# Energy diffusion and absorption in chaotic systems with rapid periodic
driving

Wade Hodson<EMAIL_ADDRESS>Department of Physics, University of Maryland,
College Park, MD 20742  Christopher Jarzynski<EMAIL_ADDRESS>Institute for
Physical Science and Technology, Department of Chemistry and Biochemistry, and
Department of Physics, University of Maryland, College Park, MD 20742

###### Abstract

When a chaotic, ergodic Hamiltonian system with $N$ degrees of freedom is
subject to sufficiently rapid periodic driving, its energy evolves
diffusively. We derive a Fokker-Planck equation that governs the evolution of
the system’s probability distribution in energy space, and we provide explicit
expressions for the energy drift and diffusion rates. Our analysis suggests
that the system generically relaxes to a long-lived “prethermal” state
characterized by minimal energy absorption, eventually followed by more rapid
heating. When $N\gg 1$, the system ultimately absorbs energy indefinitely from
the drive, or at least until an infinite temperature state is reached.

###### pacs:

Valid PACS appear here

##  I Introduction

Time-periodic driving facilitates a rich range of classical and quantum
dynamical behaviors, including synchronization and resonance Van der Pol and
Van der Mark (1927); Chacón (1994); Antonsen  _et al._ (2008); Khasseh  _et
al._ (2019), localization Dittrich  _et al._ (1993); Nag  _et al._ (2014);
Bairey  _et al._ (2017); Haldar and Das (2017), and chaos Van der Pol and Van
der Mark (1927); Ott (2002). Recent theoretical and experimental work has
aimed to identify nonequilibrium “phases of matter” that might emerge in
periodically driven systems Khemani  _et al._ (2016); Yao  _et al._ (2017);
Else  _et al._ (2017). Phenomena such as time crystallization Yao  _et al._
(2017); Else  _et al._ (2017); Zhang  _et al._ (2017); Choi  _et al._ (2017);
Russomanno  _et al._ (2017); Yao  _et al._ (2020); Kyprianidis  _et al._
(2021) and prethermalization Else  _et al._ (2017); Abanin  _et al._ (2017);
Herrmann  _et al._ (2017); Mori (2018); Mori  _et al._ (2018); Mallayya  _et
al._ (2019); Howell  _et al._ (2019); Machado  _et al._ (2019); Rajak  _et
al._ (2019); Machado  _et al._ (2020); Rubio-Abadal  _et al._ (2020); Peng
_et al._ (2021); Kyprianidis  _et al._ (2021) reveal that periodic driving can
stabilize systems in a variety of interesting and useful states.

Energy absorption poses a potential obstacle to such stabilization of
nonequilibrium states of matter. A driven open system in a nonequilibrium
steady state attains a balance in which energy absorbed from the drive is
dissipated into an environment, such as a thermal bath. But if a system is
isolated, save its interaction with the drive, then maintaining a stable state
requires the suppression of energy absorption from the drive. Much work has
been devoted to understanding energy absorption, and the conditions under
which it might be suppressed, in periodically driven, isolated classical and
quantum systems D’Alessio and Rigol (2014); Lazarides  _et al._ (2014); Abanin
_et al._ (2015); Demers and Jarzynski (2015); Ponte  _et al._ (2015); Rehn
_et al._ (2016); Mori  _et al._ (2016); Kuwahara  _et al._ (2016); Abanin  _et
al._ (2017); Mori (2018); Rajak  _et al._ (2018); Notarnicola  _et al._
(2018); Howell  _et al._ (2019); Machado  _et al._ (2019); Tran  _et al._
(2019); Rajak  _et al._ (2019) .

In this paper we study the general problem of energy absorption in isolated
classical chaotic systems subject to rapid periodic driving. We argue that the
energetic dynamics of such systems are diffusive, and we derive a Fokker-
Planck equation for the evolution of the system’s energy probability
distribution, $\eta(E,t)$. The drift and diffusion coefficients in this
equation, characterizing energy absorption and the spreading of the energy
distribution, are given explicitly in terms of the dynamics of the undriven
system – much as in the case of ordinary linear response theory Dorfman
(1999), but without the assumption of weak driving. For many-body systems our
results suggest a scenario marked by three stages: initial relaxation to an
equilibrium-like “prethermal” state Else  _et al._ (2017); Abanin  _et al._
(2017); Herrmann  _et al._ (2017); Mori (2018); Mori  _et al._ (2018);
Mallayya  _et al._ (2019); Howell  _et al._ (2019); Machado  _et al._ (2019);
Rajak  _et al._ (2019); Machado  _et al._ (2020); Rubio-Abadal  _et al._
(2020); Peng  _et al._ (2021), followed by a long interval of minimal energy
absorption, and finally rapid absorption toward an infinite-temperature state.

Our description provides a comprehensive, quantitative account of energy
absorption in rapidly and periodically driven chaotic systems. It reveals how
chaos in phase space facilitates stochastic energy evolution, how energy
diffusion leads to the breakdown of the prethermal regime, and how energy
absorption rates are determined by the underlying, undriven Hamiltonian
dynamics. Our framework also suggests a generic explanation for the
exponential-in-frequency suppression of energy absorption observed in a range
of systems Abanin  _et al._ (2015); Mori  _et al._ (2016); Kuwahara  _et al._
(2016); Else  _et al._ (2017); Abanin  _et al._ (2017); Mori (2018); Howell
_et al._ (2019); Machado  _et al._ (2019); Tran  _et al._ (2019); Rubio-Abadal
_et al._ (2020); Peng  _et al._ (2021). Finally, we argue that the classical
results that we obtain are relevant to energy absorption in quantum systems,
in an appropriate semiclassical limit.

