Energy diffusion and absorption in chaotic systems with rapid periodic driving
EEnergy diffusion and absorption in chaotic systems with rapid periodic driving
Wade Hodson ∗ Department of Physics, University of Maryland, College Park, MD 20742
Christopher Jarzynski † Institute for Physical Science and Technology, Department of Chemistry and Biochemistry,and Department of Physics, University of Maryland, College Park, MD 20742 (Dated: March 1, 2021)When a chaotic, ergodic Hamiltonian system with N degrees of freedom is subject to sufficientlyrapid periodic driving, its energy evolves diffusively. We derive a Fokker-Planck equation thatgoverns the evolution of the system’s probability distribution in energy space, and we provide explicitexpressions for the energy drift and diffusion rates. Our analysis suggests that the system genericallyrelaxes to a long-lived “prethermal” state characterized by minimal energy absorption, eventuallyfollowed by more rapid heating. When N (cid:29)
1, the system ultimately absorbs energy indefinitelyfrom the drive, or at least until an infinite temperature state is reached.
I. INTRODUCTION
Time-periodic driving facilitates a rich range of clas-sical and quantum dynamical behaviors, including syn-chronization and resonance [1–4], localization [5–8], andchaos [1, 9]. Recent theoretical and experimental workhas aimed to identify nonequilibrium “phases of matter”that might emerge in periodically driven systems [10–12]. Phenomena such as time crystallization [11–17] andprethermalization [12, 17–28] reveal that periodic drivingcan stabilize systems in a variety of interesting and usefulstates.Energy absorption poses a potential obstacle to suchstabilization of nonequilibrium states of matter. A drivenopen system in a nonequilibrium steady state attains abalance in which energy absorbed from the drive is dissi-pated into an environment, such as a thermal bath. Butif a system is isolated, save its interaction with the drive,then maintaining a stable state requires the suppressionof energy absorption from the drive. Much work hasbeen devoted to understanding energy absorption, andthe conditions under which it might be suppressed, inperiodically driven, isolated classical and quantum sys-tems [18, 20, 23–25, 29–39] .In this paper we study the general problem of energyabsorption in isolated classical chaotic systems subjectto rapid periodic driving. We argue that the energeticdynamics of such systems are diffusive, and we derivea Fokker-Planck equation for the evolution of the sys-tem’s energy probability distribution, η ( E, t ). The driftand diffusion coefficients in this equation, characteriz-ing energy absorption and the spreading of the energydistribution, are given explicitly in terms of the dynam-ics of the undriven system – much as in the case of or-dinary linear response theory [40], but without the as-sumption of weak driving. For many-body systems our ∗ [email protected] † [email protected] results suggest a scenario marked by three stages: ini-tial relaxation to an equilibrium-like “prethermal” state[12, 18–28], followed by a long interval of minimal en-ergy absorption, and finally rapid absorption toward aninfinite-temperature state.Our description provides a comprehensive, quantita-tive account of energy absorption in rapidly and peri-odically driven chaotic systems. It reveals how chaos inphase space facilitates stochastic energy evolution, howenergy diffusion leads to the breakdown of the prether-mal regime, and how energy absorption rates are de-termined by the underlying, undriven Hamiltonian dy-namics. Our framework also suggests a generic ex-planation for the exponential-in-frequency suppressionof energy absorption observed in a range of systems[12, 18, 20, 23, 24, 27, 28, 31, 35, 36, 39]. Finally, weargue that the classical results that we obtain are rel-evant to energy absorption in quantum systems, in anappropriate semiclassical limit.In Sec. II we define the problem we will study. InSec. III we argue that the energy of a rapidly drivenchaotic system evolves diffusively, and we derive theFokker-Planck equation that describes this evolution. InSec. IV we analyze energy absorption and prethermal-ization in the context of our energy diffusion model. InSec. V we briefly consider the quantum counterpart ofour classical problem, and we conclude in Sec. VI. II. SETUP
