Temperature fluctuations in mesoscopic systems

TL;DR

This paper models temperature fluctuations in mesoscopic systems using stochastic thermodynamics, revealing finite-size effects on thermodynamic quantities, fluctuation relations, and efficiency limits in heat engines.

Contribution

It introduces a stochastic differential equation framework for temperature evolution, deriving finite-size corrections to fluctuation theorems and analyzing efficiency limits in mesoscopic heat engines.

Findings

Temperature fluctuations cause deviations from extensivity in thermodynamic quantities.

Finite-size corrections to the Jarzynski equality depend on heat capacity.

Efficiency of mesoscopic Carnot engines is reduced by temperature fluctuations, preventing attainment of Carnot efficiency.

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