Nonequilibrium Overview

Nonequilibrium many-body quantum mechanics studies initial-value problems in which a prepared state evolves under a specified generator and is interrogated through chosen observables or subsystems. Its central questions are not exhausted by solving the Schrödinger equation:

• Does an observable settle near a reproducible value for most late times?
• Is that value described by a thermal ensemble?
• Which conserved quantities or dynamical constraints preserve memory of the preparation?
• How do particles, energy, correlations, entanglement, and operators spread?
• What changes when the Hamiltonian is driven or the system is coupled to an environment?

These are distinct questions. A system can equilibrate without thermalizing, thermalize locally while its global state remains pure, display a long prethermal plateau before heating, or retain memory because of integrability, localization, constraints, symmetry sectors, or a specially prepared state.

Helpful background. Density operators provide the state and reduced-state language; statistical ensembles explain the equilibrium states against which late-time measurements are compared.

The Minimal Initial-Value Problem

Let the state at time $t_0$ be a density operator

$$\rho_0\geq 0, \qquad \operatorname{Tr}\rho_0=1.$$

For a possibly time-dependent Hamiltonian $H(t)$, the propagator satisfies

$$\begin{aligned} i\hbar \frac{\partial U(t,t_0)}{\partial t} &= H(t)U(t,t_0), \\ U(t_0,t_0) &= I. \end{aligned}$$

The evolved state and the expectation value of an observable $O$ are

$$\begin{aligned} \rho(t) &= U(t,t_0)\rho_0U^\dagger(t,t_0), \\ \langle O\rangle_t &= \operatorname{Tr}\!\left[\rho(t)O\right]. \end{aligned}$$

This formula is exact, but it does not define one unique physical problem until the following data are supplied.

Preparation

State:

• whether $\rho_0$ is pure, mixed, thermal, symmetry broken, spatially inhomogeneous, or highly excited;
• its energy distribution under the evolution Hamiltonian;
• its conserved charges and symmetry sector;
• its correlation length, entanglement structure, and preparation errors.

For a pure state $|\psi_0\rangle$ and a time-independent $H$,

$$\begin{aligned} \bar E &= \langle\psi_0|H|\psi_0\rangle, \\ (\Delta E)^2 &= \langle H^2\rangle_0-\bar E^2. \end{aligned}$$

The mean energy identifies a candidate thermal energy density. The width determines how broadly the preparation samples the spectrum. Neither number alone proves thermalization.

Generator

Specify whether the evolution is:

• isolated and autonomous, with time-independent $H$;
• isolated but driven, with $H(t)$;
• periodically driven, with $H(t+T)=H(t)$;
• open, with a reduced generator that is not simply a commutator with $H$.

Also state locality, interaction range, dimensionality, conserved quantities, disorder, and explicit constraints. These features control propagation, transport, heating, and possible memory.

Probe

Thermal behavior is usually a statement about a restricted class of probes:

• a local observable $O_A$ supported in a finite region $A$;
• a few-body correlation function;
• a coarse-grained density profile;
• a reduced state $\rho_A(t)$;
• a transport current;
• an entanglement or operator-spreading diagnostic.

An observable may equilibrate while another does not. A subsystem may look thermal even though a nonlocal measurement distinguishes the exact pure state from every thermal density operator.

Limits

State the system size $L$, observation window, and order of limits. A finite-dimensional isolated system has quasiperiodic evolution and recurrences. A thermodynamic statement often concerns a window

$$\tau_{\mathrm{local}} \ll t \ll \tau_{\mathrm{rec}}(L),$$

followed by a controlled large-system limit.

Exact Closed-System Dynamics

Consider first a finite-dimensional system with a time-independent Hamiltonian. Write its spectral resolution as

$$H = \sum_E E P_E,$$

where $P_E$ projects onto the full eigenspace at energy $E$. Then

$$\rho(t) = \sum_{E,E'} e^{-i(E-E')t/\hbar} P_E\rho_0P_{E'}.$$

The expectation value is

$$\langle O\rangle_t = \sum_{E,E'} e^{-i(E-E')t/\hbar} \operatorname{Tr} \left( P_E\rho_0P_{E'}O \right).$$

Every time-dependent term is an interference term between distinct energies. Nonequilibrium dynamics is therefore governed by:

1. which energy amplitudes the preparation occupies;
2. the distribution and degeneracies of energy gaps;
3. the matrix elements of the chosen observable.

The density of states alone does not determine the dynamics.

Energy-basis form

For a nondegenerate spectrum with

$$H|n\rangle=E_n|n\rangle,$$

one obtains

$$\langle O\rangle_t = \sum_{m,n} \rho_{mn}(0) e^{-i(E_m-E_n)t/\hbar} O_{nm}.$$

The diagonal terms are constant. The off-diagonal terms oscillate with Bohr frequencies

$$\omega_{mn} = \frac{E_m-E_n}{\hbar}.$$

When many incommensurate frequencies contribute, their sum can become small for most times even though no term has been dissipated. This is dephasing.

The infinite-time average

Define the Cesàro time average

$$\overline{f} \equiv \lim_{T\to\infty} \frac{1}{T} \int_0^T f(t)\,dt,$$

when the limit exists. Since

$$\overline{ e^{-i(E-E')t/\hbar} } = \delta_{E,E'},$$

the time-averaged state is

$$\omega \equiv \overline{\rho(t)} = \sum_E P_E\rho_0P_E.$$

This is the dephased state or diagonal ensemble. The projector form is essential when $H$ has degeneracies: coherences within a degenerate energy eigenspace do not acquire a relative dynamical phase and therefore survive the time average.

For any bounded observable,

$$\overline{\langle O\rangle_t} = \operatorname{Tr}(\omega O).$$

This equality is exact under the stated finite-dimensional assumptions. It is not yet an equilibration theorem and not a thermalization theorem.

What the time average does not establish

Knowing the average does not show that $\langle O\rangle_t$ remains close to it. A signal can have the correct average while making large oscillations forever. The temporal variance

$$\sigma_O^2 \equiv \overline{ \left| \langle O\rangle_t - \operatorname{Tr}(\omega O) \right|^2 }$$

measures the residual fluctuations.