In Sec. II we define the problem we will study. In Sec. III we argue that the
energy of a rapidly driven chaotic system evolves diffusively, and we derive
the Fokker-Planck equation that describes this evolution. In Sec. IV we
analyze energy absorption and prethermalization in the context of our energy
diffusion model. In Sec. V we briefly consider the quantum counterpart of our
classical problem, and we conclude in Sec. VI.

##  II Setup

Our object of study is a classical Hamiltonian system with $N\geq 2$ degrees
of freedom. At time $t$, the microscopic state of the system is specified by a
phase space point $z_{t}\equiv z\equiv(\mathbf{q},\mathbf{p})$, where the
$N$-vectors $\mathbf{q}$ and $\mathbf{p}$ specify canonical coordinates and
momenta. The system evolves under Hamilton’s equations of motion, generated by
a Hamiltonian $H\equiv H(z,t)$:

$\frac{d\mathbf{q}}{dt}=\frac{\partial
H}{\partial\mathbf{p}},\quad\frac{d\mathbf{p}}{dt}=-\frac{\partial
H}{\partial\mathbf{q}}.$ (1)

For an ensemble of trajectories, the phase space probability distribution
$\rho(z,t)$ obeys the Liouville equation

$\frac{\partial\rho}{\partial t}=\\{H,\rho\\}=\frac{\partial
H}{\partial\mathbf{q}}\cdot\frac{\partial\rho}{\partial\mathbf{p}}-\frac{\partial
H}{\partial\mathbf{p}}\cdot\frac{\partial\rho}{\partial\mathbf{q}},$ (2)

where $\\{\cdot,\cdot\\}$ denotes the Poisson bracket Goldstein (1980). We
take $H(z,t)$ to be a periodic function of time, with period $T$, and we
decompose this Hamiltonian into its time average $H_{0}(z)\equiv
T^{-1}\int_{0}^{T}dt\,H(z,t)$, and a remainder $V(z,t)=V(z,t+T)$ with
vanishing average:

$H(z,t)=H_{0}(z)+V(z,t).$ (3)

We will refer to $H_{0}(z)$ as the “bare” or “undriven” Hamiltonian, and to
$V(z,t)$ as the “drive.” The former determines the evolution of the system in
complete isolation, that is when $V=0$. We define the system’s energy $E(t)$
to be the bare Hamiltonian evaluated at $z_{t}$: $E(t)\equiv H_{0}(z_{t})$.
When $V=0$ the energy is a constant of the motion: any trajectory $z_{t}$ is
constrained to evolve on an energy shell, that is a level surface of
$H_{0}(z)$. The evolution of the energy when $V\neq 0$ will be our central
focus. The magnitude of the drive $V(z,t)$ is arbitrary; in particular we do
not assume it to be small.

We assume that the undriven dynamics are chaotic and ergodic on each energy
shell of $H_{0}$. Such dynamics exhibit mixing as trajectories diverge from
one another exponentially with time Ott (2002). This leads to a loss of
statistical dependence between states of the system at different times, as
reflected in the decay of correlations – in effect the system loses its memory
of previously visited states. After a characteristic mixing time, any smooth
initial distribution on the energy shell evolves to a distribution that for
practical purposes is microcanonical, or thermal Reichl (1980); Dorfman
(1999). Thus chaos offers a way to understand the self-thermalizing properties
of many-body systems such as gases and liquids, while also providing low-
dimensional analogues, such as chaotic billiard systems, that are accessible
to numerical or analytical study.

We are interested in the limit of high driving frequency $\omega=2\pi/T$. When
$\omega\to\infty$ the effect of the drive averages to zero over each period,
as the system cannot appreciably react to the drive in such a short time. In
this limit the evolution generated by the driven Hamiltonian approaches the
undriven evolution: for given initial conditions $z_{0}$, and over a fixed
time interval $0\leq t\leq\tau$, the trajectory $z_{t}$ that evolves under
$H(z,t)$ converges, as $\omega\rightarrow\infty$, to the trajectory
$z_{t}^{0}$ that evolves under $H_{0}(z)$ Murdock (1999); Rahav  _et al._
(2003a). This limit can be attained regardless of the strength of the drive,
$V(z,t)$. In our case we assume $\omega$ is sufficiently high that driven and
undriven trajectories remain close over timescales characteristic of the decay
of correlations.

These considerations lead to the following picture. For sufficiently short
times, energy is approximately conserved, and the driven trajectories $z_{t}$
generated by $H(z,t)$ are similar to the undriven trajectories $z_{t}^{0}$
generated by $H_{0}(z)$, allowing us to use the latter to estimate correlation
functions that will arise in our analysis. These correlation functions, as we
shall see, will in turn describe how the system absorbs energy from the
external drive on much longer timescales.

##  III Energy diffusion

Given the assumptions mentioned in Section II, we now argue that the energy of
the driven system evolves diffusively. For simplicity, we consider
monochromatic driving,

$V(z,t)=V(z)\cos(\omega t)\,.$ (4)

Our analysis can be generalized to arbitrary time-periodic driving, by
decomposing $V(z,t)$ in a Fourier series with fundamental frequency $\omega$.