Our object of study is a classical Hamiltonian systemwith N ≥ t , the micro-scopic state of the system is specified by a phase spacepoint z t ≡ z ≡ ( q , p ), where the N -vectors q and p specify canonical coordinates and momenta. The systemevolves under Hamilton’s equations of motion, generatedby a Hamiltonian H ≡ H ( z, t ): d q dt = ∂H∂ p , d p dt = − ∂H∂ q . (1) a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b For an ensemble of trajectories, the phase space proba-bility distribution ρ ( z, t ) obeys the Liouville equation ∂ρ∂t = { H, ρ } = ∂H∂ q · ∂ρ∂ p − ∂H∂ p · ∂ρ∂ q , (2)where {· , ·} denotes the Poisson bracket [41]. We take H ( z, t ) to be a periodic function of time, with period T ,and we decompose this Hamiltonian into its time average H ( z ) ≡ T − (cid:82) T dt H ( z, t ), and a remainder V ( z, t ) = V ( z, t + T ) with vanishing average: H ( z, t ) = H ( z ) + V ( z, t ) . (3)We will refer to H ( z ) as the “bare” or “undriven” Hamil-tonian, and to V ( z, t ) as the “drive.” The former de-termines the evolution of the system in complete isola-tion, that is when V = 0. We define the system’s en-ergy E ( t ) to be the bare Hamiltonian evaluated at z t : E ( t ) ≡ H ( z t ). When V = 0 the energy is a constantof the motion: any trajectory z t is constrained to evolveon an energy shell , that is a level surface of H ( z ). Theevolution of the energy when V (cid:54) = 0 will be our centralfocus. The magnitude of the drive V ( z, t ) is arbitrary; inparticular we do not assume it to be small.We assume that the undriven dynamics are chaotic andergodic on each energy shell of H . Such dynamics ex-hibit mixing as trajectories diverge from one another ex-ponentially with time [9]. This leads to a loss of statisti-cal dependence between states of the system at differenttimes, as reflected in the decay of correlations – in effectthe system loses its memory of previously visited states.After a characteristic mixing time, any smooth initialdistribution on the energy shell evolves to a distributionthat for practical purposes is microcanonical, or ther-mal [40, 42]. Thus chaos offers a way to understand theself-thermalizing properties of many-body systems suchas gases and liquids, while also providing low-dimensionalanalogues, such as chaotic billiard systems, that are ac-cessible to numerical or analytical study.We are interested in the limit of high driving frequency ω = 2 π/T . When ω → ∞ the effect of the drive averagesto zero over each period, as the system cannot appre-ciably react to the drive in such a short time. In thislimit the evolution generated by the driven Hamiltonianapproaches the undriven evolution: for given initial con-ditions z , and over a fixed time interval 0 ≤ t ≤ τ ,the trajectory z t that evolves under H ( z, t ) converges,as ω → ∞ , to the trajectory z t that evolves under H ( z ) [43, 44]. This limit can be attained regardless ofthe strength of the drive, V ( z, t ). In our case we assume ω is sufficiently high that driven and undriven trajec-tories remain close over timescales characteristic of thedecay of correlations.These considerations lead to the following picture. Forsufficiently short times, energy is approximately con-served, and the driven trajectories z t generated by H ( z, t )are similar to the undriven trajectories z t generated by H ( z ), allowing us to use the latter to estimate correla-tion functions that will arise in our analysis. These cor-relation functions, as we shall see, will in turn describehow the system absorbs energy from the external driveon much longer timescales. III. ENERGY DIFFUSION
