arxiv: 2511.05366 · v4 · submitted 2025-11-07 · ❄️ cond-mat.stat-mech · math.DS· physics.bio-ph

Coarse-graining nonequilibrium diffusions with Markov chains

Ram\'on Nartallo-Kaluarachchi , Renaud Lambiotte , Alain Goriely This is my paper

Pith reviewed 2026-05-17 23:58 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math.DSphysics.bio-ph

keywords nonequilibrium steady statescoarse-grainingMarkov chainsentropy productionplanar diffusionsfinite-volume methodsstochastic processestrajectory inference

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The pith

Finite-volume discretization turns planar diffusions into Markov chains that preserve nonequilibrium steady states and converge in entropy production.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a finite-volume method to convert continuous planar diffusion processes into discrete-state Markov chains. This produces an approximate master equation that retains the structure of the nonequilibrium steady state from the original diffusion. The central result establishes that the entropy production rate computed from the discrete chain approaches the exact continuous value as the number of states increases without bound. The same framework also shows how to infer discrete models from observed trajectories, where the models underestimate entropy production yet still detect whether the underlying process is out of equilibrium.

Core claim

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. Discrete-state models inferred from continuous stochastic trajectories significantly underestimate the true entropy production rate but can provide tests to determine if a stationary planar diffusion is out of equilibrium.

What carries the argument

finite-volume discretization that produces a master equation for the discrete Markov chain while preserving the nonequilibrium steady-state structure of the planar diffusion

If this is right

The entropy production rate of the Markov chain approximation converges to the exact value of the continuous diffusion as the number of discrete states tends to infinity.
The discrete models enable numerical exploration of how entropy production depends on parameters in nonlinear diffusions.
Markov chains inferred from trajectory data can serve as diagnostic tests for nonequilibrium behavior even while underestimating the production rate.
The preservation of steady-state features holds for both analytically solvable cases and numerical experiments on nonlinear processes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The convergence property could be used to obtain rigorous error bounds on entropy production estimates obtained from binned trajectory data.
The method might extend to diffusions on domains with more complex geometry or to systems with multiple interacting particles.
Biological trajectory data, such as from schooling fish, could be re-discretized at varying resolutions to bracket the true entropy production value.

Load-bearing premise

The finite-volume discretization preserves the nonequilibrium steady-state structure of the underlying planar diffusion for the chosen boundary conditions and mesh.

What would settle it

Numerical computation of entropy production on successively refined meshes for an analytically solvable nonequilibrium diffusion, showing that the discrete rate remains bounded away from the continuous value, would disprove the convergence.

Figures

Figures reproduced from arXiv: 2511.05366 by Alain Goriely, Ram\'on Nartallo-Kaluarachchi, Renaud Lambiotte.

**Figure 1.** 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. 1 arXiv:2511.05366v1 [cond-mat.stat-mech] 7 Nov 2025 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗

**Figure 2.** 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. the production of entropy. The entropy production rat… view at source ↗

**Figure 3.** 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 R 2 into a set of volumes, where we approximate the flux across the boundary, resulting in a CTMC. which leads to a diagonal diffusion matrix with entries, D(x, y) = Dx (x, y) 0 0 Dy (x, … view at source ↗

**Figure 4.** 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 ω and a. Both ω and a drive irreversible dynamics, whilst σ drives r… view at source ↗

**Figure 5.** Figure 5: Discrete-approximation of NESS in the OU. For a fixed value of θ = 5, σ = 5, and a range of discretisation step-sizes, we calculate the stationary distribution of a coarsegrained 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. Given the stationary density, we can perform the H… view at source ↗

**Figure 6.** Figure 6: EPR in a coarse-grained OU process. We see that the EPR of the discretestate approximation also increases approximately quadratically with θ, as in the exact solution. Moreover, we can see that the EPR converges as the step-size goes to 0 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

**Figure 7.** Figure 7: Discrete-approximation of NESS in the Hopf oscillator. For fixed values of a ∈ {−1, 2}, ω = 2, σ = 0.5, and a range of discretisation step-sizes, we calculate the stationary distribution of a coarse-grained Hopf oscillator using the SG discretisation. Even for large stepsizes, the symmetry of the stationary density is preserved, converging to the true density as the step-size decreases. the EPR of the coa… view at source ↗

**Figure 8.** Figure 8: EPR in the coarse-grained Hopf oscillator. For fixed values of a and σ, we can increase ω to increase the EPR of the Hopf oscillator. We show that the EPR of the underlying diffusion is closely approximated by the coarse-grained process. For a = −1, we see that the EPR converges as ∆x goes to 0. For a = 2, we use larger step-sizes, and appear to be outside the asymptotic regime. Moreover, we note that for … view at source ↗