###  III.1 Argument for energy diffusion

To begin, we consider a system that evolves over a time interval $0\leq
t\leq\Delta t$, from initial conditions $z_{0}$ sampled from a microcanonical
distribution at energy $E_{0}$:

$\rho(z_{0},0)=\rho_{E_{0}}(z_{0})\equiv\frac{1}{\Sigma(E_{0})}\delta(E_{0}-H_{0}(z_{0})).$
(5)

Here

$\Sigma(E)=\frac{\partial\Omega}{\partial E}=\int dz\,\delta(H_{0}(z)-E)$ (6)

is the classical density of states;

$\Omega(E)=\int dz\,\theta(E-H_{0}(z))$ (7)

is the phase space volume enclosed by the energy shell $E$, which we assume to
be finite for all $E$; and the integrals are over phase space. Let $\Delta
E(z_{0})$ denote the net change in the system’s energy from $t=0$ to $t=\Delta
t$. By Hamilton’s equations, $\Delta E(z_{0})$ is the time integral of the
power

$\frac{dE}{dt}\equiv\frac{d}{dt}H_{0}(z_{t})=-\cos(\omega t)\dot{V}(z_{t}),$
(8)

where $\dot{V}(z)\equiv\\{V,H_{0}\\}$. The quantity $\Delta E$ can be viewed
as a random variable, whose value is determined by the sampled initial
conditions $z_{0}$. Understanding the statistics of $\Delta E$ in the high-
frequency driving regime, for an appropriate choice of $\Delta t$ (to be
clarified below), will be the key to establishing diffusion in energy space.

[We note in passing that if the undriven Hamiltonian has the form
$H_{0}=\sum_{n}({\bf p}_{n}^{2}/2m_{n})+U_{0}(\\{{\bf q}_{n}\\})$ and if the
drive $V$ depends on coordinates $\\{{\bf q}_{n}\\}$ but not momenta $\\{{\bf
p}_{n}\\}$, then (8) becomes $dE/dt=\sum_{n}{\bf F}_{n}\cdot{\bf v}_{n}$,
where ${\bf F}_{n}$ is the driving force acting on the $n$’th particle, and
${\bf v}_{n}$ is that particle’s velocity.]

We now explicitly assume the driving is rapid. To begin, we impose the
condition

$T\ll\tau_{C}(E_{0}),$ (9)

where $T=2\pi/\omega$ is the drive period and $\tau_{C}(E_{0})$ is a
characteristic timescale over which chaotic mixing on the energy shell $E_{0}$
produces the decay of correlations. Heuristically, (9) implies that a
trajectory $z_{t}$ travels only a negligible distance during one period of
driving. This condition produces the averaging over oscillations that (as
mentioned in Sec. II) results in driven trajectories $z_{t}$ resembling their
undriven counterparts $z_{t}^{0}$. Let us now choose $\Delta t$ so that over
the interval $[0,\Delta t]$ the driven trajectories in our ensemble remain
close to the initial energy shell $E_{0}$. Chaotic mixing ensures then that a
microcanonical distribution is approximately maintained. Thus for any
$t\in[0,\Delta t]$, the ensemble of points $z_{t}$ are approximately
distributed according to the initial microcanonical ensemble.

With this picture in mind, let us divide the time interval $[0,\Delta t]$ into
$M\gg 1$ subintervals of equal duration $\delta t=\Delta t/M$, and consider
$\Delta E=\sum_{i}\delta E_{i}$ as a sum of subinterval energy changes $\delta
E_{i}$, $i=1,2\cdots M$. Each increment $\delta E_{i}$ is itself a random
variable, determined by integrating the power (8) along the trajectory $z_{t}$
over the subinterval. By the arguments of the previous paragraph, the $\delta
E_{i}$’s have nearly identical, microcanonical statistics, provided we choose
$\delta t$ (and therefore also $\Delta t$) to be an integer multiple of the
driving period $T$ to ensure that each subinterval begins at the same phase of
the drive.

Chaotic mixing on the energy shell $E_{0}$ produces the decay of correlations.
Let us further choose $\delta t$ to be longer than the characteristic
correlation time $\tau_{C}(E_{0})$, so that each $\delta E_{i}$ is
approximately statistically independent from the others. The energy change
$\Delta E$ is then a sum of $M\gg 1$ approximately independent and identically
distributed increments $\delta E_{i}$: the system effectively performs a
random walk on the energy axis. By the central limit theorem $\Delta E$ is a
normally-distributed random variable, whose mean and variance grow (for fixed
$\delta t$) in proportion to the number of increments $M$, equivalently the
time elapsed $\Delta t$.

The statistical behavior just described is characteristic of a diffusive
process in energy space, motivating us to model it by a Fokker-Planck equation
Gardiner (1985). Letting

$\eta(E,t)\equiv\int dz\,\delta(H_{0}(z)-E)\rho(z,t)$ (10)

denote the energy distribution, we postulate that the time evolution of $\eta$
is given by

$\frac{\partial\eta}{\partial t}=-\frac{\partial}{\partial
E}\left(g_{1}\eta\right)+\frac{1}{2}\frac{\partial^{2}}{\partial
E^{2}}\left(g_{2}\eta\right).$ (11)

The drift and diffusion coefficients $g_{1}(E,\omega)$ and $g_{2}(E,\omega)$
characterize, respectively, the rate at which the distribution $\eta$ shifts
and spreads on the energy axis; see (13) and (14) below. These coefficients
depend on the system energy $E$ and the driving frequency $\omega$. Energy
diffusion and its description in terms of the Fokker-Planck equation have been
studied in various contexts involving externally driven Hamiltonian systems
Ott (1979); Brown  _et al._ (1987a, b); Wilkinson (1990); Linkwitz and Grabert
(1991); Jarzynski (1992, 1993); Cohen (2000); Bunin  _et al._ (2011); De
Bièvre and Parris (2011); Demers and Jarzynski (2015); De Bièvre  _et al._
(2016). Before deriving expressions for $g_{1}$ and $g_{2}$ in the high-
frequency driving regime, it is worth examining the central role that a
separation of timescales plays in our analysis.

