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Entropy production reveals hidden dynamical constraints rather than stochastic disorder

Patrick Romanescu

Abstract

Entropy production is often interpreted as a proxy for microscopic disorder or environmental roughness in stochastic systems. We test this interpretation using controlled simulations of overdamped stochastic dynamics on curved surfaces in which local noise, geometry, and forces are held fixed while global constraints are varied. Trajectories are generated for particles evolving toward a central attractor, and entropy production is quantified using both a continuum probability-current estimator and coarse-grained Markov transition statistics across multiple spatial and temporal resolutions. Across systematic sweeps of timestep size, domain extent, and boundary topology, entropy production is governed primarily by constraint-induced probability flow rather than local stochastic variability. Periodic domains that permit sustained circulation yield substantially higher entropy production than reflecting domains despite identical local stochastic structure, with the magnitude of the separation depending on domain extent. In contrast, coarse-grained estimates decrease as temporal resolution increases and rise with finer spatial binning, demonstrating that discrete estimates depend strongly on observation scale and may fail to resolve topology-induced irreversible structure. Ergo, entropy production is not a direct measure of environmental roughness or randomness. Instead, it quantifies how strongly system dynamics are driven away from reversibility by global constraints, geometry, and the space of allowed trajectories. Interpreted in this way, entropy production maps function as diagnostics of organized probability flow and provide a principled method for detecting hidden dynamical constraints from trajectory data alone.

Entropy production reveals hidden dynamical constraints rather than stochastic disorder

Abstract

Entropy production is often interpreted as a proxy for microscopic disorder or environmental roughness in stochastic systems. We test this interpretation using controlled simulations of overdamped stochastic dynamics on curved surfaces in which local noise, geometry, and forces are held fixed while global constraints are varied. Trajectories are generated for particles evolving toward a central attractor, and entropy production is quantified using both a continuum probability-current estimator and coarse-grained Markov transition statistics across multiple spatial and temporal resolutions. Across systematic sweeps of timestep size, domain extent, and boundary topology, entropy production is governed primarily by constraint-induced probability flow rather than local stochastic variability. Periodic domains that permit sustained circulation yield substantially higher entropy production than reflecting domains despite identical local stochastic structure, with the magnitude of the separation depending on domain extent. In contrast, coarse-grained estimates decrease as temporal resolution increases and rise with finer spatial binning, demonstrating that discrete estimates depend strongly on observation scale and may fail to resolve topology-induced irreversible structure. Ergo, entropy production is not a direct measure of environmental roughness or randomness. Instead, it quantifies how strongly system dynamics are driven away from reversibility by global constraints, geometry, and the space of allowed trajectories. Interpreted in this way, entropy production maps function as diagnostics of organized probability flow and provide a principled method for detecting hidden dynamical constraints from trajectory data alone.
Paper Structure (26 sections, 31 equations, 2 figures, 5 tables)

This paper contains 26 sections, 31 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Theory
  3. Embedded surface geometry and induced metric
  4. Confining potential and Riemannian gradient
  5. Riemannian Langevin diffusion in Itô form
  6. Discrete-time approximation (Euler--Maruyama)
  7. Domain restriction and boundary operators
  8. Probability current and continuum entropy production
  9. Coarse-grained Markov entropy production under spatial binning
  10. Summary of theoretical objects used in computation
  11. Numerical Experiments
  12. Domains and boundary conditions
  13. Temporal discretization and sampling
  14. Increment extraction and boundary-aware velocities
  15. Empirical steady-state density estimation on a fixed grid
  16. ...and 11 more sections

Figures (2)

  • Figure 1: Spatial diagnostics for domain $[-2.6,2.6]^2$ at $\Delta t=0.01$.Top row (A–C): reflecting boundaries.Bottom row (D–F): periodic boundaries. (A,D) Sample trajectories. (B,E) Estimated stationary density $p(x)$. (C,F) Continuum entropy production density $\sigma(x)=2\,J^\top a^{-1}J/p$ (log scale). Local stochastic dynamics are identical in both rows; only global boundary topology differs. The stationary density remains smooth across conditions, whereas $\sigma(x)$ exhibits structured heterogeneity characteristic of irreversible probability transport.
  • Figure 2: Spatial diagnostics for domain $[-3.2,3.2]^2$ at $\Delta t=0.01$.Top row (A–C): reflecting boundaries.Bottom row (D–F): periodic boundaries. (A,D) Sample trajectories. (B,E) Estimated stationary density $p(x)$. (C,F) Continuum entropy production density $\sigma(x)=2\,J^\top a^{-1}J/p$ (log scale). Increasing domain size alters the magnitude and geometry of entropy production but preserves the topology-dependent separation between rows. Regions of elevated $\sigma(x)$ coincide with sustained probability transport rather than density maxima.