Given the assumptions mentioned in Section II, we nowargue that the energy of the driven system evolves diffu-sively. For simplicity, we consider monochromatic driv-ing, V ( z, t ) = V ( z ) cos( ωt ) . (4)Our analysis can be generalized to arbitrary time-periodic driving, by decomposing V ( z, t ) in a Fourier se-ries with fundamental frequency ω . A. Argument for energy diffusion
To begin, we consider a system that evolves over a timeinterval 0 ≤ t ≤ ∆ t , from initial conditions z sampledfrom a microcanonical distribution at energy E : ρ ( z ,
0) = ρ E ( z ) ≡ E ) δ ( E − H ( z )) . (5)Here Σ( E ) = ∂ Ω ∂E = (cid:90) dz δ ( H ( z ) − E ) (6)is the classical density of states;Ω( E ) = (cid:90) dz θ ( E − H ( z )) (7)is the phase space volume enclosed by the energy shell E , which we assume to be finite for all E ; and the inte-grals are over phase space. Let ∆ E ( z ) denote the netchange in the system’s energy from t = 0 to t = ∆ t . ByHamilton’s equations, ∆ E ( z ) is the time integral of thepower dEdt ≡ ddt H ( z t ) = − cos( ωt ) ˙ V ( z t ) , (8)where ˙ V ( z ) ≡ { V, H } . The quantity ∆ E can be viewedas a random variable, whose value is determined by thesampled initial conditions z . Understanding the statis-tics of ∆ E in the high-frequency driving regime, for anappropriate choice of ∆ t (to be clarified below), will bethe key to establishing diffusion in energy space.[We note in passing that if the undriven Hamiltonianhas the form H = (cid:80) n ( p n / m n ) + U ( { q n } ) and if thedrive V depends on coordinates { q n } but not momenta { p n } , then (8) becomes dE/dt = (cid:80) n F n · v n , where F n is the driving force acting on the n ’th particle, and v n isthat particle’s velocity.]We now explicitly assume the driving is rapid. To be-gin, we impose the condition T (cid:28) τ C ( E ) , (9)where T = 2 π/ω is the drive period and τ C ( E ) is a char-acteristic timescale over which chaotic mixing on the en-ergy shell E produces the decay of correlations. Heuris-tically, (9) implies that a trajectory z t travels only anegligible distance during one period of driving. Thiscondition produces the averaging over oscillations that(as mentioned in Sec. II) results in driven trajectories z t resembling their undriven counterparts z t . Let us nowchoose ∆ t so that over the interval [0 , ∆ t ] the driven tra-jectories in our ensemble remain close to the initial energyshell E . Chaotic mixing ensures then that a microcanon-ical distribution is approximately maintained. Thus forany t ∈ [0 , ∆ t ], the ensemble of points z t are approxi-mately distributed according to the initial microcanoni-cal ensemble.With this picture in mind, let us divide the time in-terval [0 , ∆ t ] into M (cid:29) δt = ∆ t/M , and consider ∆ E = (cid:80) i δE i as a sum ofsubinterval energy changes δE i , i = 1 , · · · M . Each in-crement δE i is itself a random variable, determined byintegrating the power (8) along the trajectory z t over thesubinterval. By the arguments of the previous paragraph,the δE i ’s have nearly identical, microcanonical statistics,provided we choose δt (and therefore also ∆ t ) to be aninteger multiple of the driving period T to ensure thateach subinterval begins at the same phase of the drive.Chaotic mixing on the energy shell E produces thedecay of correlations. Let us further choose δt to belonger than the characteristic correlation time τ C ( E ),so that each δE i is approximately statistically indepen-dent from the others. The energy change ∆ E is then asum of M (cid:29) δE i : the system effectivelyperforms a random walk on the energy axis. By the cen-tral limit theorem ∆ E is a normally-distributed randomvariable, whose mean and variance grow (for fixed δt ) inproportion to the number of increments M , equivalentlythe time elapsed ∆ t .The statistical behavior just described is characteristicof a diffusive process in energy space, motivating us tomodel it by a Fokker-Planck equation [45]. Letting η ( E, t ) ≡ (cid:90) dz δ ( H ( z ) − E ) ρ ( z, t ) (10)denote the energy distribution, we postulate that thetime evolution of η is given by ∂η∂t = − ∂∂E ( g η ) + 12 ∂ ∂E ( g η ) . (11)The drift and diffusion coefficients g ( E, ω ) and g ( E, ω )characterize, respectively, the rate at which the distri-bution η shifts and spreads on the energy axis; see (13) and (14) below. These coefficients depend on the systemenergy E and the driving frequency ω . Energy diffusionand its description in terms of the Fokker-Planck equa-tion have been studied in various contexts involving ex-ternally driven Hamiltonian systems [32, 46–56]. Beforederiving expressions for g and g in the high-frequencydriving regime, it is worth examining the central role thata separation of timescales plays in our analysis.We have assumed, after (9), that ∆ t is much smallerthan the timescale τ E ( ω, E ) over which the energy of thesystem changes significantly. This condition ensures thatthe energy increments δE i have approximately identicalmicrocanonical statistics. We have also assumed thatthe interval ∆ t contains many subintervals of duration δt , and that δt > τ C ( E ), guaranteeing approximate sta-tistical independence among the increments δE i . Thusour analysis involves the hierarchy of timescales: T (cid:28) τ C ( E ) (cid:28) ∆ t (cid:28) τ E ( ω, E ) . (12)Since τ E → ∞ as ω → ∞ , this hierarchy can be satisfiedfor any particular energy shell E by setting ω sufficientlylarge. We conclude that Eq. (11) is valid over an intervalof the energy axis whose extent is determined by, andincreases with, the value of ω .The above arguments suggest that the energy diffusiondescription is valid on a coarse-grained timescale of order∆ t . On shorter timescales, computing the fine details ofthe system’s energy evolution requires the full Hamilto-nian equations of motion (1). These details vary greatlyfrom system to system. However, as we will see, thecharacteristics of the energy diffusion process ultimatelydepend only on a few key details of these system-specificdynamics, as captured in the coefficients g and g . B. Drift and diffusion coefficients
Under (11) an initial distribution η ( E,
0) = δ ( E − E )evolves after a time ∆ t (cid:28) τ E to a distribution η ( E, ∆ t )with mean and variance [45]:Mean( E ) = E + g ( E , ω )∆ t (13)Var( E ) = g ( E , ω )∆ t. (14)We can thus determine g by calculating Var( E ), the en-ergy spread acquired by an ensemble of trajectories withinitial energy E , evolved for a time ∆ t under the drivenHamiltonian. We perform this calculation in Section 1of the Appendix, obtaining, in the limit of large ω , g ( E, ω ) = 12 S ( ω ; E ) ≥ S ( ω ; E ) = (cid:90) ∞−∞ dt e − iωt C ( t ; E ) , (16)where C ( t ; E ) ≡ (cid:104) ˙ V ( z ) ˙ V ( z t ) (cid:105) − (cid:104) ˙ V ( z ) (cid:105)(cid:104) ˙ V ( z t ) (cid:105) (17)is the microcanonical autocorrelation function of the ob-servable ˙ V ( z ). Specifically, the averages denoted by (cid:104)·(cid:105) are computed by sampling initial conditions z from amicrocanonical ensemble at energy E , then evolving fortime t under H ( z ). By the Wiener-Khinchin theorem[57], the Fourier transform of C ( t ; E ) is the power spec-trum of ˙ V ( z t ) at energy E , denoted by S ( ω ; E ). Notethat (15) gives g entirely in terms of properties of the undriven system, as C ( t ; E ) and thus S ( ω ; E ) are definedin terms of the undriven trajectories z t .In solving for g we approximated driven trajectories z t by their undriven counterparts z t . As a result, weexpect that (15) contains correction terms that becomenegligible in the high-frequency limit ω → ∞ .In Section 2 of the Appendix we use Liouville’s theo-rem, which expresses the incompressibility of phase spacevolume under Hamiltonian dynamics [40], to obtain thefollowing expression for the drift coefficient g in termsof g ( E, ω ) and the density of states Σ( E ) (6): g ( E, ω ) = 12Σ ∂∂E (cid:16) g Σ (cid:17) . (18)This result is a fluctuation-dissipation relation, similar toothers previously established for various driven Hamilto-nian systems [47–49, 51–54].Using (15) and (18), the Fokker-Planck equation (11)takes the compact form ∂η∂t = 14 ∂∂E (cid:20) S Σ ∂∂E (cid:16) η Σ (cid:17)(cid:21) . (19)Eq. (19) is our main result. It describes the stochasticevolution of the system’s energy, under rapid driving, interms of quantities S ( ω ; E ) and Σ( E ) that characterizethe undriven system.As discussed earlier, we expect (19) to be valid overa region of the energy axis whose extent depends on ω .In the next section we assume ω is sufficiently large that(19) is valid over the entire energy axis [58]. IV. ENERGY ABSORPTION ANDPRETHERMALIZATIONA. Energy absorption