We have assumed, after (9), that $\Delta t$ is much smaller than the timescale
$\tau_{E}(\omega,E_{0})$ over which the energy of the system changes
significantly. This condition ensures that the energy increments $\delta
E_{i}$ have approximately identical microcanonical statistics. We have also
assumed that the interval $\Delta t$ contains many subintervals of duration
$\delta t$, and that $\delta t>\tau_{C}(E_{0})$, guaranteeing approximate
statistical independence among the increments $\delta E_{i}$. Thus our
analysis involves the hierarchy of timescales:

$T\ll\tau_{C}(E_{0})\ll\Delta t\ll\tau_{E}(\omega,E_{0}).$ (12)

Since $\tau_{E}\to\infty$ as $\omega\to\infty$, this hierarchy can be
satisfied for any particular energy shell $E_{0}$ by setting $\omega$
sufficiently large. We conclude that Eq. (11) is valid over an interval of the
energy axis whose extent is determined by, and increases with, the value of
$\omega$.

The above arguments suggest that the energy diffusion description is valid on
a coarse-grained timescale of order $\Delta t$. On shorter timescales,
computing the fine details of the system’s energy evolution requires the full
Hamiltonian equations of motion (1). These details vary greatly from system to
system. However, as we will see, the characteristics of the energy diffusion
process ultimately depend only on a few key details of these system-specific
dynamics, as captured in the coefficients $g_{1}$ and $g_{2}$.

###  III.2 Drift and diffusion coefficients

Under (11) an initial distribution $\eta(E,0)=\delta(E-E_{0})$ evolves after a
time $\Delta t\ll\tau_{E}$ to a distribution $\eta(E,\Delta t)$ with mean and
variance Gardiner (1985):

$\displaystyle\mathrm{Mean}(E)$ $\displaystyle=$ $\displaystyle
E_{0}+g_{1}(E_{0},\omega)\Delta t$ (13) $\displaystyle\mathrm{Var}(E)$
$\displaystyle=$ $\displaystyle g_{2}(E_{0},\omega)\Delta t.$ (14)

We can thus determine $g_{2}$ by calculating $\mathrm{Var}(E)$, the energy
spread acquired by an ensemble of trajectories with initial energy $E_{0}$,
evolved for a time $\Delta t$ under the driven Hamiltonian. We perform this
calculation in Section .1 of the Appendix, obtaining, in the limit of large
$\omega$,

$\displaystyle g_{2}(E,\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{2}S(\omega;E)\geq 0$ (15) $\displaystyle S(\omega;E)$
$\displaystyle=$ $\displaystyle\int_{-\infty}^{\infty}dt\,e^{-i\omega
t}C(t;E)\,,$ (16)

where

$C(t;E)\equiv\langle\dot{V}(z_{0}^{0})\dot{V}(z_{t}^{0})\rangle-\langle\dot{V}(z_{0}^{0})\rangle\langle\dot{V}(z_{t}^{0})\rangle$
(17)

is the microcanonical autocorrelation function of the observable $\dot{V}(z)$.
Specifically, the averages denoted by $\langle\cdot\rangle$ are computed by
sampling initial conditions $z_{0}^{0}$ from a microcanonical ensemble at
energy $E$, then evolving for time $t$ under $H_{0}(z)$. By the Wiener-
Khinchin theorem Kubo  _et al._ (2012), the Fourier transform of $C(t;E)$ is
the power spectrum of $\dot{V}(z_{t}^{0})$ at energy $E$, denoted by
$S(\omega;E)$. Note that (15) gives $g_{2}$ entirely in terms of properties of
the undriven system, as $C(t;E)$ and thus $S(\omega;E)$ are defined in terms
of the undriven trajectories $z_{t}^{0}$.

In solving for $g_{2}$ we approximated driven trajectories $z_{t}$ by their
undriven counterparts $z_{t}^{0}$. As a result, we expect that (15) contains
correction terms that become negligible in the high-frequency limit
$\omega\rightarrow\infty$.

In Section .2 of the Appendix we use Liouville’s theorem, which expresses the
incompressibility of phase space volume under Hamiltonian dynamics Dorfman
(1999), to obtain the following expression for the drift coefficient $g_{1}$
in terms of $g_{2}(E,\omega)$ and the density of states $\Sigma(E)$ (6):

$g_{1}(E,\omega)=\frac{1}{2\Sigma}\frac{\partial}{\partial
E}\Big{(}g_{2}\Sigma\Big{)}.$ (18)

This result is a fluctuation-dissipation relation, similar to others
previously established for various driven Hamiltonian systems Brown  _et al._
(1987a, b); Wilkinson (1990); Jarzynski (1992, 1993); Cohen (2000); Bunin  _et
al._ (2011).

Using (15) and (18), the Fokker-Planck equation (11) takes the compact form

$\frac{\partial\eta}{\partial t}=\frac{1}{4}\frac{\partial}{\partial
E}\left[S\Sigma\frac{\partial}{\partial
E}\left(\frac{\eta}{\Sigma}\right)\right].$ (19)

Eq. (19) is our main result. It describes the stochastic evolution of the
system’s energy, under rapid driving, in terms of quantities $S(\omega;E)$ and
$\Sigma(E)$ that characterize the undriven system.

As discussed earlier, we expect (19) to be valid over a region of the energy
axis whose extent depends on $\omega$. In the next section we assume $\omega$
is sufficiently large that (19) is valid over the entire energy axis 111If
$H_{0}(z)$ has a finite range, then $\omega$ can be chosen so that (11) is
valid over all allowable energies. If the range of $H_{0}(z)$ is unbounded
then the extent of validity of (11) can be made arbitrarily large, though not
necessarily infinite, by appropriate choice of $\omega$..