The energy probabilities and their effective number are

$$p_E=\operatorname{Tr}(P_E\rho_0), \qquad d_{\mathrm{eff}}=\frac{1}{\sum_E p_E^2}.$$

For a pure initial state, this equals $1/\operatorname{Tr}(\omega^2)$. For a mixed state with degeneracies, the latter quantity also counts mixing inside energy eigenspaces and need not count distinct occupied energies. For example, $\rho_0=I/d$ and $H=0$ have $d_{\mathrm{eff}}=1$, although $1/\operatorname{Tr}(\omega^2)=d$. The state is already stationary; there are no relative energy phases to dephase.

A large $d_{\mathrm{eff}}$, together with sufficiently nondegenerate energy gaps and a restricted probe, can suppress time-averaged fluctuations. Those assumptions are substantive. Large Hilbert-space dimension by itself is not enough.

A Vocabulary of Late-Time Behavior

Several terms are often used interchangeably in informal prose but answer different questions.

Dephasing

Dephasing is cancellation among oscillatory contributions with different frequencies. It can occur under exact unitary evolution and requires no environment. It explains why a sum of coherent terms may become small for most times, but by itself it does not identify a thermal ensemble.

Relaxation

Relaxation is approach from an initial value toward a late-time value or regime. It is an observational description and should name:

• the observable or reduced state;
• the target value;
• the relaxation timescale;
• whether residual oscillations or algebraic tails remain.

Relaxation need not be monotonic.

Equilibration

An observable equilibrates if it is close to a reference value for most times after a transient. One operational criterion is

$$\begin{aligned} \mathcal B_\epsilon(T) &\equiv \left\{ t\in[0,T]: \right. \\ &\qquad \left. \left| \langle O\rangle_t - \operatorname{Tr}(\omega O) \right| > \epsilon \right\}, \\ \frac{ \mu\!\left(\mathcal B_\epsilon(T)\right) }{T} &\ll 1. \end{aligned}$$

for large $T$, where $\mu$ denotes the duration of the indicated set.

Equivalently, small temporal variance implies that large deviations occupy a small fraction of times. By Markov’s inequality,

$$\overline{ \mathbf 1_{ \{ |\langle O\rangle_t-\langle O\rangle_\omega| > \epsilon \} } } \leq \frac{\sigma_O^2}{\epsilon^2}.$$

This definition permits rare recurrences.

For a subsystem $A$, use the trace distance

$$D\!\left( \rho_A(t),\omega_A \right) = \frac{1}{2} \left\| \rho_A(t)-\omega_A \right\|_1.$$

Small trace distance means that every measurement confined to $A$ has nearly the same statistics in the two states.

Thermalization

Thermalization adds a second comparison:

$$\rho_A(t) \approx \omega_A \approx \rho_A^{\mathrm{th}}$$

for the specified subsystem or observable class and for most relevant late times.

The first approximation is equilibration. The second identifies the equilibrium state with a thermal ensemble constrained by energy and any additional global charges.

Thus:

$$\text{equilibration} \not\Rightarrow \text{thermalization}.$$

An integrable system may equilibrate to a generalized ensemble rather than an ordinary Gibbs state. A localized system may equilibrate imperfectly while retaining local memory. A finite two-level system may fail even to equilibrate.

Stationarity

A density operator is stationary under an autonomous Hamiltonian when

$$[H,\rho_{\mathrm{stat}}]=0.$$

The diagonal ensemble $\omega$ is stationary. The exact state $\rho(t)$ generally does not converge to $\omega$ in trace norm; its observables can nevertheless equilibrate around the values predicted by $\omega$.

Stationarity is therefore a property of a state or effective description, while equilibration is a property of observed time dependence.

Steady states

The phrase steady state has several meanings:

• an autonomous stationary state with time-independent observables;
• a dephased late-time regime in an isolated system;
• a periodic steady regime synchronized with a drive;
• an attracting fixed point of an open-system generator.

Only the last ordinarily involves genuine attraction in state space. In a finite isolated unitary system, distinct initial states cannot contract toward one density operator because unitary evolution preserves trace distance.

Prethermalization

A prethermal state is a long-lived intermediate regime controlled by approximate conservation laws or a separation of timescales. Schematically,

$$\tau_{\mathrm{local}} \ll t \ll \tau_*,$$

with observables close to an ensemble of an effective Hamiltonian or approximately conserved charges. At $t\sim\tau_*$, slower processes may drive the system toward another state.

The prefix “pre” matters: a plateau can be physically useful and parametrically long without being the asymptotic state.

Recurrence

Finite-dimensional isolated quantum systems exhibit arbitrarily close recurrences. The recurrence time often grows extremely rapidly with size, so recurrences can be irrelevant to laboratory times while remaining decisive for mathematical statements about $t\to\infty$ at fixed $L$.

Global Unitarity and Local Thermal Behavior

Suppose the full system begins in a pure state. Unitary evolution preserves purity:

$$\operatorname{Tr}\rho(t)^2 = \operatorname{Tr}\rho_0^2 = 1.$$

It also preserves the von Neumann entropy:

$$S(\rho(t)) = -\operatorname{Tr} \left[ \rho(t)\ln\rho(t) \right] = S(\rho_0).$$

There is no contradiction between these identities and local thermalization. For a bipartition $A\cup\bar A$,

$$\rho_A(t) = \operatorname{Tr}_{\bar A}\rho(t)$$

can become highly mixed because $A$ becomes entangled with $\bar A$. The lost local information remains encoded in nonlocal correlations of the full pure state.

Local indistinguishability

A precise local claim compares reduced states:

$$D \left( \rho_A(t), \rho_A^{\mathrm{th}} \right) \ll 1.$$

This implies

$$\begin{aligned} \Delta\rho_A(t) &\equiv \rho_A(t)-\rho_A^{\mathrm{th}}, \\ \Delta_{O_A}(t) &\equiv \left| \operatorname{Tr} \left[ O_A\Delta\rho_A(t) \right] \right|, \\ \Delta_{O_A}(t) &\leq 2\lVert O_A\rVert_\infty D \left( \rho_A(t), \rho_A^{\mathrm{th}} \right). \end{aligned}$$

Every bounded observable supported in $A$ then has nearly thermal statistics. A sufficiently nonlocal measurement on the full system can still reveal purity, conserved amplitudes, or detailed preparation information.