We now consider energy absorption, focusing on many-body systems. Under what conditions does the systemabsorb energy from the rapid drive? Multiplying theFokker-Planck equation (11) by E and integrating overenergy, we obtain d (cid:104) E (cid:105) dt = (cid:104) g ( E, ω ) (cid:105) , (20)where (cid:104) f (cid:105) ≡ (cid:82) dE ηf for any f ( E ). Defining a micro-canonical temperature T µ ( E ) via1 T µ = ∂s∂E , (21) where s ( E ) ≡ k B log Σ( E ) is the microcanonical entropyand k B is Boltzmann’s constant, (18) becomes: g ( E, ω ) = 12 k B T µ (cid:20) g ( E, ω ) + k B T µ ∂g ( E, ω ) ∂E (cid:21) . (22)The expression in square brackets is an expansion of g ( E + k B T µ , ω ) for small k B T µ , truncated after first or-der. For a system with N degrees of freedom, the differ-ence between E and E + k B T µ corresponds to an energychange of k B T µ /N per degree of freedom. When N (cid:29) d (cid:104) E (cid:105) dt = (cid:68) S ( ω ; E )4 k B T µ (cid:69) . (23)For a many-body system with an unbounded phasespace, such as a gas or liquid, the density of states Σ( E )increases with energy, hence T µ ( E ) > T µ ( E ) < N classical spins described by H = B · (cid:80) n S n , T µ ( E ) < E >
0. Thus d (cid:104) E (cid:105) /dt can be negative. In thissituation we can view the normalized density of states,Σ( E ) ≡ Σ( E ) / (cid:82) dE (cid:48) Σ( E (cid:48) ), as the “infinite tempera-ture” energy distribution, obtained by considering thecanonical energy distribution η T c ( E ) ∝ Σ( E ) e − E/k B T c in the limit T c → ∞ . If g ( E ) is strictly positive for all E , ensuring that there are no insurmountable barriersalong the energy axis, then Eq. (19) describes an ergodic Markov process, and Σ( E ) is the unique stationary distri-bution to which any initial distribution evolves as t → ∞ [59, 60].We thus identify two possible energetic fates of a many-body system in the rapid driving regime. If the phasespace is unbounded, then the average energy of the sys-tem increases indefinitely, whereas if the system admits anormalized stationary distribution Σ( E ) then the systemevolves to this infinite temperature distribution. B. Prethermalization
In either case, the energy dynamics predicted by theenergy diffusion description relate to the phenomenon of prethermalization . A driven system is said to prether-malize if it reaches thermal equilibrium with respectto an effective Hamiltonian on short to intermediatetimescales, before ultimately gaining energy at far longertimes [12, 18–28]. In our case, if the system is prepared ina non-microcanonical (i.e. non-equilibrium) distributionon a particular energy shell E , then after a characteris-tic mixing time the distribution on this energy shell be-comes effectively microcanonical, i.e. prethermalizationoccurs with respect to H , at nearly constant energy.On longer timescales, the energy dynamics are governedby the Fokker-Planck equation (19), and the system ab-sorbs energy from the drive V ( t ) (23). For large ω thisabsorption can be exceedingly slow, as the power spec-trum S ( ω ; E ) decays faster than any power of ω − forany smooth H [61]. This is consistent with observedexponential-in-frequency suppression of energy absorp-tion in a range of classical and quantum model systems[12, 18, 20, 23, 24, 27, 28, 31, 35, 36, 39]. Prethermaliza-tion thus occurs when ω lies deep within the tail of thepower spectrum.Following the above-mentioned initial relaxation, en-ergy absorption is slow but does