##  IV Energy absorption and prethermalization

###  IV.1 Energy absorption

We now consider energy absorption, focusing on many-body systems. Under what
conditions does the system absorb energy from the rapid drive? Multiplying the
Fokker-Planck equation (11) by $E$ and integrating over energy, we obtain

$\frac{d\langle E\rangle}{dt}=\langle g_{1}(E,\omega)\rangle,$ (20)

where $\langle f\rangle\equiv\int dE\,\eta f$ for any $f(E)$. Defining a
microcanonical temperature $T_{\mu}(E)$ via

$\frac{1}{T_{\mu}}=\frac{\partial s}{\partial E}\,,$ (21)

where $s(E)\equiv k_{B}\log\Sigma(E)$ is the microcanonical entropy and
$k_{B}$ is Boltzmann’s constant, (18) becomes:

$g_{1}(E,\omega)=\frac{1}{2k_{B}T_{\mu}}\left[g_{2}(E,\omega)+k_{B}T_{\mu}\frac{\partial
g_{2}(E,\omega)}{\partial E}\right].$ (22)

The expression in square brackets is an expansion of
$g_{2}(E+k_{B}T_{\mu},\omega)$ for small $k_{B}T_{\mu}$, truncated after first
order. For a system with $N$ degrees of freedom, the difference between $E$
and $E+k_{B}T_{\mu}$ corresponds to an energy change of $k_{B}T_{\mu}/N$ per
degree of freedom. When $N\gg 1$ this change is negligible and (15), (20),
(22) give

$\frac{d\langle
E\rangle}{dt}=\Bigl{\langle}\frac{S(\omega;E)}{4k_{B}T_{\mu}}\Bigr{\rangle}.$
(23)

For a many-body system with an unbounded phase space, such as a gas or liquid,
the density of states $\Sigma(E)$ increases with energy, hence $T_{\mu}(E)>0$
and (15), (23) imply that the average energy of the system continually
increases with time, as expected intuitively.

If the phase space of the system is bounded, then we expect $T_{\mu}(E)<0$ at
some energies. For example, for $N$ classical spins described by $H_{0}={\bf
B}\cdot\sum_{n}{\bf S}_{n}$, $T_{\mu}(E)<0$ when $E>0$. Thus $d\langle
E\rangle/dt$ can be negative. In this situation we can view the normalized
density of states, $\overline{\Sigma}(E)\equiv\Sigma(E)/\int
dE^{\prime}\,\Sigma(E^{\prime})$, as the “infinite temperature” energy
distribution, obtained by considering the canonical energy distribution
$\eta_{T_{c}}(E)\propto\Sigma(E)e^{-E/k_{B}T_{c}}$ in the limit
$T_{c}\rightarrow\infty$. If $g_{2}(E)$ is strictly positive for all $E$,
ensuring that there are no insurmountable barriers along the energy axis, then
Eq. (19) describes an ergodic Markov process, and $\overline{\Sigma}(E)$ is
the unique stationary distribution to which any initial distribution evolves
as $t\rightarrow\infty$ Gilks  _et al._ (1995); Gallager (2013).

We thus identify two possible energetic fates of a many-body system in the
rapid driving regime. If the phase space is unbounded, then the average energy
of the system increases indefinitely, whereas if the system admits a
normalized stationary distribution $\overline{\Sigma}(E)$ then the system
evolves to this infinite temperature distribution.

###  IV.2 Prethermalization

In either case, the energy dynamics predicted by the energy diffusion
description relate to the phenomenon of prethermalization. A driven system is
said to prethermalize if it reaches thermal equilibrium with respect to an
effective Hamiltonian on short to intermediate timescales, before ultimately
gaining energy at far longer times Else  _et al._ (2017); Abanin  _et al._
(2017); Herrmann  _et al._ (2017); Mori (2018); Mori  _et al._ (2018);
Mallayya  _et al._ (2019); Howell  _et al._ (2019); Machado  _et al._ (2019);
Rajak  _et al._ (2019); Machado  _et al._ (2020); Rubio-Abadal  _et al._
(2020); Peng  _et al._ (2021). In our case, if the system is prepared in a
non-microcanonical (i.e. non-equilibrium) distribution on a particular energy
shell $E_{0}$, then after a characteristic mixing time the distribution on
this energy shell becomes effectively microcanonical, i.e. prethermalization
occurs with respect to $H_{0}$, at nearly constant energy. On longer
timescales, the energy dynamics are governed by the Fokker-Planck equation
(19), and the system absorbs energy from the drive $V(t)$ (23). For large
$\omega$ this absorption can be exceedingly slow, as the power spectrum
$S(\omega;E)$ decays faster than any power of $\omega^{-1}$ for any smooth
$H_{0}$ Bracewell (1978). This is consistent with observed exponential-in-
frequency suppression of energy absorption in a range of classical and quantum
model systems Abanin  _et al._ (2015); Mori  _et al._ (2016); Kuwahara  _et
al._ (2016); Else  _et al._ (2017); Abanin  _et al._ (2017); Mori (2018);
Howell  _et al._ (2019); Machado  _et al._ (2019); Tran  _et al._ (2019);
Rubio-Abadal  _et al._ (2020); Peng  _et al._ (2021). Prethermalization thus
occurs when $\omega$ lies deep within the tail of the power spectrum.

