Waiting-time based entropy estimators in continuous space without Markovian events

Abstract

Estimating entropy production in continuous systems that can only be observed with a limited resolution remains an open problem in stochastic thermodynamics. Extant estimators based on the measurement of waiting-time distributions require either the detection of Markovian events, which uniquely determine the state of the system, or assume a discrete underlying dynamics. We present a novel estimator that relies solely on the detection of a single particle leaving or entering regions, or crossing manifolds, in continuous space. This estimator is based on the frequency and the duration of transitions between such events. We derive this bound by introducing two kinds of discretization of space. Finally, we compare our novel bound to the TUR using simulations of a Brownian vortex and discuss its relation to other lower bounds to entropy production.

Figures (3)

• Figure 1: (a) Trajectory of an overdamped Langevin particle in a plane. The blue arrows show the steady state current at each point, given by the potential \ref{['eq:num_pot']} and non-conservative force \ref{['eq:num_force']}. Whenever the particle is in one of the colored areas, $A$ or $B$, or crosses the colored positive $y$-axis, the observer detects a signal which reveals which area the particle is in or in which direction it has crossed the positive $y$-axis, i.e., $C_+$ or $C_-$. (b) The resulting measurement. From such an infinitely long measurement, the frequency of transitions between $A_\pm,B_\pm$ and $C_\pm$ that take a certain time (like $t_{A_+ B_-}$) can be reconstructed.
• Figure 2: Two types of discretization. (a) Discretizing the distance to the manifold. In this one-dimensional model, the distance to the manifold needs to be discretized in order for the waiting-time distribution from $A$ to $B$ to be well defined. This scheme allows for the identification of $\nu^{\epsilon}_{A_+}$ as flux between the discretized states and the expansion of rates \ref{['eq:approx_rates_1d']}. (b) Discretizing the manifold itself. From this scheme, the scaling behavior of $\psi^{\epsilon}_{i_s\to j_r}(t)$ as $\epsilon\to 0$ can be deduced. The probability of choosing a geometric, i.e., undirected path that reaches a specific segment $r$ on $J$ scales as $\epsilon^{d-1}$. Starting a distance $\epsilon$ away from the initial manifold, the particle can move forward and backwards along the specific geometric path marked in purple, as indicated by the black arrows. Thus, reabsorption at the initial manifold remains possible. The probability of reaching $J$ from $I$ given such a one-dimensional geometric path scales as $\epsilon$.
• Figure 3: Quality factor of different estimators in the Brownian vortex model. (a) Quality factor as a function of the distance of the center of the two observable regions from the $x$-axis, $d_{\mathrm{off}}$, cf. \ref{['fig:kartoffelwurm']} (a). There is an optimal intermediate offset for which the quality factor $\mathcal{Q}_{\text{AB}}$ reaches its maximum. (b) Quality factor as a function of driving strength $\gamma$. All three lower bounds perform better close to equilibrium. Parameters used are $d_{AB}=2$, $r_A=r_B=0.75$, $T=6\times10^7$, $T_{\mathrm{TUR}}=100$, $\Delta t=10^{-2}$.