not vanish. As the sys-tem energy E gradually grows, the intrinsic correlationtime τ C ( E ) generically decreases with increasing particlevelocities, hence the power spectrum S ( ω ; E ) broadens.Eventually, at sufficiently large E , the drive frequencymight no longer be located in the far tail of the powerspectrum: This marks the onset of unsuppressed energyabsorption toward the infinite-temperature state. If thephase space is bounded, then ω can be chosen so thatenergy absorption is suppressed on all energy shells; inthis case energy absorption remains very slow throughoutthe system’s evolution towards the infinite-temperatureenergy distribution Σ( E ).Energy absorption from periodic driving has alsobeen studied using the Floquet-Magnus (FM) expansion,which, for time-periodic H ( z, t ), expresses the associated“Floquet” Hamiltonian H F ( z ) as a perturbative expan-sion in powers of ω − . H F is a time- independent Hamil-tonian whose dynamics coincide with those of H ( t ) atstroboscopic times t = 0 , T, T... . At high frequenciesand short timescales, the evolution obtained by truncat-ing the FM expansion at some order is expected to be agood approximation of the exact dynamics [18, 44, 62–64]. See e.g. the fourth-order (in ω − ) expression for H F ( z ) derived in [44, 62] for a system in one degreeof freedom. By contrast it appears that our results arenot obtainable via the FM expansion. For smooth H ,the coefficient g ( E, ω ) decays faster than any power of ω − at large ω (as mentioned earlier), and thus can-not be described accurately by an FM-like expansionin powers of ω − . Indeed, this might have been an-ticipated, as high frequency driving cannot induce un-bounded energy absorption unless the FM expansion di-verges [18, 21, 33, 63]. V. QUANTUM-CLASSICALCORRESPONDENCE
Energy absorption, prethermalization, and relaxationto the infinite temperature state have been documentedfor a variety of periodically driven quantum systems[18, 29, 30, 33, 34, 54]. It is instructive to ask howthe classical energy diffusion described by (19) mightemerge, in agreement with the correspondence principle,as the semiclassical limit of quantum dynamics. We nowbriefly describe a model that illustrates this correspon- dence; similar analyses may be found elsewhere in theliterature on energy diffusion [53, 65].Consider a quantum system governed by a Hamilto-nian ˆ H + ˆ V cos( ωt ), the counterpart of (3). Let us modelthe system’s evolution as a random walk in the spectrumof ˆ H , with stochastic quantum “jumps” from one energylevel to another. By Fermi’s golden rule, the transitionrate from energy E to E ± (cid:126) ω is given byΓ ± = π (cid:126) | V mn | ρ ( E n ) , (24)where V mn = (cid:104) m | ˆ V | n (cid:105) is the matrix element of ˆ V as-sociated with the energy levels E m and E n of ˆ H ; theoverbar denotes an average over a narrow range of ma-trix elements with E m ≈ E and E n ≈ E ± (cid:126) ω ; and ρ ( E ) = Σ( E ) /h N is the semiclassical density of states.As (cid:126) →
0, the spectrum of ˆ H becomes dense and ourrandom walk model leads naturally to a description interms of energy diffusion, with drift and diffusion coeffi-cients g = (Γ + − Γ − )( (cid:126) ω ) , g = (Γ + + Γ − )( (cid:126) ω ) . (25)A semiclassical estimate for matrix elements of quan-tized chaotic systems [66, 67] gives | V mn | ≈ h N − S V ( ω ; E )Σ( E ) , (26)where E ≡ ( E m + E n ) /
2. Here, S V ( ω ; E ) is the powerspectrum for the classical observable V , and is relatedto S ( ω ; E ) (the power spectrum for ˙ V ) via S = ω S V .Combining results, we find that (25) converges to theclassical results (18) and (15) as (cid:126) →
0. While this anal-ysis is based on a heuristic model that ignores quantumcoherences, it suggests that our classical energy diffusionpicture is relevant for understanding periodically drivenquantum systems; in particular it provides a semiclassi-cal explanation for the observed exponential-in-frequencysuppression of energy absorption [12, 18, 24, 27, 28, 31,35, 36, 39].