Following the above-mentioned initial relaxation, energy absorption is slow
but does not vanish. As the system energy $E$ gradually grows, the intrinsic
correlation time $\tau_{C}(E)$ generically decreases with increasing particle
velocities, hence the power spectrum $S(\omega;E)$ broadens. Eventually, at
sufficiently large $E$, the drive frequency might no longer be located in the
far tail of the power spectrum: This marks the onset of unsuppressed energy
absorption toward the infinite-temperature state. If the phase space is
bounded, then $\omega$ can be chosen so that energy absorption is suppressed
on all energy shells; in this case energy absorption remains very slow
throughout the system’s evolution towards the infinite-temperature energy
distribution $\overline{\Sigma}(E)$.

Energy absorption from periodic driving has also been 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 expansion in
powers of $\omega^{-1}$. $H_{F}$ is a time-independent Hamiltonian whose
dynamics coincide with those of $H(t)$ at stroboscopic times $t=0,T,2T...$. At
high frequencies and short timescales, the evolution obtained by truncating
the FM expansion at some order is expected to be a good approximation of the
exact dynamics Rahav  _et al._ (2003b, a); Bukov  _et al._ (2015); Abanin  _et
al._ (2017); Higashikawa  _et al._ (2018). See e.g. the fourth-order (in
$\omega^{-1}$) expression for $H_{F}(z)$ derived in Rahav  _et al._ (2003b, a)
for a system in one degree of freedom. By contrast it appears that our results
are not obtainable via the FM expansion. For smooth $H_{0}$, the coefficient
$g_{2}(E,\omega)$ decays faster than any power of $\omega^{-1}$ at large
$\omega$ (as mentioned earlier), and thus cannot be described accurately by an
FM-like expansion in powers of $\omega^{-1}$. Indeed, this might have been
anticipated, as high frequency driving cannot induce unbounded energy
absorption unless the FM expansion diverges Bukov  _et al._ (2015); Ponte  _et
al._ (2015); Abanin  _et al._ (2017); Mori  _et al._ (2018).

##  V Quantum-classical correspondence

Energy absorption, prethermalization, and relaxation to the infinite
temperature state have been documented for a variety of periodically driven
quantum systems Bunin  _et al._ (2011); Lazarides  _et al._ (2014); D’Alessio
and Rigol (2014); Ponte  _et al._ (2015); Rehn  _et al._ (2016); Abanin  _et
al._ (2017). It is instructive to ask how the classical energy diffusion
described by (19) might emerge, in agreement with the correspondence
principle, as the semiclassical limit of quantum dynamics. We now briefly
describe a model that illustrates this correspondence; similar analyses may be
found elsewhere in the literature on energy diffusion Cohen (2000); Elyutin
(2006).

Consider a quantum system governed by a Hamiltonian
$\hat{H}_{0}+\hat{V}\cos(\omega t)$, the counterpart of (3). Let us model the
system’s evolution as a random walk in the spectrum of $\hat{H}_{0}$, with
stochastic quantum “jumps” from one energy level to another. By Fermi’s golden
rule, the transition rate from energy $E$ to $E\pm\hbar\omega$ is given by

$\Gamma_{\pm}=\frac{\pi}{2\hbar}\overline{|V_{mn}|^{2}}\rho(E_{n})\,,$ (24)

where $V_{mn}=\langle m|\hat{V}|n\rangle$ is the matrix element of $\hat{V}$
associated with the energy levels $E_{m}$ and $E_{n}$ of $\hat{H}_{0}$; the
overbar denotes an average over a narrow range of matrix elements with
$E_{m}\approx E$ and $E_{n}\approx E\pm\hbar\omega$; and
$\rho(E)=\Sigma(E)/h^{N}$ is the semiclassical density of states. As $\hbar\to
0$, the spectrum of $\hat{H}_{0}$ becomes dense and our random walk model
leads naturally to a description in terms of energy diffusion, with drift and
diffusion coefficients

$g_{1}=(\Gamma_{+}-\Gamma_{-})(\hbar\omega)\quad,\quad
g_{2}=(\Gamma_{+}+\Gamma_{-})(\hbar\omega)^{2}.$ (25)

A semiclassical estimate for matrix elements of quantized chaotic systems
Feingold and Peres (1986); Wilkinson (1987) gives

$\overline{|V_{mn}|^{2}}\approx
h^{N-1}\frac{S_{V}(\omega;\overline{E})}{\Sigma(\overline{E})}\,,$ (26)

where $\overline{E}\equiv(E_{m}+E_{n})/2$. Here, $S_{V}(\omega;E)$ is the
power spectrum for the classical observable $V$, and is related to
$S(\omega;E)$ (the power spectrum for $\dot{V}$) via $S=\omega^{2}S_{V}$.
Combining results, we find that (25) converges to the classical results (18)
and (15) as $\hbar\rightarrow 0$. While this analysis is based on a heuristic
model that ignores quantum coherences, it suggests that our classical energy
diffusion picture is relevant for understanding periodically driven quantum
systems; in particular it provides a semiclassical explanation for the
observed exponential-in-frequency suppression of energy absorption Abanin  _et
al._ (2015); Mori  _et al._ (2016); Kuwahara  _et al._ (2016); Else  _et al._
(2017); Abanin  _et al._ (2017); Machado  _et al._ (2019); Tran  _et al._
(2019); Rubio-Abadal  _et al._ (2020); Peng  _et al._ (2021).

##  VI Conclusion

We have analyzed the diffusive energy dynamics of chaotic, ergodic Hamiltonian
systems under rapid periodic driving. Observing that the system’s dynamics are
only weakly affected by very rapid driving, we have established a Fokker-
Planck equation governing the evolution of the system’s energy probability
distribution. Our analysis predicts a generic, long-lived prethermal state,
and for many-body systems our results point to two possible energetic fates:
indefinite energy growth, or relaxation to the infinite-temperature
equilibrium state. In the semiclassical limit, a model of energy absorption
for periodically driven, quantized chaotic systems coincides with our purely
classical energy diffusion description.