**Figure 9.** Figure 9: Discrete-approximation of NESS in the VDP. For a fixed value of θ = 1, µ = 2 and σ = 1, we calculate the stationary distribution of a coarse-grained VDP oscillator using the SG discretisation. Even for small step-sizes, the limit cycle behaviour is present. Moreover, unlike the Hopf oscillator, the rotation is not of uniform speed leading to a build up of probability at the apexes. We consider the domain [… view at source ↗

**Figure 10.** Figure 10: Time-integration of the VDP FP equation. Using the Crank-Nicholson scheme, we integrate the FP equation over time, using our discretised Laplacian. Starting from a Gaussian initial density centred at (0, 0) with standard deviation σ = 1 in each direction, we step through with ∆t = 0.2 up to T = 1.4, we can see that the distribution equilibrates to the same distribution calculated directly in [PITH_FULL_I… view at source ↗

**Figure 11.** Figure 11: NESS of the VDP oscillator. The shape and and nature of the stationary distribution shifts in response to changes in parameters. We illustrate this by calculating the stationary distribution of the VDP for an increased value of µ = 3, θ = 5 and σ = 2 respectively. An increased value of µ promotes nonlinearity in the oscillation, whilst θ promotes circular rotation around the origin. Finally, σ increases t… view at source ↗

**Figure 12.** Figure 12: Drift field of coupled Kuramoto oscillators. We consider the drift field of a pair of Kuramoto oscillators. When the model is symmetric, the synchronised state is the only stable equilibrium, but this can be disrupted with asymmetry in the frequencies, interaction strengths or through frustration. oscillating at a natural frequency of ω > 0 where θ(t) ∈ [0, 2π). Unlike the other systems we considered, the… view at source ↗

**Figure 13.** Figure 13: NESS in the Kuramoto model. For a range of different parameter values, we can compute the stationary density, discrete flux and continuous flux field in the NESS. We can see that the coupled Kuramoto system is in a NESS with most of the flux aligned along the synchronised oscillation. Nevertheless, the shape of the stationary distribution depends on the parameters [PITH_FULL_IMAGE:figures/full_fig_p014_13.png] view at source ↗

**Figure 14.** Figure 14: EPR in coarse-grained Kuramoto oscillators. We investigate how the EPR varies as a function of the parameters. We find that coupling asymmetry increases the EPR, whilst the absolute value of the natural frequencies drives the EPR, regardless of asymmetry. Finally, we find that the EPR-maximising value of the frustration depends on the noise-intensity. Finally, we investigate the EPR as a function of the v… view at source ↗

**Figure 15.** Figure 15: Inferring NESS with Markov chain approximations. Using samples from the OU process in Eq. (46), we infer a CTMC and calculate the stationary distribution, which is a good approximation of the true density. We also compute the EPR for trajectories of different lengths and using different grid-sizes. We find that using longer trajectories and smaller grids leads to higher accuracy, but that the inferred EPR… view at source ↗

**Figure 16.** Figure 16: Surrogate testing for NESS. Using the shuffling procedure we can create surrogate Markov models from observed sequence data, and use this to perform testing to identify if real-world trajectories are from an ESS or a NESS. We illustrate this with trajectories from the OU process at θ = 5, a NESS with Φ = 25, and θ = 0, an ESS. Using a one-sided t-test, we find that at θ = 5 the process has significant (*… view at source ↗

**Figure 17.** Figure 17: Stationary dynamics in schooling fish. We consider the group polarisation vector M = (Mx, My) for populations of schooling fish of size N = 15, 30 and 60. We find that the degree of collective alignment increases as group-size decreases, which can be seen by computing the mean and standard deviation of ||M(t)|| over time. Additionally, we can infer a discrete-state Markov process from the trajectories and… view at source ↗

**Figure 18.** Figure 18: EPR for populations of schooling fish compared to surrogate models. Using the discrete-state model, we can measure the EPR of the trajectories of schooling fish. This is represented by the single orange dot in each panel. Using the surrogate model procedure, we obtain a ‘null’ distribution (the violin plot) of the EPR in the surrogate model. We perform hypothesis testing to compare the single empirical tr… view at source ↗

read the original 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 the 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Finite-volume coarse-graining of planar diffusions produces Markov chains whose entropy production converges in the fine limit, with a useful side result on trajectory inference.

read the letter

The main point is that finite-volume discretization of planar diffusions into Markov chains can be done so the entropy production converges to the continuous case as the number of states increases. They derive the approximate master equation straight from the diffusion using finite volumes and show it keeps the nonequilibrium steady state features. Convergence of the entropy production rate is shown for solvable diffusions and checked numerically on nonlinear ones. They also look at inferring these discrete models from continuous trajectories, like from schooling fish, where the discrete version underestimates the true rate but still helps test if the system is out of equilibrium. This gives a controllable way to link continuous theory with discrete models, which is handy for active matter and biological data analysis. The potential issue is whether the convergence really holds without extra assumptions. The stress test points out that for nonlinear drifts, the truncation error in the current might not go away in the entropy production expression. The paper uses illustrations but if there's no a priori bound showing independence from mesh choice or boundaries, that part could be softer than it looks. Readers working on nonequilibrium statistical mechanics or analyzing stochastic trajectories will get the most from this. It offers concrete tools and examples rather than just abstract theory. I would send it for peer review. The ideas are solid enough to warrant detailed feedback on the math and numerics.

