Force-free kinetic inference of entropy production

TL;DR

The work presents a position-trace–based framework to estimate entropy production in nonequilibrium diffusion processes without direct force measurements. Central is a variance-sum-rule reduction yielding $σ = - (D^{-1})^{ij} C^{ij}_{x}(0) + D^{ij} C^{ij}_{\nabla φ}(0)$, reinterpreted through traffic $\mathcal{T}$ and inflow $\mathcal{G}$ with $σ = 4\mathcal{T} + \mathcal{G}$; for diagonal $\mathbf{D}$, per-dimension bounds $σ_{\mathcal{S}} = \sum_{i∈\mathcal{S}} \max(4\mathcal{T}_i,0) \le σ$ provide practical nonequilibrium diagnostics under partial observations. The authors validate the approach on a linear multivariate Ornstein–Uhlenbeck process and a nonlinear hair-bundle model, demonstrating accurate recovery of $σ$ from traces and showing traffic as the dominant dissipative contributor in many regimes. They detail a robust data-analysis pipeline—derivative estimation from exponential fits and force inference from kernel-density estimates—while addressing measurement noise and partial observability. The framework offers a practical route to quantify dissipation in experiments and biology where only positional data are accessible, enabling nonequilibrium characterization without perturbations or full state observation.

Abstract

Estimating entropy production, which quantifies irreversibility and energy dissipation, remains a significant challenge despite its central role in nonequilibrium physics. We propose a novel method for estimating the mean entropy production rate $σ$ that relies solely on position traces, bypassing the need for flux or microscopic force measurements. Starting from a recently introduced variance sum rule, we express $σ$ in terms of measurable steady-state correlation functions which we link to previously studied kinetic quantities, known as traffic and inflow rate. Under realistic constraints of limited access to dynamical degrees of freedom, we derive efficient bounds on $σ$ by leveraging the information contained in the system's traffic, enabling partial but meaningful estimates of $σ$. We benchmark our results across several orders of magnitude in $σ$ using two models: a linear stochastic system and a nonlinear model for spontaneous hair-bundle oscillations. Our approach offers a practical and versatile framework for investigating entropy production in nonequilibrium systems.

Figures (16)

• Figure 1: a) Stochastic traces of $x^1_t$ and $x^2_t$, sampled at $4\,{\rm kHz}$ over a duration of 500 s. b) Corresponding correlation functions for both degrees of freedom (DOFs). c) Effective forces $-\nabla \phi$ (black arrows) pointing from regions of high $\phi$ (red areas) to regions of smaller $\phi$ (yellow areas). Parameters for panels a),b) and c) are $A_{11}=A_{22}=-20$, $A_{12}=-10$, $T=\alpha_T=1$ and $\alpha_A=0.5$. d) Heatmap of $\sigma$ as a function of the dissipative parameters $\alpha_A$ and $\alpha_T$. Numbers written over the heatmap indicate the accuracy $\pi(\overline{\sigma}) = |\sigma-\overline{\sigma}|/\Delta\overline{\sigma}$ of our estimates $\overline{\sigma}$.
• Figure 2: a) Time series of $x^1_t$ and $x^2_t$ for the hair-bundle model, recorded at a sampling rate of $100\,{\rm kHz}$ over a 10 s interval. b) Autocorrelation and cross-correlation functions computed for both degrees of freedom (DOFs). c) Effective forces $-\nabla \phi$ (black arrows) pointing from regions of higher $\phi$ (orange areas) to regions of smaller $\phi$ (yellow areas). The area where the PDF $p(\pmb{x}_t)$ is very small (and $\phi$ is very high) has been highlighted in red. Effective forces have not been shown in this area for visualization purposes. A section of the stochastic trajectory is also depicted on top of the effective potential contour plot. For panels a),b) and c), $F^{\rm max}=100$ and $S=1$. d) Relationship between $\sigma$ and $F^{\rm max}$ for different values of $S$ ($S = 0.75, 1, 1.25, 1.5, 1.75$). Solid lines represent the true $\sigma$ while markers indicate estimated values with associated uncertainties. Note how the estimation procedure demonstrates high reliability across several orders of magnitude of $\sigma$.
• Figure 3: a) Scatter plot of $4\mathcal{T}_1$ vs. $\sigma$, illustrating the lower bound \ref{['ent_bound']} for randomly sampled parameters: $A_{11} \in [-20, -1]$, $A_{22} \in [-20, -1]$, $A_{12} \in [-10, 0]$, $\alpha_A \in [-50, 1]$, and $\alpha_T \in [0.1, 100]$. Color bar indicates $\alpha_T$, showing how increasing thermal bath asymmetry tightens the bound. b) Heatmap of $\sigma$ as a function of $F^{\rm max}$ and $S$. Numerical values in the heatmap correspond to the fraction $4{\cal T}_1/\sigma$, representing the proportion of $\sigma$ estimated by the bound.
• Figure S1: Plot of ${\cal V}^{\, 1}_{\Delta x}(t) = {\rm Var}(x^{1}_t-x^1_0) = 2 (C_{x}^{11}(0)-C_{x}^{11}(t))$ with $A_{11} = -1$, $A_{22}=-1$, $A_{12}=1$, $\alpha_A=0.5$, $T=1$ and $\alpha_T \in [1,500]$. Note the deviation from normal diffusion as $\mathcal{T}_1$ (and hence $\sigma$) increases, validating enhanced diffusion as a signature of nonequilibrium in overdamped systems.
• Figure S2: 10000 realizations of the 2D model in Eq. \ref{['SDE_linear_resp']} with randomly sampled parameters in the following intervals: $A_{11} \in (-1, 1)$, $A_{22} \in (-1, 1)$, $A_{12} \in (-1, 1)$, $A_{21} \in (-1, 1)$, $T_1 \in (0.1, 2)$, $T_{2} \in (0.1, 2)$.