A central feature of our Fokker-Planck equation is that the drift and
diffusion coefficients $g_{1}$ and $g_{2}$ are determined by the undriven
dynamics. A similar situation arises in linear response theory (LRT), where
transport coefficients, such as electrical conductivities, in a system subject
to weak time-periodic driving, are expressed in terms of correlation functions
computed in the absence of driving Dorfman (1999). In LRT these results are
obtained perturbatively, through a formal expansion in powers of the driving
strength. It is unclear whether our results can similarly be obtained through
a perturbative expansion. A natural candidate for a small parameter in our
case is the inverse frequency $\omega^{-1}$, but this seems to lead to the
Floquet-Magnus expansion, which as already noted at the end of Sec. IV.2 is
somewhat at odds with our analysis. Both this discrepancy, and the question of
whether our results can be obtained through a formal perturbative expansion,
bear further investigation.

Low-dimensional billiard systems – in which a particle in a cavity alternates
between straight-line motion and specular reflection off the cavity walls –
offer an ideal testing ground for the theory presented in this paper, as
certain billiard shapes are rigorously proven Sinai (1970); Bunimovich (1979);
Wojtkowski (1986) to generate chaotic, ergodic motion. Energy absorption in
driven billiard systems, sometimes known as Fermi acceleration, is a well-
studied phenomenon Fermi (1949); Ulam (1961); Jarzynski (1993); Barnett  _et
al._ (2001); Batistić (2014); Demers and Jarzynski (2015), although much of
the existing literature focuses on the case of slow driving. In a forthcoming
work, we will present numerical evidence for the validity of the Fokker-Planck
equation (19) for a particle in a chaotic billiard subject to a spatially
uniform, rapidly time-periodic force. For this system, the driven and undriven
trajectories of the particle can be computed to machine precision and (19) can
be solved analytically, allowing for an especially precise test of the energy
diffusion description.

Our results may also be tested for previously studied many-body classical
systems, such as a many-body generalization of the kicked rotor model Chirikov
(1979); Ott (2002) that exhibits unbounded energy absorption in a range of
parameter regimes Rajak  _et al._ (2018); Notarnicola  _et al._ (2018); Rajak
_et al._ (2019). Energy absorption has also been studied in the classical
driven Heisenberg spin chain Mori (2018); Khasseh  _et al._ (2019); Howell
_et al._ (2019). For these models and others, we expect our analysis to apply
only if the time-averaged Hamiltonian $H_{0}$ generates chaotic and ergodic
dynamics.

## Acknowledgements

We gratefully acknowledge stimulating discussions with Ed Ott, David
Levermore, and Saar Rahav, and financial support from the DARPA DRINQS program
(D18AC00033).

## Appendix

Here, we obtain an expression for the energy diffusion coefficient $g_{2}$,
given by (15). We then derive the fluctuation-dissipation relation (18).

###  .1 Calculation of $g_{2}$

We begin with relation (14). According to this equation, calculating $g_{2}$
amounts to computing $\mathrm{Var}(E)$, the variance in energy acquired by an
ensemble of trajectories with initial energy $E_{0}$, evolved for a time
$\Delta t$ under the driven Hamiltonian. Specifically, we consider an ensemble
of driven trajectories evolving from microcanonically sampled initial
conditions at $t=0$. Upon integrating (8) along these trajectories, we obtain
(with no approximations so far)

$\mathrm{Var}(E)=\int_{0}^{\Delta t}\int_{0}^{\Delta
t}dt\,dt^{\prime}\cos(\omega t)\cos(\omega
t^{\prime})C_{neq}(t,t^{\prime};E_{0}),$ (A1)

where
$C_{neq}(t,t^{\prime};E_{0})\equiv\langle\dot{V}(z_{t})\dot{V}(z_{t^{\prime}})\rangle-\langle\dot{V}(z_{t})\rangle\langle\dot{V}(z_{t^{\prime}})\rangle$
is a nonequilibrium correlation function and angular brackets
$\langle\cdot\rangle$ denote an ensemble average. In the high-frequency limit
$\omega\to\infty$, as driven trajectories $z_{t}$ approach their undriven
counterparts $z_{t}^{0}$, $C_{neq}(t,t^{\prime};E_{0})$ can be replaced by the
equilibrium correlation function

$C(t^{\prime}-t;E_{0})\equiv\langle\dot{V}(z_{t}^{0})\dot{V}(z_{t^{\prime}}^{0})\rangle-\langle\dot{V}(z_{t}^{0})\rangle\langle\dot{V}(z_{t^{\prime}}^{0})\rangle,$
(A2)

which depends only on the difference $t^{\prime}-t$, due to the time-
translation symmetry of the microcanonical distribution under the undriven
dynamics.

Replacing $C_{neq}(t,t^{\prime};E_{0})$ by $C(t^{\prime}-t;E_{0})$ in (A1),
and using standard manipulations to evaluate the double integral (see, e.g.
Reif (1965)), we arrive at

$\mathrm{Var}(E)\approx\frac{1}{2}S(\omega;E_{0})\Delta t,$ (A3)

where

$S(\omega;E_{0})=\int_{-\infty}^{\infty}dt\,e^{-i\omega t}C(t;E_{0})$ (A4)

is the power spectrum of $\dot{V}(z_{t}^{0})$, which is equal to the Fourier
transform of $C(t;E_{0})$ by the Wiener-Khinchin theorem Kubo  _et al._
(2012). The approximation in (A3) contains correction terms that are sublinear
in $\Delta t$. Comparing (A3) with (14) and relabeling $E_{0}$ as $E$, we
obtain (15), our final expression for $g_{2}$.