Entanglement entropy and thermal entropy

For a globally pure state,

$$S_A(t) = S_{\bar A}(t).$$

After a global quench, $S_A(t)$ often grows as entangled excitations cross the boundary of $A$. For a small region in a large thermalizing system, its late-time entanglement entropy can approach the thermodynamic entropy of the corresponding subsystem.

This agreement is conditional and local. The Thermal Entropy versus Entanglement Entropy page develops the exact distinctions, while Volume Laws explains when extensive entanglement scaling is expected.

Coarse graining is part of the statement

Irreversibility is not generated by silently replacing $\rho(t)$ with $\omega$. The physical statement is that a restricted observer, a local subsystem, a finite-resolution instrument, or a coarse-grained description cannot resolve the off-diagonal information for most relevant times.

The exact unitary state remains reversible. The effective description becomes autonomous only after the observable class and approximation are stated.

Which Ensemble Is the Candidate?

The late-time ensemble is constrained by quantities that the evolution preserves.

Energy alone

For an isolated autonomous system,

$$\frac{d}{dt} \langle H\rangle_t = 0.$$

If energy is the only relevant extensive conserved quantity within the chosen symmetry sector and the preparation has a narrow energy-density distribution, a microcanonical state in a narrow shell around $\bar E$ is a natural candidate:

$$\rho_{\mathrm{mc}} = \frac{P_{[\bar E-\delta E,\bar E+\delta E]}} {\operatorname{Tr}P_{[\bar E-\delta E,\bar E+\delta E]}}.$$

For a small subsystem of a large short-range system away from problematic regimes, the microcanonical and canonical reduced states may agree:

$$\rho_A^{\mathrm{mc}} \approx \rho_A^\beta, \qquad \rho^\beta = \frac{e^{-\beta H}}{Z}.$$

The inverse temperature $\beta$ is fixed by matching the energy. A preparation with macroscopically separated energy peaks can instead retain a mixture of local thermal descriptions; its mean energy alone does not select one temperature. This use of ensemble equivalence has assumptions involving subsystem size, interaction range, phase coexistence, and the thermodynamic limit; see Ensemble Equivalence.

Global charges and symmetry sectors

If $H$ commutes with particle number $N$, total magnetization, or another charge $Q_a$, the candidate state must reproduce those values. A generalized grand-canonical form is

$$\rho_{\mathrm{th}} = \frac{1}{Z} \exp \left[ -\beta \left( H-\sum_a\mu_a Q_a \right) \right].$$

It is usually cleaner to resolve exact symmetry sectors before testing spectral statistics or ETH. Mixing sectors can create spurious degeneracies and false indications of nonchaotic behavior.

Extensively many conserved quantities

An integrable model can possess an extensive family of independent or quasilocal charges $Q_j$. An ordinary Gibbs state retains too little information. A generalized Gibbs candidate has the form

$$\rho_{\mathrm{GGE}} = \frac{1}{Z_{\mathrm{GGE}}} \exp \left( -\sum_j\lambda_j Q_j \right),$$

with multipliers chosen to match the initial charge expectations.

This formula is only a template. Which charges are complete, local, or quasilocal is model dependent, and the order of limits can matter.

What the diagonal ensemble remembers

The diagonal ensemble retains all initial populations in the energy basis:

$$\omega = \sum_E P_E\rho_0P_E.$$

A thermal or generalized ensemble is a compressed description of $\omega$ for a specified observable class. Thermalization succeeds when the omitted microscopic information is irrelevant to those observables in the controlled limit.

This hierarchy is useful:

$$\begin{gathered} \text{exact state }\rho(t) \\ \Downarrow\ \text{time averaging or equilibration} \\ \text{diagonal ensemble }\omega \\ \Downarrow\ \text{local ensemble equivalence} \\ \text{thermal or generalized ensemble}. \end{gathered}$$

The second arrow is the nontrivial thermalization step.

Why Generic Systems May Thermalize

The eigenstate thermalization hypothesis, or ETH, proposes that matrix elements of few-body observables in generic nonintegrable many-body energy eigenstates have a smooth thermodynamic structure. At overview depth, its two consequences are:

1. diagonal matrix elements $O_{nn}$ vary smoothly with energy, so a narrow energy distribution produces a thermal-looking mean;
2. off-diagonal matrix elements are sufficiently small and irregular that temporal fluctuations are suppressed in large systems.

This links the diagonal ensemble to a thermal ensemble for local probes. It is not a theorem for every local Hamiltonian, every eigenstate, or every initial condition.

One should distinguish:

• weak ETH: almost all eigenstates in an energy window are thermal for the chosen local probes;
• strong ETH: every eigenstate in the relevant window is thermal;
• dynamical thermalization: the chosen preparation has enough weight on appropriate eigenstates and sufficiently small late-time fluctuations.

Rare nonthermal eigenstates may invalidate strong ETH while leaving most preparations thermal. Conversely, a specially prepared state concentrated on rare eigenstates can fail to thermalize even when most eigenstates satisfy ETH.

Spectral level statistics, eigenvector structure, entanglement, and operator growth provide related diagnostics, but none alone is a definition of thermalization.

How Thermalization Can Fail

Failure is not one phenomenon. Different mechanisms preserve different kinds of information.

Integrability

Integrable systems have many conserved quantities that constrain late-time local observables. They may dephase and equilibrate while disagreeing with an ordinary Gibbs ensemble. A suitable generalized Gibbs ensemble can recover the local stationary data in models where the relevant charge set is known.

The one-dimensional quantum Newton’s-cradle experiment provided an influential example of strikingly weak relaxation in a nearly integrable Bose gas over thousands of collisions.

Localization and disorder

Disorder can inhibit transport and preserve local memory. Interactions can produce slow entanglement growth even when particle or energy transport is strongly suppressed. The stability and dimensional scope of many-body localization in thermodynamic systems remain active research questions; finite-size numerics and finite-time experiments must not be promoted automatically to an asymptotic phase statement.

Kinetic constraints and Hilbert-space fragmentation

Local constraints can split the accessible Hilbert space into dynamically disconnected or weakly connected sectors. A system can then fail to explore the full symmetry sector even without conventional integrability or quenched disorder.

