Relaxation and Thermalization

An isolated quantum system evolves unitarily. Its exact state does not forget its initial condition, its von Neumann entropy does not increase, and a finite system generally recurs. Yet local observables can settle near reproducible values, small subsystems can become nearly indistinguishable from thermal states, and equilibrium statistical mechanics can predict late-time measurements with striking accuracy.

Thermalization of local probes involves two questions: do their fluctuations become small for most times, and do their mean values agree with a thermal ensemble? Either comparison can fail.

For a subsystem $A$, the central error decomposition is

$$D\left( \rho_A(t), \rho_A^{\mathrm{th}} \right) \le{} D\left( \rho_A(t), \omega_A \right) + D\left( \omega_A, \rho_A^{\mathrm{th}} \right),$$

where $\omega$ is the dephased state and

$$D(\rho,\sigma) = \frac{1}{2} \|\rho-\sigma\|_1$$

is trace distance. The first term is the dynamical equilibration error. The second is the ensemble-identification error. Calling the first term small does not make the second small.

Helpful background. Density operators and thermal states explain the comparison ensembles; the nonequilibrium overview introduces unitary dephasing and the distinction between a global state and its local measurements.

Unitary evolution and energy dephasing

Let $H$ be time independent after the preparation. In finite dimension, write its spectral resolution using distinct energies:

$$H = \sum_E E P_E.$$

The state evolves as

$$\rho(t) = e^{-iHt/\hbar} \rho_0 e^{iHt/\hbar}.$$

For a bounded observable $O$,

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

The time dependence is carried entirely by unequal-energy phases. The exact state retains them; a restricted observable may cease to resolve their coherent sum.

The dephased state

The infinite-time average is

$$\begin{aligned} \omega &\equiv \overline{\rho(t)}, \\ \omega &= \lim_{T\to\infty} \frac{1}{T} \int_0^T \rho(t)\,dt, \\ \omega &= \sum_E P_E\rho_0P_E. \end{aligned}$$

Therefore

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

The projector form matters. Coherences inside a degenerate energy eigenspace have no relative phase and survive. “Delete every off-diagonal matrix element in a chosen eigenbasis” is basis dependent and generally wrong in a degenerate subspace.

The dephased state is also called the diagonal ensemble when a nondegenerate energy basis is understood. It stores every populated final-energy sector and, within degeneracies, all stationary coherence. It is not generally microcanonical or canonical, nor does it maximize entropy subject to energy alone. With nondegenerate energies it does maximize entropy when every energy population is fixed, a much stronger set of constraints.

Time average is not convergence

A time average can exist while the instantaneous signal keeps making large excursions. Define

$$\delta_O(t) \equiv \langle O(t)\rangle - \operatorname{Tr}(\omega O)$$

and the infinite-time variance

$$\sigma_O^2 \equiv \overline{ |\delta_O(t)|^2 }.$$

Small $\sigma_O$ means the observable remains near its time average for most times. The equality

$$\overline{\delta_O(t)}=0$$

alone says nothing about fluctuation size.

Relaxation, Equilibration, and Thermalization

Relaxation

Relaxation describes an observable or reduced state approaching a late-time regime after a transient. A complete statement names:

• the observable or subsystem;
• the proposed late-time value or manifold;
• the time window;
• the relaxation scale;
• residual oscillations or long-time tails;
• finite-size and boundary effects.

Relaxation need not be monotonic or exponential. Damped oscillations, algebraic decay, stretched exponentials, plateaus, and multistage relaxation are all possible.

Equilibration

An observable equilibrates when it is close to a reference value for most late times. For tolerance $\epsilon$, define the bad-time set

$$\mathcal B_\epsilon(T) = \left\{ t\in[0,T]: |\delta_O(t)|>\epsilon \right\}.$$

An operational criterion is

$$\frac{ \mu\left[\mathcal B_\epsilon(T)\right] }{T} \ll1$$

for a declared large-$T$ window, where $\mu$ is duration. Rare recurrences are allowed.

For a subsystem,

$$\rho_A(t) = \operatorname{Tr}_{\bar A}\rho(t), \qquad \omega_A = \operatorname{Tr}_{\bar A}\omega,$$

and equilibration can be tested through

$$D\left( \rho_A(t),\omega_A \right) \ll1$$

for most times.

Thermalization

Thermalization adds ensemble identification:

$$\omega_A \approx \rho_A^{\mathrm{th}},$$

or, for a restricted observable class $\mathcal C$,

$$\operatorname{Tr}(\omega O) \approx \operatorname{Tr}(\rho^{\mathrm{th}}O), \qquad O\in\mathcal C.$$

A thermalization claim must therefore specify:

1. the candidate ensemble;
2. the conserved constraints used to fix it;
3. the observable class or subsystem geometry;
4. the preparation class;
5. the time and size limits;
6. the numerical or experimental tolerance.

