19) T L k (1 + τ κ ) (1 + τ κ ) ( m κ κ + k + k ) m τ ( κ λ − κ κ ( κ λ −
1) ( κ − κ ) ( κ − κ ) ( κ − κ ) − (cid:18) e ( κ λ − θλ − (cid:19) (cid:18) e ( κ λ − θλ − (cid:19) T L k (1 + τ κ ) (1 + τ κ ) ( m κ κ + k + k ) m τ ( κ λ − κ κ ( κ λ −
1) ( κ − κ ) ( κ − κ ))
1) ( κ − κ ) ( κ − κ )) ( κ − κ ))