The correct ensemble must be built on the actually accessible sector, not on the entire Hilbert space.

Quantum many-body scars

Some nonintegrable models contain atypical eigenstates embedded among otherwise thermal eigenstates. Initial states with large overlap on this sparse structure can show long-lived revivals and nonthermal dynamics.

Scars are a state-selective exception, not a blanket statement that the full spectrum is nonthermal. Persistent oscillations also require controls against finite-size effects, approximate symmetries, and ordinary weakly coupled modes.

Exact symmetries and degeneracies

If data from different symmetry sectors are mixed, conserved sector weights create apparent memory. Degenerate energies preserve within-eigenspace coherences in the diagonal ensemble. Both effects are exact and should be separated from more exotic mechanisms.

Small effective dimension

A preparation occupying only a few energy eigenstates need not have small temporal fluctuations: there are few oscillatory terms to cancel. This is a warning, not a criterion for failure. A single ETH eigenstate is stationary and can already be locally thermal; a superposition of two such eigenstates can also have tiny fluctuations if the relevant off-diagonal matrix elements are small.

Hydrodynamic slow modes

Conserved densities can relax diffusively, superdiffusively, ballistically, or anomalously. Slow relaxation does not by itself imply failure of ETH. It may be the expected long-wavelength hydrodynamics of a thermalizing system.

Spreading Is Not One Velocity

After a local perturbation or global quench, several fronts can be defined.

Particle, spin, and energy transport

Conserved densities obey continuity equations such as

$$\partial_t n(\mathbf x,t) + \boldsymbol{\nabla}\cdot \mathbf j(\mathbf x,t) = 0.$$

Their long-wavelength dynamics can be ballistic, diffusive, subdiffusive, or superdiffusive depending on conservation laws, dimensionality, interactions, disorder, and integrability.

Transport asks how a conserved quantity moves. It is not identical to correlation or information spreading.

Correlation spreading

For bounded local operators $A_X$ and $B_Y$ separated by distance $d(X,Y)$, a Lieb–Robinson bound for a finite-range lattice Hamiltonian with uniformly bounded interactions has the schematic form

$$\begin{aligned} \ell_{XY}(t) &\equiv d(X,Y)-v_{\mathrm{LR}}|t|, \\ \left\| [A_X(t),B_Y] \right\| &\leq C \exp \left[ -\mu\ell_{XY}(t) \right]. \end{aligned}$$

Outside the effective cone, influence is exponentially suppressed. The velocity $v_{\mathrm{LR}}$ is generally an upper bound, not the observed quasiparticle, correlation, transport, or butterfly velocity. The precise theorem and assumptions live at Lieb–Robinson Bound.

This commutator controls the influence of a local perturbation. Equal-time connected correlations after a global quench are a different quantity: in an initially clustering state, both measured operators can spread toward one another, producing a separation scale of order $2v_{\mathrm{LR}}t$. Pre-existing long-range correlations also need to be included.

Entanglement spreading

Entanglement entropy across a cut can grow even when a conserved density barely moves. In one-dimensional generic systems after a global quench, a roughly linear growth regime is common before finite-size or subsystem-volume saturation:

$$S_A(t) \sim s_{\mathrm{ent}}v_E t.$$

Here $s_{\mathrm{ent}}$ is an entropy per unit length, $v_E$ has units of velocity, and the formula describes one boundary cut. An interval with two independently contributing boundaries can acquire an additional factor of two. This is a scaling template, not a universal law: integrability, localization, conservation laws, long-range interactions, criticality, and inhomogeneity can change the form.

Operator spreading and scrambling

In the Heisenberg picture,

$$W(t) = e^{iHt/\hbar} W e^{-iHt/\hbar}$$

can acquire support on an expanding region. A squared commutator with a distant probe detects when the evolved operator becomes noncommuting there. Scrambling asks a stronger question about the delocalization of initially accessible quantum information.

Fast operator growth does not by itself prove local thermalization, and local thermalization does not require that every scrambling diagnostic has saturated.

A practical ordering

In a given model one may find different characteristic speeds:

$$\begin{gathered} v_{\mathrm{transport}}, \quad v_{\mathrm{corr}}, \quad v_E, \\ v_B, \quad v_{\mathrm{LR}}. \end{gathered}$$

There is no general rule that makes them equal. Report which front was measured and how it was extracted.

Driven Isolated Systems

For a closed system with explicit time dependence, energy need not be conserved. The instantaneous energy obeys

$$\frac{d}{dt} \langle H(t)\rangle = \left\langle \frac{\partial H(t)}{\partial t} \right\rangle.$$

To derive this, write

$$\begin{aligned} E(t) &\equiv \operatorname{Tr} \left[ \rho(t)H(t) \right], \\ \dot E(t) &= \operatorname{Tr} \left[ \dot\rho(t)H(t) \right] + \operatorname{Tr} \left[ \rho(t)\dot H(t) \right]. \end{aligned}$$

The first term vanishes under

$$\dot\rho = -\frac{i}{\hbar} [H,\rho]$$

because the trace of a commutator is zero. The second term is the power injected by the drive.

Periodic driving

For

$$H(t+T)=H(t),$$

one-period evolution defines the Floquet operator

$$\begin{aligned} U_F &\equiv U(T,0), \\ &= \mathcal T \exp \left[ -\frac{i}{\hbar} \int_0^T H(t)\,dt \right]. \end{aligned}$$

Stroboscopic dynamics is

$$\rho(nT) = U_F^n\rho_0(U_F^\dagger)^n.$$

For generic interacting systems with a bounded local Hilbert space and no mechanism preventing absorption, periodic driving can lead toward an effectively infinite-temperature state at asymptotically late times. This statement has important exceptions and assumptions:

• exact or emergent conserved quantities can restrict heating;
• integrability can change the outcome;
• many-body localization has been proposed as a route to avoiding generic heating in suitable regimes, with stability questions requiring care;
• unbounded local spectra require a separate analysis;
• finite-frequency experiments may access only a long preasymptotic window.

