Quasistatic response for nonequilibrium processes: evaluating the Berry potential and curvature

TL;DR

The paper develops a geometric framework for quasistatic perturbations of steady nonequilibrium processes, showing that excess observables (e.g., entropy flux, heat, dynamical activity) acquire a Berry-phase structure in Markov jump dynamics under slow cyclic protocols. It defines the excess current via $H^{\\text{exc}}(\\Gamma)=\\int_{\\Gamma} d\\lambda \cdot R(\\lambda)$ with the Berry potential $R(\\lambda)=\\langle \nabla_\\lambda V_\\lambda\\rangle_\\lambda^{\\mathrm s}$ where $V_\\lambda$ solves a Poisson equation, and introduces the Berry curvature $\\Omega_{\\mu\\nu}=\\partial_\\mu R_\\nu-\\partial_\\nu R_\\mu$ as the measure of Maxwell-relations violations and frenetic contributions. The work connects these geometric quantities to equilibrium limits (where $\\Omega=0$ and Clausius relations hold), presents Aharonov–Bohm–type effects where a nonzero Berry phase exists without local curvature, and derives sufficient low-temperature conditions under which all Berry potentials and curvatures vanish, extending the Third Law to driven open systems. Overall, the study provides a rigorous, gauge-invariant description of nonequilibrium geometric responses with implications for response theory and thermodynamics of driven systems.

Abstract

We investigate how introducing slow, time-dependent perturbations to a steady, nonequilibrium process alters the expected (excess) values of important observables, such as the dynamical activity and entropy flux. When we make a cyclic thermodynamic transformation, the excesses are described in terms of a (geometric) Berry phase with corresponding Berry potential and Berry curvature quantifying the response. Focussing on Markov jump processes, we show how a non-zero Berry curvature leads to a breakdown of the thermodynamic Maxwell relations and of the Clausius heat theorem. We also present a variant of the Aharonov-Bohm effect in which the parameters follow a curve with vanishing Berry curvature, but the system still experiences a nonzero Berry phase. Finally, we identify (sufficient) no-localization conditions in terms of mean first-passage times under which the corresponding Berry potentials and curvatures vanish at absolute zero, extending, for arbitrary driving, e.g. the case of vanishing heat capacity as for the Nernst postulate.

Figures (13)

• Figure 1: Protocol $\Gamma = (\lambda(\varepsilon t), 0\leq t\leq \tau/\varepsilon)$ in parameter space $\cal M$. The closed loop $\Gamma = \partial \mathscr{S}$ traces the boundary of a surface $\mathscr{S}$.
• Figure 2: Heat conducting three-level system with energy gaps $\delta_1, \delta_2$. The transitions between $(z,y)$ and between $(x,z)$ are thermal from left and right heat baths at inverse temperatures $\beta_1, \beta_2$. The externally applied switch $x\leftrightarrow y$ has constant rate $\alpha$.
• Figure 3: The quasipotentials $V^{\cal P_i}_\lambda$ (used in \ref{['mC']}) of the heat flux as a function of the inverse temperatures with $\delta_1=1, \delta_2=2$. The solid lines have $\alpha=1$ and the dashed lines correspond to $\alpha=0$, keeping the same colors. We work here, and in the rest of the manuscript, with dimensionless units.
• Figure 4: The response $R_\alpha^{\cal P_i}$ of the excess heat to each of the heat baths upon changing the switching rate $\alpha$ in Fig. \ref{['ziaex']} around $\alpha=1$. We see the dependence on the bath temperatures for $\delta_1=1, \delta_2=2$.
• Figure 5: Graph (with state space $K=\{x,y,z,u,w\}$) consisting of a triangle connected to a line.

Theorems & Definitions (5)

• Example 1: Thermal response
• Example 2: Excess reactivity
• Example 3: Excess current
• Example 1: Berry curvature of the excess heat
• Example 2: Berry curvature of excess reactivity