An integrable system may equilibrate but require a generalized ensemble. A localized system may retain local memory. A small system may not equilibrate at all. A thermalizing Hamiltonian can still have exceptional initial states.

Initial-state independence

A stronger form of thermalization compares several preparations. Let $\rho_0$ and $\rho_0'$ have the same energy density and the same relevant conserved densities. Local thermalization predicts, within its domain,

$$\operatorname{Tr} \left[ \omega(\rho_0)O_A \right] - \operatorname{Tr} \left[ \omega(\rho_0')O_A \right] \longrightarrow0$$

as the thermodynamic limit is taken with fixed local region $A$.

Testing one initial state cannot establish this independence. It may only show that one trajectory agrees with one ensemble.

Why the Global State Does Not Become Gibbsian

Suppose $\rho^{\mathrm{th}}$ is stationary:

$$[H,\rho^{\mathrm{th}}]=0.$$

Unitary invariance of trace distance gives

$$\begin{aligned} D\left( \rho(t),\rho^{\mathrm{th}} \right) &= D\left( e^{-iHt/\hbar}\rho_0e^{iHt/\hbar}, \rho^{\mathrm{th}} \right) \\ &= D\left( \rho_0,\rho^{\mathrm{th}} \right). \end{aligned}$$

Thus the global state cannot approach a stationary thermal state in trace norm unless it was already equally close. If $\rho_0$ is pure, then

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

for all time, whereas a finite-temperature Gibbs state is mixed.

Partial trace changes the conclusion because it discards access to correlations with the complement:

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

The local distance can become small even while the global distance remains constant. Information about the preparation is redistributed into nonlocal correlations rather than destroyed.

Global and subsystem entropy

Global unitary evolution preserves von Neumann entropy:

$$S(\rho(t)) = S(\rho_0).$$

For a globally pure state,

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

can nevertheless grow as $A$ entangles with its complement. In a thermalizing regime and for a sufficiently small subsystem, the leading late-time entanglement entropy can agree with thermodynamic entropy at the same energy density.

This is a local statement with a subsystem-size limit. Thermal Entropy versus Entanglement Entropy owns the exact distinctions, and Volume Laws owns the scaling regimes.

Effective Dimension and Equilibration Bounds

How many energies participate?

For any initial density operator, define the weight in each distinct energy eigenspace:

$$p_E = \operatorname{Tr}(P_E\rho_0).$$

The effective dimension is

$$d_{\mathrm{eff}}\equiv\frac{1}{\sum_Ep_E^2}.$$

For a pure initial state, $P_E\rho_0P_E$ has rank at most one in each energy block, so $d_{\mathrm{eff}}=1/\operatorname{Tr}(\omega^2)$. That equality can fail for mixed states with degenerate energies: mixing within a stationary block increases inverse purity without introducing any new dynamical frequency.

It obeys

$$1 \le d_{\mathrm{eff}} \le N_{\mathrm{occ}},$$

where $N_{\mathrm{occ}}$ is the number of occupied distinct energies. Equal weight on $N$ energies gives $d_{\mathrm{eff}}=N$; concentration on one energy gives $d_{\mathrm{eff}}=1$.

A large Hilbert-space dimension does not imply a large effective dimension. The preparation may occupy only a special low-dimensional sector.

Energy-gap degeneracy

Dephasing is controlled by energy differences. Define the maximal degeneracy of a nonzero gap,

$$g \equiv \max_{\nu\ne0} \#\left\{ (E,E'): E-E'=\nu \right\}.$$

For finite-dimensional autonomous evolution, the infinite-time fluctuation bound of Short and Farrelly gives

$$\sigma_O^2 \le \frac{ g\, \|O\|_\infty^2 }{ d_{\mathrm{eff}} }.$$

When nonzero gaps are nondegenerate, $g=1$. The result says that a preparation spread over many energies and a spectrum without large resonant gap families suppress time-averaged fluctuations of bounded observables.

The hypotheses are real:

• gap multiplicities are counted in the actual invariant space occupied by the preparation; restricting to a single symmetry sector is justified only when the state is supported there;
• integrable and free systems can have extensive resonances;
• an observable can have special matrix elements;
• $d_{\mathrm{eff}}$ depends on the preparation;
• the bound may be loose by many orders of magnitude.

