Nonequilibrium Macroscopic Response Relations for Counting Statistics

Abstract

Understanding how macroscopic nonequilibrium systems respond to changes in external or internal parameters remains a fundamental challenge in physics. In this work, we report a parameter transitional symmetry valid for macroscopic dynamics arbitrarily far from equilibrium. The symmetry leads to exact response relations and gives meaningful expansions in both linear and short-time regimes. This framework provides a universal description of macroscopic response phenomena arbitrarily far from equilibrium - including non-stationary processes and time-dependent attractors. The theory is validated and demonstrated numerically using the Willamowski-Rossler model, which exhibits rich dynamical behaviors including limit cycles and chaos.

Figures (3)

• Figure 1: Parameter translational symmetry between macroscopic fluctuating trajectories generated with different dynamical parameters.
• Figure 2: Stochastic trajectories of the Willamowski-Rössler model. (a) Limit cycle attractor. (b) Chaotic (strange) attractor.
• Figure 3: Numerical validation of the macroscopic response relations. (a-d) The positivity of the response measure $\mathscr{R}(t)$ (\ref{['eq: second law like']}) for limit cycle (a,b) and chaotic (c,d) attractors under different perturbations. (e-h) The time evolution of the normalized mean currents $\langle\iota_{\rho}\rangle/R_{\rho}$ (\ref{['eq: steady-state relation']}), showing convergence to a common value $t$ in the steady state for both limit cycles (e,f) and chaos (g,h). (f) and (h) are obtained after pre-equilibration time $t_\text{pre-eq} = 10$. Color legend: red for $\rho = 1$, orange for $\rho = 2$, yellow for $\rho = 3$, green for $\rho = 4$, blue for $\rho = 5$, brown for $\rho = -1$, purple for $\rho = -5$.