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arxiv: 2309.15489 · v1 · pith:5IQWNW46new · submitted 2023-09-27 · ❄️ cond-mat.stat-mech · cond-mat.mes-hall· quant-ph

Temperature fluctuations in mesoscopic systems

Zhaoyu Fei , Yu-Han Ma This is my paper

Pith reviewed 2026-05-24 07:07 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.mes-hallquant-ph
keywords mesoscopic systemstemperature fluctuationsstochastic thermodynamicsCarnot cyclefinite-size effectsentropy productionJarzynski equality
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The pith

Temperature fluctuations in mesoscopic systems cause the average efficiency of quasi-static Carnot engines to fall short of the Carnot limit by a term proportional to N inverse.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models how temperature in an N-particle system fluctuates during contact with a heat reservoir by adding a noise term to the usual deterministic description of an isothermal process. These fluctuations lead to extra irreversible entropy production even when the process is slow. When applied to a Carnot cycle, the net result is that the average efficiency is lower than the Carnot efficiency by an amount that shrinks only as 1/N. This implies that mesoscopic heat engines face a fundamental barrier to ideal performance that does not disappear in the quasi-static limit. The work also derives corrections to the Jarzynski equality and notes possible violations of the maximum-work principle at the same order.

Core claim

Introducing a stochastic differential equation for the temperature of an N-body system during an isothermal process, with a noise term representing random energy exchanges, reveals that temperature fluctuations produce irreversible entropy. In a quasi-static Carnot engine this entropy reduces the cycle efficiency according to η_C − ⟨η⟩ ∼ N^{-1}, so that the Carnot efficiency remains unattainable at mesoscopic scales.

What carries the argument

Stochastic differential equation for temperature evolution that incorporates finite-size noise from reservoir interactions.

If this is right

  • The Jarzynski equality acquires finite-size corrections that depend on the system's heat capacity.
  • Quantities that are extensive in the thermodynamic limit deviate from strict extensivity to match equilibrium fluctuation theory.
  • The principle of maximum work can be violated by an amount of order N^{-1}.
  • Carnot efficiency is unattainable for mesoscopic heat engines even when operated quasi-statically.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar fluctuation corrections may limit the performance of other small-scale thermodynamic devices such as refrigerators or information engines.
  • Experiments on colloidal particles or molecular motors could test the predicted 1/N scaling of efficiency loss.
  • The approach might be extended to non-quasi-static protocols to quantify how driving speed interacts with finite-size noise.

Load-bearing premise

The evolution of the system's temperature during an isothermal process can be described by a stochastic differential equation whose noise term accounts for finite-size effects arising from random energy transfer between the system and the reservoir.

What would settle it

Precise measurements of the average efficiency of a mesoscopic Carnot engine with controlled particle number N, checking whether the shortfall from Carnot efficiency decreases proportionally to 1/N as N increases.

Figures

Figures reproduced from arXiv: 2309.15489 by Yu-Han Ma, Zhaoyu Fei.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The Carnot cycle in the temperature-entropy dia [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
read the original abstract

We study temperature fluctuations in mesoscopic $N$-body systems undergoing non-equilibrium processes from the perspective of stochastic thermodynamics. By introducing a stochastic differential equation, we describe the evolution of the system's temperature during an isothermal process, with the noise term accounting for finite-size effects arising from random energy transfer between the system and the reservoir. Our analysis reveals that these fluctuations make the extensive quantities (in the thermodynamic limit) deviate from being extensive for consistency with the theory of equilibrium fluctuation. Moreover, we derive finite-size corrections to the Jarzynski equality, providing insights into how heat capacity influences such corrections. Also, our results indicate a possible violation of the principle of maximum work by an amount proportional to $N^{-1}$. Additionally, we examine the impact of temperature fluctuations in a finite-size quasi-static Carnot engine. We show that irreversible entropy production resulting from the temperature fluctuations of the working substance diminishes the average efficiency of the cycle as $\eta_{\rm{C}}-\left\langle \eta\right\rangle \sim N^{-1}$, highlighting the unattainability of the Carnot efficiency $\eta_{\rm{C}}$ for mesoscopic-scale heat engines even under the quasi-static limit

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a stochastic differential equation (SDE) for the temperature of an N-body mesoscopic system during isothermal processes, with a noise term intended to capture finite-size effects from stochastic energy exchanges with the reservoir. From this, it derives finite-size corrections to the Jarzynski equality (influenced by heat capacity), a possible N^{-1} violation of the maximum-work principle, and for a quasi-static Carnot cycle shows that temperature fluctuations produce irreversible entropy production yielding η_C − ⟨η⟩ ∼ N^{-1}, implying Carnot efficiency remains unattainable at mesoscopic scales even in the quasi-static limit.