###  .2 Calculation of $g_{1}$

We now derive (18), which expresses a fluctuation-dissipation relation between
the drift and diffusion coefficients $g_{1}$ and $g_{2}$. To do so, we first
note that the constant function $\rho(z)=1$ is a stationary solution to the
Liouville equation (2). This reflects the incompressibility of phase space
volume under Hamiltonian dynamics (Liouville’s theorem) Dorfman (1999). Since
$\rho=1$ is stationary under the dynamics in phase space, the corresponding
(unnormalized) distribution in energy space should be stationary under the
Fokker-Planck equation. This energy distribution, obtained by marginalizing
over the constant solution $\rho=1$, is the density of states $\Sigma(E)$ –
see (6). Setting $\eta(E,t)=\Sigma(E)$ as a stationary solution of the Fokker-
Planck equation (11), we have

$0=-\frac{\partial}{\partial
E}\left[g_{1}\Sigma-\frac{1}{2}\frac{\partial}{\partial
E}\left(g_{2}\Sigma\right)\right].$ (A5)

Thus the quantity in square brackets is constant as a function of $E$. We
label this constant by $\alpha$:

$\alpha\equiv g_{1}\Sigma-\frac{1}{2}\frac{\partial}{\partial
E}\left(g_{2}\Sigma\right).$ (A6)

We now aim to show that $\alpha=0$, which then immediately implies the
fluctuation-dissipation relation (18).

To proceed, we first use (A6) to eliminate $g_{1}$ from the Fokker-Planck
equation (11), obtaining

$\frac{\partial\eta}{\partial t}=-\alpha\frac{\partial}{\partial
E}\left(\frac{\eta}{\Sigma}\right)+\frac{1}{2}\frac{\partial}{\partial
E}\left[g_{2}\Sigma\frac{\partial}{\partial
E}\left(\frac{\eta}{\Sigma}\right)\right].$ (A7)

In the main text, in arguing that the system energy evolves diffusively, we
considered trajectories with a common initial energy $E_{0}$, and we arrived
at the hierarchy of timescales (12) required for the validity of the energy
diffusion picture: $T\ll\tau_{C}(E_{0})\ll\Delta t\ll\tau_{E}(\omega,E_{0})$.
Since $\tau_{E}\rightarrow\infty$ as $\omega\rightarrow\infty$, this hierarchy
suggests that for a given, sufficiently large value of $\omega$, there is a
range of energies over which (A7) is valid. This range can be enlarged by
increasing the value of $\omega$, but there might exist no value $\omega^{*}$
such that (A7) is valid over the entire energy axis for all
$\omega>\omega^{*}$. Thus let us fix the value of $\omega$ and let $[a,b]$
denote a finite interval of the energy axis, such that (A7) is valid for
energies $a\leq E\leq b$. The existence of such an interval is sufficient to
establish that $\alpha=0$, as we now show.

Figure 1:  Schematic depiction of phase space. On the left, the initial
distribution is uniform, $\rho(z,0)=c$ (shaded), up to a cutoff energy
$E_{0}$. The figure on the right shows the distribution a short time later,
after evolution under $H(z,t)$; $\rho(z,\delta t)=c$ in the shaded region.
Only in the annular region between the two energy shells $a$ and $b$ (dashed
circles) does $\rho(z,t)$ vary with time for $t\in[0,\delta t]$. Asterisks
depict initial and final conditions for a representative trajectory.

Consider an ensemble of trajectories evolving under $H(z,t)$, from an initial
phase space distribution that is uniform up to a cutoff $E_{0}\in(a,b)$:

$\rho(z,0)=c\,\theta(E_{0}-H_{0}(z)),$ (A8)

where $\theta(\cdot)$ is the unit step function and $c^{-1}=\Omega(E_{0})$ is
the volume of phase space enclosed by the energy shell $E_{0}$ (which was
assumed finite in Section III). The corresponding energy distribution is

$\displaystyle\eta(E,0)$ $\displaystyle=$ $\displaystyle\int
dz\,\delta(E-H_{0})\rho(z,0)$ (A9) $\displaystyle=$ $\displaystyle
c\,\Sigma(E)\theta(E_{0}-E).$

As this ensemble of trajectories evolves in time, the value of the density
$\rho$ at any $(z,t)$ is either $c$ or 0, by Liouville’s theorem. For a
sufficiently short but finite interval $0\leq t\leq\delta t$, $\rho(z,t)$
remains constant outside the region of phase space between the two energy
shells $H_{0}=a$ and $H_{0}=b$ (see Fig. 1), hence

$\eta(E,t)=\begin{cases}c\,\Sigma(E)&\text{if $E\leq a$}\\\ 0&\text{if $E\geq
b$}\end{cases}$ (A10)

We emphasize that (A10) is exact, and a direct consequence of Liouville’s
theorem.

(A10) implies that for $t\in[0,\delta t]$ there is no net flow of probability
into or out of the energy interval $[a,b]$:

$0=\frac{d}{dt}\int_{a}^{b}\eta(E,t)\,dE.$ (A11)

We can use (A7), which is valid in $[a,b]$, along with (A10) to evaluate the
right side of (A11), obtaining

$0=\left[-\alpha\left(\frac{\eta}{\Sigma}\right)+\frac{1}{2}g_{2}\Sigma\frac{\partial}{\partial
E}\left(\frac{\eta}{\Sigma}\right)\right]_{a}^{b}=\alpha c,$ (A12)

hence $\alpha=0$. By (A6), this establishes the fluctuation-dissipation
relation (18).

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