VI. CONCLUSION
We have analyzed the diffusive energy dynamics ofchaotic, ergodic Hamiltonian systems under rapid peri-odic driving. Observing that the system’s dynamics areonly weakly affected by very rapid driving, we have es-tablished a Fokker-Planck equation governing the evolu-tion of the system’s energy probability distribution. Ouranalysis predicts a generic, long-lived prethermal state,and for many-body systems our results point to two pos-sible energetic fates: indefinite energy growth, or relax-ation to the infinite-temperature equilibrium state. Inthe semiclassical limit, a model of energy absorption forperiodically driven, quantized chaotic systems coincideswith our purely classical energy diffusion description.A central feature of our Fokker-Planck equation is thatthe drift and diffusion coefficients g and g are deter-mined by the undriven dynamics. A similar situationarises in linear response theory (LRT), where transportcoefficients, such as electrical conductivities, in a systemsubject to weak time-periodic driving, are expressed interms of correlation functions computed in the absenceof driving [40]. In LRT these results are obtained per-turbatively, through a formal expansion in powers of thedriving strength. It is unclear whether our results cansimilarly be obtained through a perturbative expansion.A natural candidate for a small parameter in our case isthe inverse frequency ω − , but this seems to lead to theFloquet-Magnus expansion, which as already noted atthe end of Sec. IV B is somewhat at odds with our anal-ysis. Both this discrepancy, and the question of whetherour results can be obtained through a formal perturba-tive expansion, bear further investigation.Low-dimensional billiard systems – in which a particlein a cavity alternates between straight-line motion andspecular reflection off the cavity walls – offer an idealtesting ground for the theory presented in this paper, ascertain billiard shapes are rigorously proven [68–70] togenerate chaotic, ergodic motion. Energy absorption indriven billiard systems, sometimes known as Fermi accel-eration , is a well-studied phenomenon [32, 52, 71–74], al-though much of the existing literature focuses on the caseof slow driving. In a forthcoming work, we will presentnumerical evidence for the validity of the Fokker-Planckequation (19) for a particle in a chaotic billiard subject toa spatially uniform, rapidly time-periodic force. For thissystem, the driven and undriven trajectories of the par-ticle can be computed to machine precision and (19) canbe solved analytically, allowing for an especially precisetest of the energy diffusion description.Our results may also be tested for previously stud-ied many-body classical systems, such as a many-bodygeneralization of the kicked rotor model [9, 75] that ex-hibits unbounded energy absorption in a range of pa-rameter regimes [25, 37, 38]. Energy absorption has alsobeen studied in the classical driven Heisenberg spin chain[4, 20, 23]. For these models and others, we expect ouranalysis to apply only if the time-averaged Hamiltonian H generates chaotic and ergodic dynamics. ACKNOWLEDGEMENTS
We gratefully acknowledge stimulating discussionswith Ed Ott, David Levermore, and Saar Rahav, andfinancial support from the DARPA DRINQS program(D18AC00033).
APPENDIX
Here, we obtain an expression for the energy diffu-sion coefficient g , given by (15). We then derive the fluctuation-dissipation relation (18).
1. Calculation of g We begin with relation (14). According to this equa-tion, calculating g amounts to computing Var( E ), thevariance in energy acquired by an ensemble of trajectorieswith initial energy E , evolved for a time ∆ t under thedriven Hamiltonian. Specifically, we consider an ensem-ble of driven trajectories evolving from microcanonicallysampled initial conditions at t = 0. Upon integrating (8)along these trajectories, we obtain (with no approxima-tions so far)Var( E ) = (cid:90) ∆ t (cid:90) ∆ t dt dt (cid:48) cos( ωt ) cos( ωt (cid:48) ) C neq ( t, t (cid:48) ; E ) , (A1)where C neq ( t, t (cid:48) ; E ) ≡ (cid:104) ˙ V ( z t ) ˙ V ( z t (cid:48) ) (cid:105) − (cid:104) ˙ V ( z t ) (cid:105)(cid:104) ˙ V ( z t (cid:48) ) (cid:105) isa nonequilibrium correlation function and angular brack-ets (cid:104)·(cid:105) denote an ensemble average. In the high-frequencylimit ω → ∞ , as driven trajectories z t approach their un-driven counterparts z t , C neq ( t, t (cid:48) ; E ) can be replaced bythe equilibrium correlation function C ( t (cid:48) − t ; E ) ≡ (cid:104) ˙ V ( z t ) ˙ V ( z t (cid:48) ) (cid:105) − (cid:104) ˙ V ( z t ) (cid:105)(cid:104) ˙ V ( z t (cid:48) ) (cid:105) , (A2)which depends only on the difference t (cid:48) − t , due to thetime-translation symmetry of the microcanonical distri-bution under the undriven dynamics.Replacing C neq ( t, t (cid:48) ; E ) by C ( t (cid:48) − t ; E ) in (A1), andusing standard manipulations to evaluate the double in-tegral (see, e.g. [76]), we arrive atVar( E ) ≈ S ( ω ; E )∆ t, (A3)where S ( ω ; E ) = (cid:90) ∞−∞ dt e − iωt C ( t ; E ) (A4)is the power spectrum of ˙ V ( z t ), which is equal to theFourier transform of C ( t ; E ) by the Wiener-Khinchintheorem [57]. The approximation in (A3) contains cor-rection terms that are sublinear in ∆ t . Comparing (A3)with (14) and relabeling E as E , we obtain (15), ourfinal expression for g .