Why energy gaps enter the bound

For a pure state, choose one occupied vector per distinct energy and write $\langle O(t)\rangle-\langle O\rangle_\omega$ as a sum of terms $c_m^*c_nO_{mn}e^{i(E_m-E_n)t/\hbar}$ with $m\ne n$. Time averaging its squared magnitude keeps terms with equal gaps. Cauchy–Schwarz bounds the square of each group by its multiplicity $g$ times the sum of individual squared magnitudes:

$$\sigma_O^2\leq g\sum_{m\ne n}p_mp_n|O_{mn}|^2.$$

For Hermitian $O$, use $p_mp_n\leq(p_m^2+p_n^2)/2$ and $\sum_n|O_{mn}|^2\leq\|O\|_\infty^2$. This gives the stated bound. A purification of $\rho_0$, evolved by $H\otimes I$, extends it to mixed states while preserving the distinct energies and their probabilities. Large $d_{\mathrm{eff}}$ is sufficient in this bound, not necessary: an already stationary state has zero fluctuations even if $d_{\mathrm{eff}}=1$.

Fraction of bad times

At every finite averaging time $T$, Markov’s inequality gives the exact statement

$$\frac{\mu[\mathcal B_\epsilon(T)]}{T} \leq\frac{1}{T\epsilon^2}\int_0^T|\delta_O(t)|^2\,dt.$$

Taking a long-time upper limit then gives

$$\limsup_{T\to\infty} \frac{\mu[\mathcal B_\epsilon(T)]}{T} \leq\frac{\sigma_O^2}{\epsilon^2} \leq\frac{g\|O\|_\infty^2}{\epsilon^2d_{\mathrm{eff}}}.$$

An infinite-time variance cannot be substituted for the finite-window integral without controlling the approach to that limit. This is a “most times” statement, allowing rare recurrences.

What these bounds do not provide

Infinite-time fluctuation bounds do not generally determine:

• when equilibration first occurs;
• whether the initial transient is short;
• whether a long prethermal plateau precedes the final regime;
• whether the equilibrium value is thermal;
• whether conserved densities relax diffusively or anomalously;
• whether the same result holds uniformly for all local observables.

Finite-time results require information about how densely energy gaps cluster within a frequency resolution of order $1/T$. A spectrum can have small infinite-time variance and still equilibrate on an impractically long timescale.

Subsystems and Local Observables

Trace distance is an all-measurements criterion

For states on subsystem $A$,

$$D(\rho_A,\sigma_A) = \max_{0\le M\le I} \left| \operatorname{Tr} \left[ M(\rho_A-\sigma_A) \right] \right|.$$

Thus small trace distance means that no measurement confined to $A$ distinguishes the states with large probability advantage.

For any bounded $O_A$,

$$\left| \operatorname{Tr} \left[ O_A(\rho_A-\sigma_A) \right] \right| \le 2 \|O_A\|_\infty D(\rho_A,\sigma_A).$$

The converse requires an informationally complete set of probes. Agreement of one observable does not imply small trace distance.

Subsystem equilibration bounds

Choose a complete operator basis on a subsystem of Hilbert-space dimension $d_A$. Applying observable equilibration bounds to every basis element and combining them with norm inequalities yields the infinite-time bound

$$\overline{ D\left( \rho_A(t),\omega_A \right) } \leq \frac{ d_A\sqrt g }{ 2\sqrt{d_{\mathrm{eff}}} },$$

using the energy-probability definition of $d_{\mathrm{eff}}$ above. Other bounds based on the environment can improve this estimate, but no change of definition is needed for this inequality.

The qualitative condition is robust:

$$d_{\mathrm{eff}} \gg d_A^2g.$$

A fixed small region can equilibrate in a large system even when a region occupying a finite fraction of the system does not satisfy the same bound.

Fixed region versus finite fraction

Local thermalization usually means

$$V\to\infty \quad\text{with}\quad |A|\ \text{fixed},$$

or at least

$$\frac{|A|}{V}\to0.$$

If $A$ contains half the system, global conservation and purity impose order-one constraints. Canonical and microcanonical reduced states can also differ at finite subsystem fraction. The subsystem scaling must be stated explicitly.

Selecting the Thermal Candidate

The candidate ensemble must respect conserved information relevant to the measured region. Exact sector restrictions and thermodynamic charge densities play different roles.

Step 1: Restrict exact sectors

If

$$[H,Q]=0$$

and the preparation lies in a definite sector $q$, the evolution remains there. A thermal comparison should use a sector-projected state such as

$$\rho_{\beta,q} = \frac{ P_qe^{-\beta H} }{ \operatorname{Tr} \left( P_qe^{-\beta H} \right) }.$$

Parity, particle number, total magnetization, crystal momentum, and superselection charges are common examples. Mixing disconnected sectors can produce wrong finite-size predictions and misleading level statistics.

