Fluctuation theorem for time-averaged work

TL;DR

The paper addresses connecting time-averaged work in adiabatically driven, thermally isolated systems to isothermal-like quasistatic work via a fluctuation theorem. It defines the time-averaged work $\langle\overline{W}\rangle$ and proves $\langle e^{-\beta \overline{W}}\rangle = e^{-\beta \langle W_{\rm qs}\rangle}$, with $\Delta F$ and an effective Hamiltonian formalism providing a tractable link to isothermal quantities. Numerical evidence for a driven classical harmonic oscillator supports the theorem, and an optimality-based argument yields a strong inequality $\langle W\rangle(\tau) \ge \langle W_{\rm qs}\rangle$. The work grounds a pathway from adiabatic to isothermal perspectives in non-equilibrium thermodynamics and characterizes the limits of shortcuts to adiabaticity for thermally isolated systems.

Abstract

There is evidence that taking the time average of the work performed by a thermally isolated system effectively "transforms" the adiabatic process into an isothermal one. This approach allows inherent quantities of adiabatic processes to be accessed through the definitions of isothermal processes. A fluctuation theorem is then established, linking the time-averaged work to the quasistatic work. Numerical evidence supporting this equality is provided for a classical harmonic oscillator with a driven linear equilibrium position parameter. Furthermore, the strong inequality for the averaged work is derived from the deduced fluctuation theorem using optimality arguments.

Figures (1)

• Figure 1: Fluctuation theorem \ref{['eq:jarzynskiequality']} for the classical harmonic oscillator, with a driven linear equilibrium position parameter. It was used datasets with $10^5$ values of $\exp{[-\beta(\overline{W}-\langle{W}_{\rm qs}\rangle)]}$ sampled according with the canonical ensemble. It was used $\beta=1$, $\lambda_0=1$, $\delta\lambda=0.5$.