Coarse-graining nonequilibrium diffusions with Markov chains

TL;DR

Coarse-graining nonequilibrium planar diffusions into discrete-state Markov chains is achieved via a finite-volume discretisation of the Fokker-Planck equation. The paper proves that the entropy production rate $\Phi$ of the discrete approximation converges to the continuous diffusion value as the grid is refined, and validates this with solvable models (Ornstein–Uhlenbeck, Hopf oscillator) and unsolvable ones (stochastic van der Pol, frustrated Kuramoto). It extends to statistical inference from trajectories, showing that EPR inferred from coarse-grained trajectories underestimates the true EPR, yet can still test for NESS via surrogate testing and is useful for hypothesis testing on real trajectories, such as schooling fish. The work thus provides a principled bridge between continuous nonequilibrium diffusion thermodynamics and discrete-state representations, with practical implications for data-driven analyses of nonequilibrium systems.

Abstract

We investigate nonequilibrium steady-state dynamics in both continuous- and discrete-state stochastic processes. Our analysis focuses on planar diffusion dynamics and their coarse-grained approximations by discrete-state Markov chains. Using finite-volume approximations, we derive an approximate master equation directly from the underlying diffusion and show that this discretisation preserves key features of the nonequilibrium steady-state. In particular, we show that the entropy production rate of the approximation converges as the number of discrete states goes to the limit. These results are illustrated with analytically solvable diffusions and numerical experiments on nonlinear processes, demonstrating how this approach can be used to explore the dependence of entropy production rate on model parameters. Finally, we address the problem of inferring discrete-state Markov models from continuous stochastic trajectories. We show that discrete-state models significantly underestimate the true entropy production rate. However, we also show that they can provide tests to determine if a stationary planar diffusion is out of equilibrium. This property is illustrated with both simulated data and empirical trajectories from schooling fish.

Figures (18)

• Figure 1: Coarse-graining a stochastic trajectory. A stochastic trajectory from a diffusion can be modelled as a sequence of discrete states by performing a discretisation of state space.
• Figure 2: Nonequilibrium steady-state. Processes in a NESS are characterised by the presence of stationary probability flux. A process can be decomposed using the HHD into a reversible component where the drift balances the diffusive fluctuations to maintain the process at stationarity, and the irreversible component drives rotation around the stationary density.
• Figure 3: Approximating a diffusion as a Markov process. We aim to derive an approximation of a continuous diffusion as a discrete-state Markov chain using a finite-volume approximation. We use a rectangular grid to coarse-grain $\mathbb{R}^2$ into a set of volumes, where we approximate the flux across the boundary, resulting in a CTMC.
• Figure 4: Hopf oscillator in a nonequilibrium steady-state. The stationary density and flux show that the Hopf oscillator converges to a NESS. For $a<0$, this is a distribution peaked at the origin. For $a>0$, this is a 'Mexican-hat' distribution. Both show rotational probability flux due to the oscillatory dynamics. The EPR varies as a function of $\omega$ and $a$. Both $\omega$ and $a$ drive irreversible dynamics, whilst $\sigma$ drives reversible diffusion.
• Figure 5: Discrete-approximation of NESS in the OU. For a fixed value of $\theta =5$, $\sigma = 5$, and a range of discretisation step-sizes, we calculate the stationary distribution of a coarse-grained OU process using the SG discretisation. Even for large step-sizes, the symmetry of the stationary density is preserved, converging to the true density as the step-size decreases.