Step 2: Match energy

For an isolated autonomous system,

$$\operatorname{Tr}[\rho(t)H] = \bar E$$

is constant. If the preparation has a narrow energy-density distribution, a microcanonical shell is a natural first candidate:

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

The shell width must be:

• broad enough to contain many levels;
• narrow on thermodynamic energy scales;
• compatible with exact sectors;
• tested for finite-size stability.

Matching only the mean energy is insufficient for a preparation with macroscopically separated energy peaks: it can approach a mixture of local thermal states at different temperatures. Choose and vary the shell width independently of the observable being tested.

Step 3: Use ensemble equivalence locally

For a large short-range system away from ensemble-inequivalent regimes, the reduced microcanonical and canonical states can agree on a small region:

$$\left( \rho_{\mathrm{mc}} \right)_A \approx \left( \rho_\beta \right)_A,$$

where $\beta$ is fixed by

$$\operatorname{Tr} \left( \rho_\beta H \right) = \bar E.$$

This does not mean the global density operators are equal at finite size. Moreover, the comparison is with $\operatorname{Tr}_{\bar A}(e^{-\beta H}/Z)$, not automatically $e^{-\beta H_A}/Z_A$: interactions across the boundary can modify the reduced Gibbs state. Ensemble Equivalence owns the locality, convexity, and thermodynamic-limit assumptions.

Step 4: Retain additional extensive charges

If mutually commuting extensive charges $Q_a$ remain relevant, a generalized Gibbs form is

$$\rho_{\mathrm{G}} = \frac{1}{Z} \exp \left[ -\beta H - \sum_a \lambda_aQ_a \right].$$

The multipliers are fixed by

$$\operatorname{Tr} (\rho_{\mathrm G}Q_a) = \operatorname{Tr} (\rho_0Q_a).$$

For particle number, one commonly writes $\lambda_N=-\beta\mu$. Mutual commutation of the $Q_a$ is not needed to maximize entropy with fixed expectations: the same exponential follows whenever the constraint problem has a full-rank solution. If each $Q_a$ commutes with $H$, this state is also stationary. Noncommuting charges do not, however, admit joint sharp-value sectors, so sector-restricted and thermodynamic interpretations require care.

Integrable systems can possess extensively many local or quasilocal charges. Their detailed generalized Gibbs construction belongs to Integrability and Generalized Gibbs Ensembles Preview. The lesson here is simpler: energy-only Gibbs predictions are invalid if another retained charge changes local observables.

Exact symmetry labels and thermodynamic constraints differ

Not every conserved label needs an intensive multiplier. A finite-system parity or momentum sector may affect spectral statistics and finite-size averages but become locally invisible in the thermodynamic limit. Conserved densities such as energy, particle number, or magnetization generally remain thermodynamically relevant.

Ask whether changing a charge density at fixed energy density changes local equilibrium predictions.

Conserved Densities and Hydrodynamic Slow Modes

For a locally conserved density $q(\mathbf x,t)$,

$$\partial_tq + \nabla\cdot\mathbf j_q = 0.$$

Local equilibration can occur before the conserved profile becomes spatially uniform. In that regime, a local-equilibrium description uses slowly varying fields:

$$\begin{aligned} \rho_{\mathrm{loc}} &\propto \exp \left[ - \int d^dx\, \beta(\mathbf x) \kappa(\mathbf x) \right], \\ \kappa(\mathbf x) &\equiv h(\mathbf x) - \sum_a \mu_a(\mathbf x)q_a(\mathbf x). \end{aligned}$$

The fields $\beta(\mathbf x)$ and $\mu_a(\mathbf x)$ then evolve through hydrodynamic equations.

For diffusion,

$$\partial_t q = D\nabla^2q,$$

and a Fourier mode relaxes as

$$q_{\mathbf k}(t) = q_{\mathbf k}(0) e^{-Dk^2t}.$$

The longest wavelength in a box of size $L$ has a relaxation scale

$$\tau_{\mathrm{diff}} \sim \frac{L^2}{D}.$$

This divergence does not by itself signal failure of thermalization. It is the expected slow relaxation of a conserved mode. Ballistic, superdiffusive, subdiffusive, or anomalous transport produces different scalings. Transport Coefficients Preview owns the response and transport conventions.

Long-time tails

Coupled conserved modes can generate algebraic late-time decay,

$$\delta O(t) \sim t^{-\alpha},$$

rather than a simple exponential. Fitting a short window to $e^{-t/\tau}$ can therefore misidentify both the mechanism and asymptotic scale.

Kinematic Typicality Is Not Dynamics

Canonical typicality concerns most pure states in a large constrained subspace. It does not claim that a particular Hamiltonian trajectory samples those states.

