Temperature fluctuations in mesoscopic systems

Temperature is a fundamental concept in thermodynamics. In macroscopic thermodynamics, systems possess their own intrinsic temperature which equals the reservoir temperature when they equilibrate. In stochastic thermodynamics for simple systems at the microscopic level, thermodynamic quantities other than temperature (a deterministic parameter of the reservoir) are stochastic. To bridge the disparity in the perspectives about temperature between the micro- and macroregimes, we assign a generic mesoscopic $N$-body system an intrinsic fluctuating temperature $T$ in this work. We simplify the complicated dynamics of numerous particles to one stochastic differential equation with respect to $T$, where the noise term accounts 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. Moreover, we derive finite-size corrections, characterized by heat capacity of the system, to the Jarzynski equality. A possible violation of the principle of maximum work that scales with $N^{-1}$ is also discussed. Additionally, we examine the impact of temperature fluctuations in a finite-size 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 ${\eta }_{\mathrm{C}}-\langle \eta \rangle \sim N^{-1}$, highlighting the unattainability of the Carnot efficiency ${\eta }_{\mathrm{C}}$ for mesoscopic heat engines even under the quasistatic limit. Our general framework paves the way for further exploration of nonequilibrium thermodynamics and the corresponding finite-size effects in a mesoscopic regime.