High-frequency prethermalization

When the photon energy $\hbar\Omega$, with $\Omega=2\pi/T$, is large compared with a local interaction scale $J$, a suitably truncated effective Hamiltonian $H_*$ can be approximately conserved:

$$U_F \approx \exp \left( -\frac{i}{\hbar} H_*T \right)$$

with errors controlled for local observables or energy density up to an appropriate time. This is not a system-size-independent approximation in the operator norm of the full many-body propagator. In local bounded systems, rigorous results can make the heating scale $\tau_*$ exponentially long in a suitable frequency-to-local-scale ratio:

$$\tau_* \gtrsim \tau_0 \exp \left( c\frac{\hbar\Omega}{J} \right),$$

with model- and theorem-dependent constants and norms.

The system can then relax with respect to $H_*$ before eventually absorbing substantial energy. This is Floquet prethermalization.

Driven steady regimes

An isolated periodically driven system has invertible unitary evolution. It therefore does not possess an attracting density operator in the same sense as a dissipative system. Nevertheless, local observables may:

• dephase to stroboscopically stationary values;
• synchronize into a $T$-periodic regime;
• remain in a long-lived prethermal plateau;
• heat slowly or rapidly.

The word “steady” should identify which of these is meant.

Open and Driven–Dissipative Systems

If the many-body system exchanges information or energy with an environment, its reduced state can obey a nonunitary equation. A Markovian example is

$$\begin{aligned} \dot\rho &= -\frac{i}{\hbar} [H,\rho] + \sum_\alpha \mathcal D[L_\alpha]\rho, \\ \mathcal D[L]\rho &\equiv L\rho L^\dagger - \frac{1}{2} \left\{ L^\dagger L,\rho \right\}. \end{aligned}$$

A steady state satisfies

$$\mathcal L(\rho_{\mathrm{ss}})=0.$$

Unlike closed unitary evolution, a dissipative semigroup can contract distinguishability and attract many initial states toward the same $\rho_{\mathrm{ss}}$.

Bath-induced thermalization, a nonequilibrium current-carrying steady state, and closed-system ETH thermalization are three different mechanisms. They can share a Gibbs-looking density operator in a limit without sharing the same dynamics or assumptions.

The system–bath approximations, complete positivity, detailed balance, and Liouvillian spectrum belong to Open Quantum Systems and the Markovian master-equation chapters.

Relaxation, transport, and recurrence times

Nonequilibrium behavior is rarely characterized by one relaxation time. A useful hierarchy is:

Scale	Typical role	What can set it
$\tau_{\mathrm{micro}}$	local oscillation or collision time	inverse hopping, exchange, interaction, or gap
$\tau_{\mathrm{deph}}$	cancellation of occupied Bohr frequencies	energy-width and gap distribution
$\tau_{\mathrm{local}}$	local equilibration	interactions, quasiparticle motion, operator growth
$\tau_{\mathrm{tr}}(L)$	transport across size $L$	ballistic, diffusive, or anomalous hydrodynamics
$\tau_{\mathrm{ent}}(L_A)$	entanglement across region $A$	entanglement velocity and subsystem geometry
$\tau_*$	prethermal lifetime	weak integrability breaking or high-frequency drive
$\tau_{\mathrm{heat}}$	substantial drive-induced heating	absorption rate and resonances
$\tau_{\mathrm{rec}}(L)$	finite-size recurrence	full many-body gap arithmetic

For a local energy scale $J$, dimensional analysis gives

$$\tau_{\mathrm{micro}} \sim \frac{\hbar}{J}.$$

This does not determine transport or thermalization times. A conserved density can require a size-dependent time

$$\tau_{\mathrm{tr}}(L) \sim \begin{cases} L/v, & \text{ballistic},\\ L^2/D, & \text{diffusive}, \end{cases}$$

where $v$ is a transport velocity and $D$ a diffusion constant.

Plateaus require two inequalities

To identify a prethermal or equilibrium plateau, show both:

$$t \gg \tau_{\mathrm{approach}}$$

and

$$t \ll \tau_{\mathrm{escape}}.$$

A short, nearly flat trace does not establish either inequality. The lower inequality shows that the transient has ended; the upper inequality separates the plateau from recurrence, heating, finite-size saturation, or drift.

Order of limits

The expressions

$$\lim_{t\to\infty} \lim_{L\to\infty} \langle O\rangle_{L,t}$$

and

$$\lim_{L\to\infty} \lim_{t\to\infty} \langle O\rangle_{L,t}$$

need not agree. In the first expression, the thermodynamic limit is taken before the late-time limit. In the second, each finite system is evolved arbitrarily long first, exposing recurrences and exact finite-size structure.

Finite-size scaling at one fixed time is also insufficient. The useful object is often a joint window

$$\tau_{\mathrm{local}}(L) \ll t \ll \tau_{\mathrm{rec}}(L)$$

whose width grows with $L$.

Experimental and numerical windows

Experiments have decoherence, loss, calibration drift, and finite observation time. Numerics have finite size, bond dimension, timestep, truncation, and sampling error. A late-time claim should identify which boundary terminates the trustworthy window.

For tensor-network evolution, entanglement growth can make the reachable time shrink relative to the physical relaxation time. A simulation that ends before the relaxation time cannot decide whether the system thermalizes.

Worked Example: A Two-Level Quench That Does Not Equilibrate

Take

$$H = \frac{\hbar\Omega}{2} \sigma_x, \qquad \rho_0 = |\uparrow_z\rangle \langle\uparrow_z|.$$

The propagator is

$$U(t) = \cos\!\left(\frac{\Omega t}{2}\right)I - i\sin\!\left(\frac{\Omega t}{2}\right)\sigma_x.$$

The $z$-polarization is

$$\langle\sigma_z\rangle_t = \cos(\Omega t).$$

Its infinite-time average is zero:

$$\overline{\langle\sigma_z\rangle_t} = 0.$$

But its temporal variance is

$$\overline{ \left| \langle\sigma_z\rangle_t \right|^2 } = \frac{1}{2}.$$

The signal spends an order-one fraction of time far from its average. The diagonal ensemble predicts the mean but not equilibration.

The initial state has equal populations in the two energy eigenstates, so

$$d_{\mathrm{eff}}=2.$$

This example isolates the role of many frequencies: exact unitary evolution and a well-defined diagonal ensemble do not guarantee small fluctuations.