Let

$$\begin{aligned} \mathcal H &= \mathcal H_A\otimes\mathcal H_B, \\ \dim\mathcal H_A &=d_A, \qquad \dim\mathcal H_B=d_B. \end{aligned}$$

For a Haar-random pure state on the full tensor product,

$$\mathbb E \operatorname{Tr}\rho_A^2 = \frac{ d_A+d_B }{ d_Ad_B+1 }.$$

The maximally mixed state on $A$ has purity $1/d_A$. Hence

$$\mathbb E \left\| \rho_A-\frac{I_A}{d_A} \right\|_2^2 = \frac{ d_A^2-1 }{ d_A(d_Ad_B+1) }.$$

Using $\|X\|_1\le\sqrt{d_A}\|X\|_2$ and concavity gives

$$\mathbb E D\left( \rho_A,\frac{I_A}{d_A} \right) \le \frac{1}{2} \sqrt{ \frac{ d_A^2-1 }{ d_Ad_B+1 } }.$$

When $d_B\gg d_A$, a typical pure state is locally close to infinite temperature. A constrained energy shell replaces $I_A/d_A$ by the reduced shell state and can lead to a finite-temperature canonical state under additional assumptions.

This is a concentration-of-measure result. To use it dynamically, one still needs to show that the preparation and Hamiltonian produce the relevant local typicality or eigenstate structure.

How eigenstate thermalization selects the mean

The eigenstate thermalization hypothesis proposes a structured form for matrix elements of few-body observables in generic nonintegrable systems. At the level needed here, its diagonal implication is

$$O_{nn} \approx O_{\mathrm{th}}(E_n)$$

within a symmetry sector and a narrow energy-density window.

Assume the energy spectrum in the occupied symmetry sector is nondegenerate and the initial state has a narrow energy-density distribution. Then

$$\operatorname{Tr}(\omega O) = \sum_n p_nO_{nn} \approx O_{\mathrm{th}}(\bar E).$$

With degeneracies, the exact time average retains within-energy coherence and must be evaluated using the projectors $P_E$. Choosing a basis that diagonalizes $\omega$ inside each energy block makes a diagonal sum possible, but the ETH hypothesis must hold in that basis or be stated for the whole block.

Off-diagonal ETH structure can also suppress fluctuations. The ETH page develops the ansatz, its symmetry assumptions, and the distinction between typical and exceptional eigenstates.

ETH is neither the definition of equilibration nor a theorem for every Hamiltonian. It is one mechanism for making the ensemble-identification error small for suitable observables and preparations.

Dephasing Does Not Require Chaos

Consider $L$ independent spins with

$$H = \frac{\hbar}{2} \sum_{j=1}^{L} \omega_j\sigma_j^z$$

and initial state

$$|\psi_0\rangle = |+\rangle_x^{\otimes L}.$$

The spatially averaged transverse magnetization is

$$M_x(t) = \frac{1}{L} \sum_{j=1}^{L} \cos(\omega_jt).$$

If the empirical frequency distribution approaches $p(\omega)$,

$$M_x(t) \longrightarrow \int d\omega\, p(\omega)\cos(\omega t).$$

For a Gaussian distribution with mean $\omega_0$ and width $\Delta$,

$$p(\omega) = \frac{1}{ \sqrt{2\pi}\Delta } \exp \left[ - \frac{ (\omega-\omega_0)^2 }{ 2\Delta^2 } \right],$$

the integral is

$$M_x(t) = e^{-\Delta^2t^2/2} \cos(\omega_0t).$$

The averaged signal relaxes by inhomogeneous dephasing. Yet:

• every spin remains pure;
• the evolution creates no entanglement;
• each $\sigma_j^z$ is conserved;
• one site continues to precess forever;
• a finite discrete frequency set recurs.

This is relaxation of one coarse observable without ordinary many-body thermalization. Chaos is not required for dephasing, and dephasing is not enough for Gibbs behavior.

Persistent oscillations in a two-level example

Let

$$H = E_0|0\rangle\langle0| + E_1|1\rangle\langle1|$$

with $E_1\ne E_0$, and

$$|\psi_0\rangle = \frac{ |0\rangle+|1\rangle }{ \sqrt2 }.$$

For

$$O = |0\rangle\langle1| + |1\rangle\langle0|,$$

one finds

$$\langle O(t)\rangle = \cos \left[ \frac{ (E_1-E_0)t }{ \hbar } \right].$$

The time average is zero, but

$$\sigma_O^2 = \frac12.$$

Here $d_{\mathrm{eff}}=2$, so there is no large dimension to suppress fluctuations. The example is the cleanest warning that a correct diagonal-ensemble average is not the same as equilibration.

