Thermodynamic optimization equalities in weakly driven processes

TL;DR

This work derives thermodynamic optimization equalities from the Euler–Lagrange equation for weakly driven classical open systems, revealing that at optimality the rates satisfy $\dot{U}_{\mathcal{S}}(t)=2\dot{W}^*(t)=2\dot{Q}(t)$. By embedding this within a linear-response framework using the relaxation function $\Psi(t)$ and the response function $\phi(t)$, the authors analyze optimization of work, heat, and internal energy, showing that the optimal-heat protocol is adiabatic with $Q^*(\tau)=0$, while internal-energy optimization can involve jumps and leads to specific limiting values as $\tau\to0^+$. A practical contribution is a convergence criterion for genetic-programming optimization based on the thermodynamic equalities, demonstrated on an overdamped Brownian-motion model where the work and heat optimizations agree with exact results, whereas energy optimization exhibits no nontrivial optimum. These results link optimal control in non-equilibrium thermodynamics to directly testable equalities, providing a robust stopping condition for numerical optimization and insights into how optimal protocols affect different thermodynamic channels.

Abstract

Equalities are generally more suitable for experimental verification than inequalities. In this work, I derive valid equalities from the Euler-Lagrange equation for the optimization of macroscopic thermodynamic averages in weakly driven classical open systems. These equalities show that optimization occurs when work and heat become path-independent. I illustrate their applicability by employing them as a convergence criterion in the global optimization technique of genetic programming. Moreover, due to fluctuation-dissipation relations for internal energy, work, and heat, analogous results hold for their variances.

Figures (2)

• Figure 1: Graphic (a): Comparison between numerical optimal protocols for work and exact result for switching times $\tau/\tau_R=0.1,1,10$. Graphics (b)-(d): Comparison between $\dot{U}_{\mathcal{S}}$ and $2\dot{Q}$ for different switching times at the moment where the code stopped according to our convergence criterion. They are measured in unit $\Psi_0 \delta\lambda^2/\tau_R$.
• Figure 2: Graphic (a): Comparison between numerical optimal protocols for heat and exact result for switching times $\tau/\tau_R=0.1,1,10$. Graphics (b)-(d): Comparison between $\dot{U}_{\mathcal{S}}$ and $2\dot{Q}$ for different switching times at the moment where the code stopped according to our convergence criterion. They are measured in unit $\Psi_0 \delta\lambda^2/\tau_R$.