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Finite reservoirs and irreversibility corrections to Hamiltonian systems statistics

Matteo Colangeli, Antonio Di Francesco, Lamberto Rondoni

arXiv:2305.14922v1cond-mat.stat-mechmath-phmath.MP

TL;DR

The paper investigates whether canonical statistics describe exponential observables in finite reservoirs and under processes that break time-reversal symmetry. Using solvable oscillator models, simulations, and an ideal-gas expansion, it finds protocol-dependent averages and argues that standard statistical mechanics needs adjustment in these settings.

  • Problem

    Canonical statistics assume infinite reservoirs, but finite baths and short experiments may matter for exponential observables that depend on distribution tails.

    Source: §1 · pp. 1–2

  • Method

    The paper analyzes driven Hamiltonian systems using solvable oscillator models, simulations, and a model of irreversible ideal-gas expansion.

    Source: §1 · p. 2

  • Results

    Finite-size effects and emergent irreversibility lead to protocol-dependent averages of exponential quantities such as exp(−βWJ).

    Source: §5 · p. 16

  • Takeaways & Limitations

    The standard statistical-mechanical formalism should be adapted for finite-size and irreversible settings involving exponential quantities.

    Source: §5 · p. 16

    For the ideal-gas expansion, the initial canonical distribution ceases to describe the system once particles can enter the other half of the box.

    Source: §4 · p. 15

Abstract

from arXiv · show

We consider several Hamiltonian systems perturbed by external agents, that preserve their Hamiltonian structure. We investigate the corrections to the canonical statistics resulting from coupling such systems with possibly large but finite reservoirs, and from the onset of processes breaking the time reversal symmetry. We analyze exactly solvable oscillators systems, and perform simulations of relatively more complex ones. This indicates that the standard statistical mechanical formalism needs to be adjusted, in the ever more investigated nano-scale science and technology. In particular, the hypothesis that heat reservoirs be considered infinite and be described by the classical ensembles is found to be critical when exponential quantities are considered, since the large size limit may not coincide with the infinite size canonical result. Furthermore, process-dependent emergent irreversibility affects ensemble averages, effectively frustrating, on a statistical level, the time reversal invariance of Hamiltonian dynamics, that is used to obtain numerous results.

1. Introduction

The paper asks whether canonical statistics remain reliable for exponential observables and small, rapidly driven systems, where finite reservoirs and short observation times make distribution tails and irreversibility relevant. It examines finite-size effects and protocol-dependent failures of the Jarzynski equality (JE) in driven Hamiltonian systems, including an irreversible gas expansion.

1. Introduction

  • Finite reservoirs and short experiments matter especially for exponential observables, because they depend on probability-distribution tails rather than only the bulk.The introduction notes that small systems may also be nonergodic and that thermal baths are necessarily finite.Source: §1 · pp. 1–2
  • The JE relies on microscopic time-reversal symmetry, and prior work cited here attributes some JE failures to process-dependent irreversibility rather than system size.The introduction contrasts those cases with highly nonequilibrium overdamped Langevin systems where the JE can remain valid.Source: §1 · p. 2
  • Finite-size effects can make JE violations strongly protocol-dependent, reaching 110% in the examples; analogous results occur for infinite baths at small temperatures.The paper highlights protocol speed and resonant periodic driving as ways to enhance the finite-size effects.Source: §1 · p. 2
  • The paper also examines ideal-gas expansion and concludes that an irreversible process can invalidate canonical statistics for a particle system.The paper’s section overview identifies free expansion as an example.Source: §1 · p. 2

2. Derivation of the Jarzynski equality

The derivation considers a Hamiltonian system and environment initially in canonical equilibrium, then drives the isolated combined system through a fixed parameter protocol. It obtains the Jarzynski equality, whose protocol independence follows from the canonical ensemble but may fail for finite ensembles.

