Restoring detailed balance in non-Hermitian Markov processes

TL;DR

The paper addresses non-Hermitian Markov generators that violate detailed balance and exhibit nonmonotonic entropy production. It introduces Dyson maps to transform $H$ into a Hermitian $\mathcal{H}$, ensuring nonnegative entropy production and spectral invariance, with a metric operator $\Omega$ linking original and transformed measurements. The authors derive a constructive scheme combining algebraic constraints and a universal numerical algorithm to compute Dyson maps, and illustrate the method on a network-based SIS model showing monotone entropy growth in the transformed dynamics while preserving essential statistical structure. They also discuss the connection to quantum dynamics via a Wick rotation and acknowledge limitations for complex spectra, outlining potential applications to long-time dynamics and efficient simulations.

Abstract

Stochastic processes out-of-equilibrium often involve asymmetric contributions that break detailed balance and lead to non-monotonic entropy production, limiting thermodynamic interpretations and inference techniques. Here we use Dyson maps to restore monotonic entropy growth in those processes, allowing the use of standard tools from statistical physics, providing a general and computationally tractable method applicable to a broad class of Markovian systems.

Figures (6)

• Figure 1: Time evolution of the SIS-process ($\beta=0.5,\gamma=0.05$). (top) Entropies of the original system $H$ ($S_\mathrm{Renyi}$ and $S_\mathrm{Shannon}$) and for the Dyson mapping $\mathcal{H}$ ($S'_\mathrm{Renyi}$) (bottom) Renyi entropy of a homotopic map between $H_\mathrm{SIS}$ and $\mathcal{H_\mathrm{SIS}}$, $((1-\alpha)I + \alpha \eta) H ((1-\alpha)I + \alpha \eta)^{-1}$.
• Figure 2: SIS-dynamics on a static contact network. The colors indicate the state of the vertex. Two possible transitions are given, an infection (top), and a recovery (bottom). Right: structure of the stochastic generator $H_\mathrm{SIS}$ of the process, with the constant recovery transitions (blue) above the diagonal, the infection transitions (red) below it, indicating the asymmetry of the stochastic process.
• Figure 3: Mean and standard deviation of number of infectious vertices in the SIS system ($\beta=0.5,\gamma=0.05$), obtained by integration of $H_\mathrm{SIS}$ (lines) and $\mathcal{H}_\mathrm{SIS}$ (markers).
• Figure 4: Phase diagram. Average and standard deviation of infectious vertices, as well as Rényi entropy of original and transformed system for the ratio $\beta/\gamma$.
• Figure 5: Values of $P_\ell$ and $\phi_\ell$ in the steady state for $\beta/\gamma=10^{-2}$ (top), $\beta/\gamma=10^{-\frac{1}{2}}$ (middle) and $\beta/\gamma=20$ (bottom), obtained by solving Eq. \ref{['eq:master']} (green) and Eq. \ref{['eq:entropy-derivative']} (purple), along with the Kolmogorov–Smirnov ($KS$) distance between them.