Different observables have different relaxation times

No single thermalization time applies to every probe. Useful scales include:

• Microscopic, $\tau_{\mathrm{micro}}$: local oscillation or collision time, controlled by local couplings and gaps.
• Dephasing, $\tau_{\mathrm{deph}}$: destructive phase interference, controlled by the occupied energy differences.
• Local, $\tau_{\mathrm{local}}$: equilibration of a local reduced state, controlled by locality, scattering, and operator growth.
• Transport, $\tau_{\mathrm{tr}}$: redistribution of conserved densities, controlled by the transport law and system size.
• Prethermal, $\tau_{\mathrm{pre}}$: lifetime of a prethermal plateau, controlled by weak integrability breaking or scale separation.
• Recurrence, $\tau_{\mathrm{rec}}$: finite-size recurrence time, controlled by the detailed many-body spectrum.

Their ordering is model and probe dependent. A local nonconserved observable can equilibrate before global density homogenizes. A current can remain constrained by conservation after local correlations look stationary. In suitable high-frequency or strong-scale-separation regimes, the plateau lifetime can grow exponentially in a large ratio of energy scales.

Energy width is not a universal relaxation rate

For a pure state, the final-energy variance controls the short-time survival probability:

$$\mathcal L(t) = 1 - \frac{ (\Delta E)^2t^2 }{ \hbar^2 } + O(t^4).$$

It does not generally imply

$$\tau_{\mathrm{relax}} = \frac{\hbar}{\Delta E}.$$

Local relaxation depends on observable matrix elements, energy-gap structure, conservation laws, and spatial propagation. A global overlap can decay rapidly while a local density changes slowly.

Finite Size, Recurrence, and Order of Limits

For a finite-dimensional isolated system, expectation values are quasiperiodic. Under broad conditions, the state returns arbitrarily close to its initial state. Permanent pointwise convergence is therefore not the generic finite-size statement.

The thermodynamic local limit often intends

$$\lim_{t\to\infty} \left[ \lim_{V\to\infty} \langle O_A(t)\rangle_V \right],$$

with fixed $A$. Taking $V\to\infty$ first sends boundary reflections and recurrences to later times.

By contrast, a finite-size diagonal-ensemble calculation takes

$$\lim_{V\to\infty} \left[ \lim_{T\to\infty} \frac{1}{T} \int_0^T \langle O(t)\rangle_V\,dt \right].$$

The two procedures need not agree automatically. Near phase transitions, in systems with symmetry breaking, or with nonuniform conserved modes, additional limits are involved.

Boundary-reflection window

For a local probe a distance $r$ from a boundary and a characteristic front velocity $v$,

$$t_{\mathrm{boundary}} \sim \frac{r}{v}.$$

This estimates the arrival of a disturbance originating at the boundary. A signal launched at the probe and reflected back travels a distance $2r$, giving about $2r/v$. Choose the appropriate source–probe geometry before fitting bulk behavior.

Mechanisms and Their Limits

Generic nonintegrable systems

In many nonintegrable lattice models:

• local interactions spread operators and entanglement;
• phases from many energy gaps dephase;
• exact conserved densities relax hydrodynamically;
• eigenstate matrix elements become smooth within sectors;
• small subsystems approach microcanonical or canonical predictions.

This is a broad empirical and theoretical picture, not one universal theorem with one timescale.

Integrable systems

Integrable systems can dephase and relax while retaining memory in many conserved mode occupations or quasilocal charges. An energy-only Gibbs state is then generally insufficient. Relaxation without ordinary thermalization is not a contradiction.

Many-body localization

In an effective localized description, quasilocal integrals of motion retain local memory while interactions can produce slow entanglement growth. The many-body localization page distinguishes this useful description from the unsettled thermodynamic stability of particular disordered models.

Fragmentation, constraints, and scars

Kinetic constraints or gauge constraints can split a nominal symmetry sector into disconnected dynamical components. Quantum many-body scars can support atypical revivals for special preparations even when nearby states look thermal. The accessible component, not merely the formal Hilbert space, determines the effective dimension.

Near-integrability and prethermalization

A weak perturbation of an integrable or otherwise structured Hamiltonian can produce:

$$\begin{gathered} \text{rapid constrained relaxation} \\ \Downarrow \\ \text{long plateau} \\ \Downarrow \\ \text{final thermalization} \end{gathered}$$

Calling the plateau the final ensemble without a perturbation- and time-dependent analysis can hide the eventual drift.

Driven and open systems

A periodically driven isolated system does not conserve the undriven energy and can heat, prethermalize, or localize depending on frequency and structure. A system coupled to a bath evolves nonunitarily and can genuinely approach a global stationary mixed state.

Bath-induced thermalization and closed-system local thermalization can yield similar reduced density operators while obeying different equations, conserved quantities, and reversibility properties.