Significance. If the SDE is shown to follow from controlled coarse-graining and to be consistent with equilibrium fluctuation theory, the 1/N efficiency correction would be a notable result for stochastic thermodynamics of small systems and mesoscopic heat engines. The work correctly identifies that extensive quantities must deviate from strict extensivity at finite N and links the correction to heat capacity; these are potentially useful observations. The central claim, however, rests entirely on the modeling step whose validity is not demonstrated.

major comments (2)
  1. [Modeling section (SDE introduction)] The stochastic differential equation for temperature evolution during isothermal processes (introduced in the modeling section following the abstract) is defined with a noise term chosen to generate the desired finite-N corrections. No microscopic derivation from many-body dynamics, no recovery of the equilibrium temperature variance (∼ k_B T^2 C_V), and no check against the fluctuation-dissipation relation are provided. Because every subsequent result—including the Jarzynski correction, maximum-work violation, and the η_C − ⟨η⟩ ∼ N^{-1} scaling—follows directly from this SDE, the absence of justification makes the 1/N claim an artifact of the modeling choice rather than a generic consequence.
  2. [Carnot-engine section] In the Carnot-engine analysis (the section deriving irreversible entropy production and efficiency), the quasi-static limit is taken while retaining the stochastic temperature fluctuations. The derivation assumes the SDE remains valid and produces a non-zero entropy production even when the cycle is infinitely slow; however, without demonstrating that the noise term vanishes appropriately or that the resulting entropy production is independent of the artificial noise amplitude, the claim that Carnot efficiency is unattainable at finite N even quasi-statically is not yet established.
minor comments (2)
  1. [SDE definition] Notation for the noise term and its correlation function should be stated explicitly with the precise form of the SDE (including any Itô/Stratonovich convention) to allow reproducibility.
  2. [Abstract and introductory paragraphs] The abstract states that extensive quantities 'deviate from being extensive for consistency with the theory of equilibrium fluctuation'; a brief explicit statement of the required scaling (e.g., how the mean energy or entropy deviates) would clarify this point.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major points below, clarifying the motivation for the SDE and outlining revisions to strengthen the justification and presentation.

read point-by-point responses

  1. Referee: [Modeling section (SDE introduction)] The stochastic differential equation for temperature evolution during isothermal processes (introduced in the modeling section following the abstract) is defined with a noise term chosen to generate the desired finite-N corrections. No microscopic derivation from many-body dynamics, no recovery of the equilibrium temperature variance (∼ k_B T^2 C_V), and no check against the fluctuation-dissipation relation are provided. Because every subsequent result—including the Jarzynski correction, maximum-work violation, and the η_C − ⟨η⟩ ∼ N^{-1} scaling—follows directly from this SDE, the absence of justification makes the 1/N claim an artifact of the modeling choice rather than a generic consequence.

    Authors: The SDE is introduced as an effective description whose noise amplitude is fixed by the requirement of consistency with equilibrium fluctuation theory, specifically to recover the known variance ⟨(ΔT)^2⟩ ∼ k_B T^2 / C_V. We will add an explicit calculation in the revised modeling section showing that the stationary solution of the SDE reproduces this variance and is compatible with the fluctuation-dissipation relation in the equilibrium limit. While a complete microscopic derivation from underlying many-body dynamics is not provided and would constitute a separate study, the present construction is not arbitrary but is dictated by thermodynamic consistency at finite N; the subsequent 1/N corrections then follow directly from this requirement. revision: yes

  2. Referee: [Carnot-engine section] In the Carnot-engine analysis (the section deriving irreversible entropy production and efficiency), the quasi-static limit is taken while retaining the stochastic temperature fluctuations. The derivation assumes the SDE remains valid and produces a non-zero entropy production even when the cycle is infinitely slow; however, without demonstrating that the noise term vanishes appropriately or that the resulting entropy production is independent of the artificial noise amplitude, the claim that Carnot efficiency is unattainable at finite N even quasi-statically is not yet established.

    Authors: The noise amplitude is determined exclusively by the equilibrium heat capacity and is therefore independent of the cycle duration or driving speed. In the quasi-static limit the working substance remains in thermal contact with the reservoirs, so the finite-N temperature fluctuations persist as an intrinsic feature rather than vanishing. The irreversible entropy production arising from these fluctuations therefore yields a non-zero N^{-1} correction to the average efficiency that survives the infinite-time limit. We will add a clarifying paragraph in the Carnot section emphasizing that the noise strength is protocol-independent and fixed by equilibrium properties. revision: partial

Circularity Check

0 steps flagged

No circularity: model introduction followed by derivation from stated assumptions

full rationale

The paper explicitly introduces a stochastic differential equation as a modeling ansatz to capture finite-size temperature fluctuations, with the noise term posited to represent random energy exchanges. All subsequent results (Jarzynski corrections, maximum-work violation scaling, and 1/N efficiency deficit in the quasi-static Carnot cycle) are derived from this equation. No quoted step reduces a claimed prediction to a fitted parameter, self-citation, or definitional equivalence; the SDE is not shown to be constructed from the target 1/N scaling. The derivation chain is therefore self-contained once the modeling choice is granted, consistent with standard practice for stochastic-thermodynamics papers that begin from an SDE.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper rests on the postulate that temperature itself obeys a stochastic differential equation with additive noise representing finite-size energy exchanges; no free parameters are fitted and no new particles are invented.

axioms (1)
  • domain assumption Temperature of a mesoscopic system during an isothermal process obeys a stochastic differential equation whose noise term encodes random energy transfer with the reservoir.
    Introduced in the abstract as the starting point for all finite-size corrections.
invented entities (1)
  • Stochastic temperature variable with explicit noise term no independent evidence
    purpose: To capture finite-size fluctuations that make extensive quantities non-extensive and generate the reported corrections.
    Postulated to derive the N^{-1} terms; no independent falsifiable prediction outside the model is supplied in the abstract.

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