2. Derivation of the Jarzynski equality

  • The setup treats the system and environment together as a Hamiltonian system, initially in canonical equilibrium with a bath at temperature T.The Hamiltonian is divided into system energy, environment energy, and their interaction; an external agent controls the system parameter.Source: §2 · p. 3
  • The derivation repeats a fixed parameter protocol from λ(0)=A to λ(τ)=B, sampling different initial conditions from the canonical distribution and computing work for each realization.The system is isolated from the bath at t=0; deterministic dynamics make work depend on the sampled initial condition.Source: §2 · p. 3
  • The JE relates an exponential-work average to the equilibrium free-energy difference and is protocol-independent under the canonical ensemble, but finite support can make its right-hand side protocol-dependent.The text notes that the appropriate probability distribution for finite environments is not generally known.Source: §2 · p. 3

3. Models and methods

The section tests how finite-support initial ensembles alter Jarzynski exponential-work averages across three driven oscillator models. Its central result is that finite reservoirs make the average protocol-dependent, with resonances and reservoir bounds shaping the deviations.

3. Models and methods

  • The study compares a linearly driven oscillator, a periodically forced oscillator, and a periodically forced system coupled to a second oscillator.Only the system oscillator is driven in the coupled model.Source: §3 · p. 4

3.1. Single oscillator under linear protocol

  • With truncated canonical support, finite-size correction factors depend on protocol speed; at fixed bounds, sufficiently fast protocols can reduce the exponential-work average below its infinite-size value.At fixed speed, increasing both position and momentum bounds recovers the infinite-size result.Source: §3.1 · p. 5

3.2. Single Oscillator with Periodic Forcing

  • For periodic forcing, the bounded-ensemble average depends sensitively on protocol time and can approach 0 or 2 for sufficiently large times, even with large position bounds.At resonance, the oscillator’s position amplitude grows linearly with time, contributing a time-dependent term to the correction.Source: §3.2 · p. 7

3.3. Coupled Oscillators with Periodic Forcing

  • The coupled-oscillator model is evaluated numerically by sampling truncated-canonical initial states, integrating their dynamics, and estimating the work average from the resulting trajectories.The authors use simulations because analytical solutions become practically involved for unequal masses and spring constants.Source: §3.3 · pp. 7–8

3.4. Results

  • Finite-reservoir effects make the exponential-work average protocol-dependent, unlike the infinite-reservoir result.For the constant-speed oscillator, the apparent speed-independence at larger finite bounds shifts to higher speeds rather than holding generally.Source: §3.4 · p. 8
  • At resonance γ=ω, undamped periodic forcing grows the oscillator’s amplitude linearly, increasing work and driving the finite-ensemble average toward zero; higher β widens the affected frequency interval.A larger global stiffness appears to stabilize the average’s dependence on γ.Source: §3.4 · pp. 9–11
  • For equal-mass oscillators with low interaction stiffness, increasing inverse temperature can make the finite-ensemble average vanish across tested forcing frequencies.Higher inverse temperature narrows the distribution, producing finite-size effects even for an infinite bath; the authors describe the finite-ensemble result as physically relevant.Source: §3.4 · p. 11
  • With an environment ten times more massive than the system, changing interaction stiffness yields similar results at L=M=1 but noticeably different behavior at L=M=5.Both weakly and strongly coupled cases show resonance-region oscillations; at L=M=5, peaks exceed 1 and finite-size effects are magnified.Source: §3.4 · pp. 11–12

4. Irreversible Expansion of an Ideal Gas

The section models an ideal gas released from the left half of a box and tracks how the moving wall’s protocol affects the final particle distribution. The resulting protocol dependence persists even with an ideal infinite thermostat, so the canonical prediction can fail for this irreversible process.