Separating fluctuations from an incorrect ensemble

For a concrete finite-window comparison, report both the mean and the fluctuations. For a Hermitian observable, let $m=\overline{\langle O\rangle}$ and $O_{\mathrm{th}}$ be its candidate thermal value. Over the same averaging window,

$$\overline{(\langle O\rangle-O_{\mathrm{th}})^2} =\overline{(\langle O\rangle-m)^2} +(m-O_{\mathrm{th}})^2.$$

Thus a small fluctuating part can coexist with a large systematic mismatch. Fix the ensemble parameters from conserved quantities, compare several probes, and vary the window, system size, and numerical accuracy separately. Preparations with the same narrow energy and charge densities give a further test of initial-state independence.

Trace distance obeys both a triangle inequality and its reverse. If

$$D(\rho_A(t),\omega_A)\leq0.01,\qquad D(\omega_A,\rho_A^{\mathrm{th}})=0.20,$$

then

$$0.19\leq D(\rho_A(t),\rho_A^{\mathrm{th}})\leq0.21.$$

The subsystem is close to its time average throughout that window and demonstrably far from this thermal candidate. Exercise 4 applies this distinction.

An exact two-qubit example with $\rho_0=0.69|00\rangle\langle00|+0.02|01\rangle\langle01|+0.29|11\rangle\langle11|$ and $H=\hbar g(|01\rangle\langle10|+|10\rangle\langle01|)$. Its first-qubit population is $p_0(t)=0.70+0.01\cos(2gt)$. Fluctuations about $\omega_A=\operatorname{diag}(0.70,0.30)$ stay small, but the proposed $\rho_A^{\mathrm{th}}=I/2$ remains far away. This illustrates ensemble mismatch, not thermodynamic thermalization of two qubits.

Numerical and Experimental Pitfalls

Exact diagonalization

Exact diagonalization gives the full finite-size spectrum but is especially vulnerable to:

• unresolved symmetry sectors;
• sparse energy shells;
• boundary recurrences;
• atypical initial states;
• finite-size drift of level statistics and matrix elements.

The diagonal ensemble can be computed exactly, but its agreement with a thermal ensemble must still be extrapolated.

Tensor-network time evolution

Bond-dimension growth can bias late-time observables toward artificially simple states. Monitor:

$$\begin{gathered} \text{discarded weight}, \qquad \chi, \qquad \Delta t, \\ \text{energy drift}, \qquad \text{charge drift}. \end{gathered}$$

Convergence of one local observable does not guarantee convergence of entanglement or correlations.

Experiments

Apparent damping can result from:

• shot-to-shot parameter variation;
• spatial inhomogeneity;
• ensemble averaging;
• particle loss;
• technical noise;
• unresolved detection bandwidth;
• environmental decoherence.

These effects may coexist with intrinsic many-body relaxation. Varying system size, isolation time, disorder, and preparation can help separate them.

Exercises

1. Dephasing with degeneracy

Let

$$H = \sum_EEP_E.$$

Derive the infinite-time average of $\rho(t)$ and explain why coherence within one $P_E$ block survives.

Solution

Insert the spectral resolution:

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

The time average of the phase vanishes when $E\ne E'$ and equals one when $E=E'$. Hence

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

All vectors in one degenerate eigenspace acquire the same phase. Their relative coherence is stationary and cannot be removed by time averaging.

2. Effective dimension

A pure state has final-energy probabilities

$$p_1=\frac12, \qquad p_2=p_3=p_4=p_5=\frac18.$$

Compute $d_{\mathrm{eff}}$. Compare it with the number of occupied energies.

Solution

The purity of the dephased state is

$$\begin{aligned} \operatorname{Tr}\omega^2 &= \left(\frac12\right)^2 + 4\left(\frac18\right)^2, \\ &= \frac14+\frac1{16}, \\ &= \frac5{16}. \end{aligned}$$

Therefore

$$d_{\mathrm{eff}} = \frac{16}{5} = 3.2.$$

Five energies are occupied, but the uneven weights reduce the effective dimension below five.

3. Fraction of bad times

Suppose

$$\sigma_O^2 \le 10^{-6}$$

for a dimensionless bounded observable. Bound the fraction of times for which $|\delta_O(t)|>10^{-2}$.

Solution

Markov’s inequality in the long-time limit gives

$$\limsup_{T\to\infty}\frac{\mu(\mathcal B_\epsilon(T))}{T} \leq \frac{\sigma_O^2}{\epsilon^2}.$$

With $\epsilon=10^{-2}$,

$$\limsup_{T\to\infty}\frac{\mu(\mathcal B_\epsilon(T))}{T} \leq \frac{10^{-6}}{10^{-4}} = 10^{-2}.$$

The upper asymptotic fraction of times outside the tolerance is at most one percent. This does not set the duration of the initial transient. The bound need not be tight.