2. Calculation of g We now derive (18), which expresses a fluctuation-dissipation relation between the drift and diffusion co-efficients g and g . To do so, we first note that theconstant function ρ ( z ) = 1 is a stationary solution to theLiouville equation (2). This reflects the incompressibil-ity of phase space volume under Hamiltonian dynamics(Liouville’s theorem) [40]. Since ρ = 1 is stationary un-der the dynamics in phase space , the corresponding (un-normalized) distribution in energy space should be sta-tionary under the Fokker-Planck equation. This energydistribution, obtained by marginalizing over the constantsolution ρ = 1, is the density of states Σ( E ) – see (6).Setting η ( E, t ) = Σ( E ) as a stationary solution of theFokker-Planck equation (11), we have0 = − ∂∂E (cid:20) g Σ − ∂∂E ( g Σ) (cid:21) . (A5)Thus the quantity in square brackets is constant as afunction of E . We label this constant by α : α ≡ g Σ − ∂∂E ( g Σ) . (A6)We now aim to show that α = 0, which then immediatelyimplies the fluctuation-dissipation relation (18).To proceed, we first use (A6) to eliminate g from theFokker-Planck equation (11), obtaining ∂η∂t = − α ∂∂E (cid:16) η Σ (cid:17) + 12 ∂∂E (cid:20) g Σ ∂∂E (cid:16) η Σ (cid:17)(cid:21) . (A7)In the main text, in arguing that the system energyevolves diffusively, we considered trajectories with a com-mon initial energy E , and we arrived at the hierarchyof timescales (12) required for the validity of the energydiffusion picture: T (cid:28) τ C ( E ) (cid:28) ∆ t (cid:28) τ E ( ω, E ). Since τ E → ∞ as ω → ∞ , this hierarchy suggests that for agiven, sufficiently large value of ω , there is a range ofenergies over which (A7) is valid. This range can be en-larged by increasing the value of ω , but there might existno value ω ∗ such that (A7) is valid over the entire energyaxis for all ω > ω ∗ . Thus let us fix the value of ω andlet [ a, b ] denote a finite interval of the energy axis, suchthat (A7) is valid for energies a ≤ E ≤ b . The existenceof such an interval is sufficient to establish that α = 0,as we now show.Consider an ensemble of trajectories evolving under H ( z, t ), from an initial phase space distribution that isuniform up to a cutoff E ∈ ( a, b ): ρ ( z,
0) = c θ ( E − H ( z )) , (A8)where θ ( · ) is the unit step function and c − = Ω( E ) isthe volume of phase space enclosed by the energy shell E (which was assumed finite in Section III). The corre-sponding energy distribution is η ( E,
0) = (cid:90) dz δ ( E − H ) ρ ( z, c Σ( E ) θ ( E − E ) . (A9)As this ensemble of trajectories evolves in time, thevalue of the density ρ at any ( z, t ) is either c or 0, byLiouville’s theorem. For a sufficiently short but finiteinterval 0 ≤ t ≤ δt , ρ ( z, t ) remains constant outside the FIG. 1. Schematic depiction of phase space. On the left,the initial distribution is uniform, ρ ( z,
0) = c (shaded), upto a cutoff energy E . The figure on the right shows thedistribution a short time later, after evolution under H ( z, t ); ρ ( z, δt ) = c in the shaded region. Only in the annular regionbetween the two energy shells a and b (dashed circles) does ρ ( z, t ) vary with time for t ∈ [0 , δt ]. Asterisks depict initialand final conditions for a representative trajectory. region of phase space between the two energy shells H = a and H = b (see Fig. 1), hence η ( E, t ) = (cid:40) c Σ( E ) if E ≤ a E ≥ b (A10)We emphasize that (A10) is exact, and a direct conse-quence of Liouville’s theorem.(A10) implies that for t ∈ [0 , δt ] there is no net flow ofprobability into or out of the energy interval [ a, b ]:0 = ddt (cid:90) ba η ( E, t ) dE. (A11)We can use (A7), which is valid in [ a, b ], along with (A10)to evaluate the right side of (A11), obtaining0 = (cid:20) − α (cid:16) η Σ (cid:17) + 12 g Σ ∂∂E (cid:16) η Σ (cid:17)(cid:21) ba = αc, (A12)hence α = 0. 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