4. Irreversible Expansion of an Ideal Gas

  • The gas begins with uniformly distributed positions in the left half and Gaussian velocities; particles move freely and reflect elastically at the box boundaries.The fraction reaching the right half is calculated from initial positions and velocities, with the many-particle prediction applying for large N.Source: §4 · pp. 12–13
  • The analytical prediction for NR(τ) sums contributions from the initial-position and velocity ranges that place particles in the right half at time τ.The derivation integrates over the velocity intervals, then initial positions, and assumes sufficiently many particles.Source: §4 · p. 13
  • The characteristic time 2L/⟨|v|⟩ estimates relaxation toward uniform particle distribution; larger β, or lower temperature, means longer relaxation.The usual mean time based on ⟨1/v⟩ does not exist in one dimension.Source: §4 · p. 13
  • Changing the wall protocol duration changes the final right-half population and free-energy difference, even with an ideal infinite thermostat, contrary to the Jarzynski equality.For sufficiently large τ, NL = NR, which differs from the initial NL = N, NR = 0.Source: §4 · p. 14
  • A finite, sufficiently high barrier can approximate initial confinement for a chosen finite duration and tolerance, preserving the protocol-dependent free-energy result.The initial canonical distribution ceases to describe the evolving state once particles are allowed to access the full box.Source: §4 · pp. 14–15

5. Concluding remarks

The concluding remarks report finite-reservoir and irreversibility effects in driven oscillators and an ideal-gas expansion. Across these examples, protocol-dependent exponential averages challenge canonical predictions when finite size or broken time-reversal symmetry matters.

Concluding remarks

  • A constant-speed trap recovers the infinite-bath result when the protocol is not too fast, while faster driving produces larger deviations, especially for smaller integration bounds and lower temperatures.The canonical result returns as L and M grow, but larger bounds are required at smaller kp or β.Source: §5 · p. 15
  • For a periodically driven oscillator, finite-bath averages can deviate strongly from 1, even reaching 0 near resonance; recovering the infinite-bath result is harder over more periods.The infinite-bath result equals 1 over a multiple of the driving period.Source: §5 · p. 15
  • In a coupled-oscillator model, numerical results show deviations near specific forcing frequencies, while sufficiently high driving frequencies yield the infinite-bath value apart from oscillations.The finite-bath result has no analytical expression in this case; the conclusion is based on finite ensembles and was robust to ensemble-size variation.Source: §5 · pp. 15–16
  • The ideal-gas example changes accessible phase space instantaneously, so the initial ensemble no longer describes the evolving state and the standard formalism does not apply directly.The authors identify a finite-potential-barrier and finite-time formulation for the experiment.Source: §5 · p. 16
  • Finite size and emergent irreversibility can both produce protocol-dependent averages of exponential quantities, motivating caution with canonical results when large fluctuations matter.The conclusion is limited to cases where standard conditions are severely violated and observables require accurate tail characterization.Source: §5 · pp. 15–16

A. Some explicit calculations for Sec. 3.1

This appendix gives explicit calculations for the linear-protocol oscillator using canonical and truncated canonical distributions. It sets up the bounded averages and separates the integrals used in their evaluation.

A. Some explicit calculations for Sec. 3.1

  • The calculation uses the canonical distribution for a configuration of the system plus environment, with its initial partition function.The notation defines P0(Γ) as the canonical distribution and identifies Z0 as the partition function.Source: A. Some explicit calculations for Sec.… · p. 17
  • For the single oscillator under a linear protocol, the appendix considers canonical distributions truncated at distance L and maximum momentum M.The truncation sets the distribution to zero when |x| > L or |p| > M.Source: A. Some explicit calculations for Sec.… · p. 17
  • The bounded average of exp(−βWJ) is evaluated by separating its integral into parts and computing them individually.The appendix proceeds from the separated integrals to the resulting expression.Source: A. Some explicit calculations for Sec.… · pp. 17–18

B. Some explicit calculations for Sec. 3.2

This appendix section presents explicit calculations for a single oscillator subject to periodic forcing. The supplied passages identify an intermediate expression and a subsequent result.

B. Some explicit calculations for Sec. 3.2

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