Worked Example: Dephasing Before Recurrence

Consider a normalized many-frequency signal

$$A_N(t) = \frac{1}{N} \sum_{k=1}^{N} \cos(\omega_k t).$$

Take $\Delta>0$. For a dense set of frequencies distributed uniformly over

$$\omega \in [\omega_0-\Delta,\omega_0+\Delta],$$

the continuum limit gives

$$\begin{aligned} A_\infty(t) &= \frac{1}{2\Delta} \int_{\omega_0-\Delta}^{\omega_0+\Delta} \cos(\omega t)\,d\omega \\ &= \cos(\omega_0t) \frac{\sin(\Delta t)}{\Delta t}. \end{aligned}$$

The envelope decays as $1/t$:

$$\left| A_\infty(t) \right| \leq \frac{1}{|\Delta t|}$$

away from $t=0$. No microscopic oscillator has decayed; the sum has dephased.

At finite $N$, the signal can recur. For the explicit equally spaced frequencies

$$\omega_k=\omega_0+\Delta\frac{2k-1-N}{N}, \qquad k=1,\ldots,N,$$

the sum is

$$A_N(t)=\cos(\omega_0t) \frac{\sin(\Delta t)}{N\sin(\Delta t/N)},$$

with removable singularities evaluated by continuity. For $N=9$ and $\omega_0=4\Delta$, all nine phases align again at $t=9\pi/\Delta$, where $A_9=1$, while $A_\infty=0$. The first dephasing time remains of order $1/\Delta$ as the recurrence is pushed to later times. This explicit sequence illustrates why taking the continuum limit first can remove recurrences.

The worked frequency-sum example with $\omega_0=4\Delta$. The solid curve is the exact nine-frequency sum; the dashed curve is its continuum limit. Their early-time agreement does not persist near the finite-system recurrence at $\Delta t=9\pi$. No individual oscillator loses coherence.

Worked Example: Global Purity and Local Entropy

Begin with two qubits in $|00\rangle$ and evolve under

$$H = \hbar g \left( |00\rangle\langle11| + |11\rangle\langle00| \right).$$

The state is

$$|\psi(t)\rangle = \cos(gt)|00\rangle - i\sin(gt)|11\rangle.$$

It remains globally pure:

$$S \left( |\psi(t)\rangle\langle\psi(t)| \right) = 0.$$

The reduced state of the first qubit is

$$\rho_1(t) = \begin{pmatrix} \cos^2(gt) & 0\\ 0 & \sin^2(gt) \end{pmatrix}.$$

Its entropy is

$$S_1(t) = h_2 \left( \sin^2(gt) \right),$$

where

$$h_2(p) = -p\ln p - (1-p)\ln(1-p).$$

At $gt=\pi/4$, the reduced state is maximally mixed even though the full state is pure. In a many-body system, analogous entanglement with a large complement can make a small region locally thermal-looking for long times.

This two-qubit model demonstrates the mechanism but not thermodynamic irreversibility: it has exact revivals on the timescale $1/g$.

Worked Example: A Conserved Charge Rejects a Candidate Ensemble

Suppose

$$[H,Q]=0.$$

Then

$$\langle Q\rangle_t = \langle Q\rangle_0$$

for all $t$. Consider a candidate state $\rho_{\mathrm{cand}}$ with

$$\operatorname{Tr} \left( \rho_{\mathrm{cand}}Q \right) \neq \langle Q\rangle_0.$$

The candidate cannot describe thermalization for an observable class containing $Q$. More quantitatively,

$$\begin{aligned} \langle Q\rangle_{\mathrm{cand}} &\equiv \operatorname{Tr} \left( \rho_{\mathrm{cand}}Q \right), \\ \Delta_Q &\equiv \left| \langle Q\rangle_0 - \langle Q\rangle_{\mathrm{cand}} \right|, \\ \Delta_Q &\leq 2\lVert Q\rVert_\infty D \left( \rho(t), \rho_{\mathrm{cand}} \right). \end{aligned}$$

A nonzero charge mismatch therefore lower-bounds the trace distance:

$$D \left( \rho(t), \rho_{\mathrm{cand}} \right) \geq \frac{ \left| \langle Q\rangle_0 - \langle Q\rangle_{\mathrm{cand}} \right| }{ 2\lVert Q\rVert_\infty }.$$

In large systems one normally applies this logic to charge densities or finite regions, since the norm of an extensive $Q$ grows with system size.

Worked Example: Work Done by a Parameter Ramp

Let

$$H(t) = H_0+\lambda(t)V.$$

The power is

$$\frac{dE}{dt} = \dot\lambda(t) \langle V\rangle_t.$$

The net energy change is

$$\Delta E = \int_{t_i}^{t_f} \dot\lambda(t) \langle V\rangle_t\,dt.$$

The result depends on the evolving expectation value, not only on the endpoints of $\lambda$. Two ramps with the same endpoints can inject different energies because the state follows different trajectories.

For an instantaneous quench, the state is continuous across the quench while the Hamiltonian changes. The post-quench conserved energy is

$$E_f = \operatorname{Tr} \left( \rho_0 H_f \right),$$

not generally the ground-state energy of $H_f$ and not the expectation value of the pre-quench Hamiltonian.

Comparing late-time data with an ensemble

A thermalization comparison uses the same energy and conserved charges for every tested observable. Determine one temperature and the necessary chemical potentials, then compare several local quantities without refitting those parameters. Agreement with one conserved density is particularly weak: that agreement can be imposed by construction.

For an observable, a useful error is

$$\delta_O(t)= \frac{|\langle O\rangle_t-\langle O\rangle_{\mathrm{ens}}|} {O_{\mathrm{scale}}},$$

where $O_{\mathrm{scale}}>0$ is chosen independently of the discrepancy. For reduced states, trace distance bounds all local measurement differences. With natural logarithms, quantum Pinsker gives

$$D(\rho,\sigma)\leq \sqrt{\tfrac12 D_{\mathrm{rel}}(\rho\Vert\sigma)}.$$

Relative entropy is finite only when $\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma$; otherwise this bound is uninformative.

Vary the observation window, system size, and numerical accuracy separately. A stable time average with large fluctuations shows neither equilibration nor thermalization. A plateau matching Gibbs predictions supports thermalization for the tested probes; the mechanism still needs its own check. ETH concerns eigenstate matrix elements, integrability concerns conserved charges, and prethermalization concerns approximate conservation with a parametrically long escape time. The relaxation and thermalization page develops these comparisons quantitatively.

