Entropy Production Rate in Stochastically Time-evolving Asymmetric Networks

Abstract

Fluctuations in parameters that are typically treated as fixed play a crucial role in the behavior of complex systems. However, to date, we lack a general non-equilibrium thermodynamic treatment of such a complex system. In this Letter, to address this problem, we develop a framework in which fluctuating interactions between units of nonlinear network systems are modeled as uncorrelated colored noise (i.e., annealed disorder) with a correlation time. This approach enables us to quantify how the entropy production rate (EPR) depends on both the characteristic time-scale and the strength of the disorder. Using dynamical mean field theory, we derive an exact expression for EPR at any transient time that is validated by simulations of the full non-linear dynamics. At stationarity, a relation between EPR and autocorrelation is established and then used to analytically study the particular case of linear systems.

Figures (12)

• Figure 1: Phase diagram of the mean $\langle x \rangle$ (a), second moment $\langle x x\rangle$ (b), variance $\langle x x\rangle - \langle x\rangle^2$ (c), and entropy production rate (EPR) (d) in the parameter space $(g,\mu)$ obtained from the dynamical mean-field theory (DMFT) equation \ref{['effective']}. Comparison with the phase diagrams obtained from the full dynamics \ref{['eq:eqn1']} is shown in Figs. \ref{['fig:DMFT-SP-tau1']}, \ref{['fig:DMFT-SP-tau10']}, and \ref{['fig:DMFT-SP-tau100']}. Here and in what follows, unless specified, we focus on $F(z) = \tanh(z)$, a standard form of activation function in modeling neural networks. Parameters: noise strength $\sigma = 0.01$, time step $dt = 10^{-2}$, total simulation time $t=50$, DMFT iterations 1000, and each point is averaged over 2000 number of realizations.
• Figure 2: Entropy production rate (EPR) as a function of Var$(x)\equiv \langle xx\rangle -\langle x\rangle^2$. The color intensity increases with increasing $\tau_0$. Symbols corresponding to the same color (or $\tau_0$) are obtained from Fig. \ref{['fig:DMFT-PDF']} for all combinations of $(g,\mu)$. The red straight dashed line is to guide the eye only.
• Figure 3: a) Correlation function $C_x(\tau)$\ref{['Bessel']} as a function of time $\tau$ for two different values of $g$ and $\sigma=1$. b) EPR \ref{['final']} as function of $g$. The black dashed line is for the quenched case $\tau_0\to \infty$. Inset: Data for $\tau_0=1$ for $g\geq 1$. In both panels, the blue color intensity increases with $\tau_0=1,10,100$.
• Figure S1: Comparison of average (a), second moment (b), and entropy production rate (EPR) (c) obtained using full dynamics \ref{['eq:eqn1']} and effective dynamics (DMFT equation) \ref{['effective']}, each as a function of time for three different correlation times $\tau_0=1,10,100$. For full dynamics: we show the scaled average (scaled by system's size $N$). Solid: DMFT. Dashed: Full dynamics. Other parameters: Number of particles $N = 2000$, discretization time $dt = 0.01$, temperature $T = 5\times 10^{-5}$, coupling parameter $\mu = 1$, number of realizations for full dynamics: 100, and number of iterations for DMFT: 1000.
• Figure S2: Comparison of scaled average, scaled second moment, and entropy production rate (EPR) obtained using full dynamics \ref{['eq:eqn1']} (panel a) and effective dynamics (DMFT equation) \ref{['effective']} (panel b) in $g,\mu$ plane at time $t=50$ and $\tau_0=1$. All other parameters are same as in Fig. \ref{['TD-comp-FD-and-DMFT']}.