Referee Report

1 major / 2 minor

Summary. The manuscript investigates nonequilibrium steady-state dynamics for planar diffusions and their coarse-grained representations as discrete-state Markov chains. Using finite-volume discretizations, it derives an approximate master equation from the underlying Fokker-Planck equation and claims to prove that the entropy production rate of the resulting Markov chain converges to the continuous value as the number of discrete states tends to infinity. The convergence is illustrated on analytically solvable cases and nonlinear numerical examples; the work also treats inference of discrete models from continuous trajectories, showing that such models underestimate entropy production yet can still serve as a diagnostic for nonequilibrium, with demonstrations on simulated data and empirical schooling-fish trajectories.

Significance. If the convergence result is placed on a rigorous footing, the paper would supply a principled route to coarse-grain continuous nonequilibrium processes while retaining thermodynamic consistency, which is useful for modeling and inference in statistical mechanics and biophysics. The combination of solvable benchmarks, numerical tests, and real-data application is a concrete strength that demonstrates practical utility.

major comments (1)

[Derivation of the master equation and convergence claim] The central convergence claim for the entropy production rate (abstract and the section deriving the master equation from the finite-volume scheme): no a-priori error estimate is supplied showing that the discrete entropy-production functional converges to the continuous one. For a Cartesian mesh and nonlinear drift the local truncation error on the probability current is O(h); it is not demonstrated that the resulting perturbation vanishes in the functional ∫|j|^2/p as h→0 independently of the chosen boundary conditions or mesh regularity.

minor comments (2)

[Numerical experiments] Specify the precise mesh geometry, boundary-condition implementation, and quadrature rules employed in the finite-volume discretization for the nonlinear test cases.
[Inference from trajectories] Add a short paragraph clarifying the inference procedure (maximum-likelihood or otherwise) used to obtain the discrete transition rates from continuous trajectories, including any regularization or binning details.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for a more rigorous treatment of the convergence claim. We address the major comment point by point below and will revise the manuscript to strengthen the result.

read point-by-point responses

Referee: [Derivation of the master equation and convergence claim] The central convergence claim for the entropy production rate (abstract and the section deriving the master equation from the finite-volume scheme): no a-priori error estimate is supplied showing that the discrete entropy-production functional converges to the continuous one. For a Cartesian mesh and nonlinear drift the local truncation error on the probability current is O(h); it is not demonstrated that the resulting perturbation vanishes in the functional ∫|j|^2/p as h→0 independently of the chosen boundary conditions or mesh regularity.

Authors: We agree that the current manuscript does not supply an explicit a-priori error estimate establishing convergence of the discrete entropy-production functional to its continuous counterpart. The derivation obtains the master equation by integrating the Fokker-Planck equation over control volumes, which preserves the divergence structure of the probability current. Convergence is shown analytically for linear solvable cases and numerically for nonlinear examples, but a general proof that the O(h) local truncation error on the current produces a vanishing perturbation in ∫|j|^2/p is absent. In the revised version we will add a dedicated error-analysis subsection. Under standard assumptions (Lipschitz drift, uniformly elliptic and bounded diffusion coefficient, Cartesian mesh, and either periodic or no-flux boundary conditions), we will prove that the discrete current converges to the continuous current in a norm sufficient to guarantee that the entropy-production difference tends to zero as the mesh size h→0. Mesh-regularity requirements will be stated explicitly. This addition will place the convergence claim on a firmer footing while preserving the existing numerical and applied results. revision: yes

Circularity Check

0 steps flagged

Derivation of master equation and entropy-production convergence is self-contained

full rationale

The paper derives an approximate master equation from the continuous planar diffusion via finite-volume discretization and then proves that the entropy production rate of this discrete approximation converges to the continuous value as the number of states tends to infinity. This is presented as a direct consequence of the discretization preserving the nonequilibrium steady-state structure, supported by analytic examples and numerical checks rather than by fitting parameters to the target quantity. No step reduces the claimed convergence to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation; the central result is a mathematical property of the scheme under the stated boundary conditions and mesh assumptions. The inference section on underestimation from trajectories is a separate empirical observation, not part of the convergence derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard stochastic calculus and finite-volume discretization of Fokker-Planck operators; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The underlying process admits a unique nonequilibrium steady state whose probability current can be discretized without destroying the divergence-free property on the grid.
Invoked when the authors state that the discretization preserves key features of the nonequilibrium steady-state.

pith-pipeline@v0.9.0 · 5485 in / 1227 out tokens · 19439 ms · 2026-05-17T23:58:27.769170+00:00 · methodology

discussion (0)

Reference graph

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