Exercises

Exercise 1: Degenerate diagonal ensemble

Let

$$\begin{aligned} H &= E_0 \left( |1\rangle\langle1| + |2\rangle\langle2| \right) + E_3|3\rangle\langle3|, \\ E_3 &\neq E_0. \end{aligned}$$

The initial state is

$$|\psi_0\rangle = \frac{1}{\sqrt3} \left( |1\rangle+|2\rangle+|3\rangle \right).$$

Find the time-averaged state. Which coherence survives?

Solution

The spectral projectors are

$$P_0 = |1\rangle\langle1| + |2\rangle\langle2|, \qquad P_3 = |3\rangle\langle3|.$$

The time-averaged state is

$$\omega = P_0\rho_0P_0 + P_3\rho_0P_3.$$

Expanding,

$$\omega = \frac13 \left( |1\rangle\langle1| + |2\rangle\langle2| + |1\rangle\langle2| + |2\rangle\langle1| \right) + \frac13 |3\rangle\langle3|.$$

The coherence between $|1\rangle$ and $|2\rangle$ survives because those vectors have the same energy. Coherences connecting either of them to $|3\rangle$ oscillate with nonzero frequency and vanish in the time average.

Exercise 2: Time average is not equilibration

For the two-level quench with

$$\langle\sigma_z\rangle_t = \cos(\Omega t),$$

find the fraction of one period for which

$$\left| \langle\sigma_z\rangle_t \right| > \frac12.$$

Interpret the result.

Solution

Set $\theta=\Omega t$. Over one period $0\leq\theta<2\pi$, the condition is

$$|\cos\theta|>\frac12.$$

It holds on intervals of total length

$$\frac{4\pi}{3}.$$

The fraction is therefore

$$\frac{4\pi/3}{2\pi} = \frac23.$$

Although the infinite-time average is zero, the observable differs from that value by more than $1/2$ for two thirds of all times. The system does not equilibrate relative to this tolerance and observable.

Exercise 3: Width controls dephasing

For

$$A_\infty(t) = \cos(\omega_0t) \frac{\sin(\Delta t)}{\Delta t},$$

estimate the dephasing time and explain what happens as $\Delta\to0$.

Solution

The envelope first becomes small once

$$\Delta t \gtrsim 1.$$

Thus

$$\tau_{\mathrm{deph}} \sim \frac{1}{\Delta}.$$

As $\Delta\to0$, the frequency distribution collapses to one frequency. Using

$$\lim_{\Delta\to0} \frac{\sin(\Delta t)}{\Delta t} = 1,$$

one obtains

$$A_\infty(t) \to \cos(\omega_0t).$$

The dephasing time diverges and persistent oscillations return. A broad distribution of occupied frequencies, not merely long evolution, produces the decay.

Exercise 4: Local entropy from a pure state

For

$$|\psi(t)\rangle = \cos(gt)|00\rangle - i\sin(gt)|11\rangle,$$

find the times at which the first qubit is maximally mixed. What are the global and local entropies then?

Solution

The reduced state is maximally mixed when

$$\cos^2(gt) = \sin^2(gt) = \frac12.$$

Hence

$$gt = \frac{\pi}{4} + \frac{k\pi}{2}, \qquad k\in\mathbb Z.$$

At those times,

$$\rho_1 = \frac{I}{2}, \qquad S(\rho_1)=\ln2.$$

The full state remains pure, so

$$S \left( |\psi\rangle\langle\psi| \right) = 0.$$

The local entropy is entanglement entropy, not an increase of the global fine-grained entropy.

Exercise 5: Ensemble rejection by charge mismatch

Let $Q$ be a conserved operator with

$$\lVert Q\rVert_\infty=5.$$

The initial state has $\langle Q\rangle_0=2$, while a proposed ensemble has $\langle Q\rangle_{\mathrm{ens}}=-1$. Give a lower bound on the trace distance between the exact state and that ensemble.

Solution

The expectation mismatch is

$$\left| 2-(-1) \right| = 3.$$

Using

$$D(\rho,\sigma) \geq \frac{ |\langle Q\rangle_\rho-\langle Q\rangle_\sigma| }{ 2\lVert Q\rVert_\infty },$$

one finds

$$D \left( \rho(t),\rho_{\mathrm{ens}} \right) \geq \frac{3}{10}.$$

The proposed ensemble is operationally distinguishable by measuring $Q$ and cannot describe thermalization for an observable class that includes this charge.

Exercise 6: Energy injection under a cyclic protocol

For

$$H(t) = H_0+\lambda(t)V,$$

suppose $\lambda(t_i)=\lambda(t_f)=0$. Must $\Delta E$ vanish? Explain from the work formula.

Solution

No. The energy change is

$$\Delta E = \int_{t_i}^{t_f} \dot\lambda(t) \langle V\rangle_t\,dt.$$

Although the parameter returns to its initial value, $\langle V\rangle_t$ depends on the evolving state and generally has a phase lag relative to the drive. The integral can therefore be nonzero.

Only under additional conditions, such as a perfectly adiabatic cyclic protocol that returns the state to the same energy eigenspace up to phase, is zero net work guaranteed. Generic cyclic driving can heat the system.

Exercise 7: Distinguish three late-time claims

A finite isolated chain shows small fluctuations of a local density around the value predicted by its diagonal ensemble. That value disagrees with a Gibbs ensemble but agrees with a GGE. Classify the behavior.

Solution

The local density equilibrates because it remains close to its diagonal-ensemble value for most late times.

It does not thermalize to the ordinary Gibbs ensemble because that ensemble gives the wrong late-time value.

It is consistent with generalized thermalization or equilibration to a GGE, provided the proposed conserved charges are specified and the agreement extends beyond one fitted observable. The observation alone does not prove integrability; conserved-charge structure and broader tests are still required.

Exercise 8: Design a thermalization test

You simulate a nonintegrable spin chain after a product-state quench. List a minimal set of controls needed before claiming local thermalization.