4. Split the thermalization error

Suppose

$$D(\rho_A(t),\omega_A)\le0.01$$

for most late times and

$$D(\omega_A,\rho_A^{\mathrm{th}})=0.20.$$

What can be concluded about equilibration and thermalization?

Solution

The first distance shows that $A$ equilibrates closely to its dephased reduced state. The triangle inequality and its reverse give

$$0.19\leq D(\rho_A(t),\rho_A^{\mathrm{th}})\leq0.21.$$

More importantly, the ensemble-identification error is already $0.20$, so the dephased state is not close to the proposed thermal state at the stated accuracy. The system equilibrates locally but does not thermalize to that candidate ensemble.

5. Gaussian inhomogeneous dephasing

Evaluate

$$M_x(t) = \int_{-\infty}^{\infty} d\omega\, p(\omega)\cos(\omega t)$$

for a Gaussian $p(\omega)$ with mean $\omega_0$ and standard deviation $\Delta$. Explain why the result is not proof of thermalization.

Solution

The characteristic function of the Gaussian is

$$\int d\omega\, p(\omega)e^{i\omega t} = e^{i\omega_0t} e^{-\Delta^2t^2/2}.$$

Taking the real part gives

$$M_x(t) = e^{-\Delta^2t^2/2} \cos(\omega_0t).$$

The decay comes from phase cancellation among independent frequencies. Every $\sigma_j^z$ is conserved, no interactions or entanglement are present, and a local spin does not approach a Gibbs state. The result demonstrates observable relaxation by dephasing only.

6. A missing conserved charge

Two initial states have equal $\langle H\rangle$ but different particle numbers $N_1\ne N_2$, and $[H,N]=0$. Can both thermalize to the same canonical state $e^{-\beta H}/Z$?

Solution

Not if particle-number density affects the tested local observables. Unitary evolution preserves each initial number sector or number distribution. A correct comparison must either restrict to fixed-$N$ sectors or use

$$\rho_{\beta,\mu} \propto e^{-\beta(H-\mu N)}$$

with $\mu$ chosen to reproduce the conserved number. Equal mean energy alone is insufficient.

7. Canonical typicality scaling

For $d_A=4$ and $d_B=10^4$, use

$$\mathbb E D\left( \rho_A,\frac{I_A}{d_A} \right) \le \frac12 \sqrt{ \frac{d_A^2-1}{d_Ad_B+1} }$$

to estimate the mean distance bound.

Solution

Substitution gives

$$\mathbb E D \le \frac12 \sqrt{ \frac{15}{40001} } \approx 9.7\times10^{-3}.$$

Thus a typical pure state on the full $4\times10^4$ tensor product is locally close to the maximally mixed state on $A$. This says nothing by itself about whether a specified Hamiltonian trajectory reaches typical states.

8. Design a thermalization test

A finite spin-chain simulation shows that one local magnetization approaches a canonical value. Design a stronger test with at least six controls.

Solution

A credible test would:

1. resolve all exact symmetry sectors;
2. fix temperature from the conserved energy rather than magnetization;
3. compare several local observables and connected correlations;
4. compare the diagonal ensemble with microcanonical and canonical predictions;
5. vary system size and stop before boundary reflections;
6. report late-time fluctuations and the bad-time fraction;
7. vary initial states at fixed energy and charge densities;
8. test microcanonical shell-width sensitivity;
9. verify time-step, truncation, and bond-dimension convergence;
10. check for slow conserved modes or a prethermal drift.

The original trace does not isolate equilibration, ensemble identification, finite-size effects, or numerical error.

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Further Connections

• Statistical Ensembles Overview — equilibrium ensemble inputs and domains.
• Thermal Density Operators — Gibbs-state structure without a dynamical claim.
• Number Operators and Conserved Quantities — exact charges and sector decomposition.
• Time-Dependent Correlations — two-time stationarity, spectra, and nonequilibrium center time.
• Glasses and Spin Glasses — waiting-time aging, weak ergodicity breaking, and failed stationarity beyond generic equilibration tests.
• Quantum Thermalization — experimental comparisons between local relaxation and constrained thermal predictions.
• Hydrodynamics and Effective Theory Preview — how local equilibration and conserved slow modes become a controlled long-wavelength EFT.
• Real-Time Thermal Dynamics Preview — forward–backward contour evolution for thermal initial-value problems.
• Many-Body Entanglement Overview — reduced states and state-class diagnostics.
• Operator Entanglement and Scrambling Preview — why operator spreading and thermalization are related but inequivalent.
• Scrambling and OTOCs Preview — thermal OTOC conventions, butterfly fronts, and recovery-aware scrambling tests.
• Thermodynamic Limit — recurrence, subsystem fraction, and noncommuting limits.