Solution

A credible minimum is:

1. Resolve the exact symmetry sector and record the initial energy density and variance.
2. Construct a microcanonical or canonical comparison state with the same energy and charges.
3. Compare several local observables and connected correlations using one common temperature.
4. Show approach to a plateau and quantify temporal fluctuations over several windows.
5. Repeat for increasing system sizes and vary boundary conditions where practical.
6. Converge numerical controls such as timestep, Krylov tolerance, or tensor-network bond dimension.
7. Check that the observed time window lies before finite-size recurrence or truncation failure.
8. Separate the thermalization observation from any stronger ETH claim; the latter requires eigenstate-matrix-element tests in resolved sectors.

No single item proves the result, but together they address preparation, ensemble, observable class, time window, size scaling, and numerical reliability.

References

• D. A. Abanin, W. De Roeck, W. W. Ho, and F. Huveneers, “Effective Hamiltonians, prethermalization, and slow energy absorption in periodically driven many-body systems,” Physical Review B 95, 014112, 2017, doi:10.1103/PhysRevB.95.014112.
• D. M. Basko, I. L. Aleiner, and B. L. Altshuler, “Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states,” Annals of Physics 321, 1126–1205, 2006, doi:10.1016/j.aop.2005.11.014.
• H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, “Probing many-body dynamics on a 51-atom quantum simulator,” Nature 551, 579–584, 2017, doi:10.1038/nature24622.
• P. Calabrese and J. Cardy, “Time dependence of correlation functions following a quantum quench,” Physical Review Letters 96, 136801, 2006, doi:10.1103/PhysRevLett.96.136801.
• M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P. Schauß, T. Fukuhara, C. Gross, I. Bloch, C. Kollath, and S. Kuhr, “Light-cone-like spreading of correlations in a quantum many-body system,” Nature 481, 484–487, 2012, doi:10.1038/nature10748.
• L. D’Alessio and M. Rigol, “Long-time behavior of isolated periodically driven interacting lattice systems,” Physical Review X 4, 041048, 2014, doi:10.1103/PhysRevX.4.041048.
• L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, “From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics,” Advances in Physics 65, 239–362, 2016, doi:10.1080/00018732.2016.1198134.
• J. M. Deutsch, “Quantum statistical mechanics in a closed system,” Physical Review A 43, 2046–2049, 1991, doi:10.1103/PhysRevA.43.2046.
• D. V. Else, B. Bauer, and C. Nayak, “Prethermal phases of matter protected by time-translation symmetry,” Physical Review X 7, 011026, 2017, doi:10.1103/PhysRevX.7.011026.
• A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, “Quantum thermalization through entanglement in an isolated many-body system,” Science 353, 794–800, 2016, doi:10.1126/science.aaf6725.
• H. Kim and D. A. Huse, “Ballistic spreading of entanglement in a diffusive nonintegrable system,” Physical Review Letters 111, 127205, 2013, doi:10.1103/PhysRevLett.111.127205.
• T. Kinoshita, T. Wenger, and D. S. Weiss, “A quantum Newton’s cradle,” Nature 440, 900–903, 2006, doi:10.1038/nature04693.
• A. Lazarides, A. Das, and R. Moessner, “Equilibrium states of generic quantum systems subject to periodic driving,” Physical Review E 90, 012110, 2014, doi:10.1103/PhysRevE.90.012110.
• E. H. Lieb and D. W. Robinson, “The finite group velocity of quantum spin systems,” Communications in Mathematical Physics 28, 251–257, 1972, doi:10.1007/BF01645779.
• N. Linden, S. Popescu, A. J. Short, and A. Winter, “Quantum mechanical evolution towards thermal equilibrium,” Physical Review E 79, 061103, 2009, doi:10.1103/PhysRevE.79.061103.
• T. Mori, T. Kuwahara, and K. Saito, “Rigorous bound on energy absorption and generic relaxation in periodically driven quantum systems,” Physical Review Letters 116, 120401, 2016, doi:10.1103/PhysRevLett.116.120401.
• A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, “Colloquium: nonequilibrium dynamics of closed interacting quantum systems,” Reviews of Modern Physics 83, 863–883, 2011, doi:10.1103/RevModPhys.83.863.
• P. Reimann, “Foundation of statistical mechanics under experimentally realistic conditions,” Physical Review Letters 101, 190403, 2008, doi:10.1103/PhysRevLett.101.190403.
• M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, “Relaxation in a completely integrable many-body quantum system: an ab initio study of the dynamics of the highly excited states of 1D lattice hard-core bosons,” Physical Review Letters 98, 050405, 2007, doi:10.1103/PhysRevLett.98.050405.
• M. Rigol, V. Dunjko, and M. Olshanii, “Thermalization and its mechanism for generic isolated quantum systems,” Nature 452, 854–858, 2008, doi:10.1038/nature06838.
• A. J. Short and T. C. Farrelly, “Quantum equilibration in finite time,” New Journal of Physics 14, 013063, 2012, doi:10.1088/1367-2630/14/1/013063.
• M. Srednicki, “Chaos and quantum thermalization,” Physical Review E 50, 888–901, 1994, doi:10.1103/PhysRevE.50.888.
• C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, “Weak ergodicity breaking from quantum many-body scars,” Nature Physics 14, 745–749, 2018, doi:10.1038/s41567-018-0137-5.
• J. von Neumann, “Proof of the ergodic theorem and the $H$-theorem in quantum mechanics,” European Physical Journal H 35, 201–237, 2010 translation of the 1929 article, doi:10.1140/epjh/e2010-00008-5.

Further Connections

• Thermal Density Operators defines Gibbs states, temperature, and energy matching without assuming a dynamical route to equilibrium.
• Thermodynamic Limit develops size scaling, boundary effects, and noncommuting limits.
• Time-Dependent Correlations supplies the correlators used to track propagation, stationarity, and spectral relaxation.
• Real-Time Thermal Dynamics Preview explains why nonequilibrium initial data require more than equilibrium imaginary-time correlators.
• Schwinger–Keldysh Bridge carries that initial-value logic into doubled fields, influence actions, and nonequilibrium QFT.
• Many-Body Entanglement Overview organizes subsystem entropies and mixed-state measures.
• Operator Entanglement and Scrambling Preview separates operator support, operator entanglement, OTOCs, and recoverability.
• Closed versus Open Quantum Systems distinguishes unitary joint evolution, reduced dynamics, and dynamics conditioned on a measurement record.