Using precision coefficients on recurrence times and integrated currents to lower bound the average dissipation rate

TL;DR

This paper extends the Thermodynamic Uncertainty Relation (TUR) by incorporating the statistics of recurrence times for forward and backward transitions along an observable channel in a continuous-time Markov jump process. Using a transition-based framework, it introduces an effective affinity $\mathcal{A}$ and defines a refined bound on the stationary entropy production rate $\sigma^{\rm ss}$ in terms of the long-time precision $\mathcal{T}_\infty$ of the integrated current and the recurrence-time precisions, yielding $\sigma^{\rm ss} \geq 2J \coth^{-1}\left(J\mathcal{T}_\infty - \Delta \mathcal{P}^\tau\right)$. This augmented TUR saturates for unicyclic networks and remains tight far from equilibrium, outperforming the standard TUR in many regimes and enabling dissipation inference from observable currents and recurrence statistics, with potential applications to nanoscale molecular machines and biochemical sensing networks. Practically, the framework connects experimentally accessible fluctuations to a fundamental energetic cost, offering design principles for optimizing precision versus dissipation in nonequilibrium processes.

Abstract

For continuous-time Markov jump processes on irreducible networks and time-independent rate constants, we employ a transition-based formalism to express the long-time precision of a single integrated current over an observable channel in terms of precisions of the recurrence times of the forward and backward jumps, and of an effective affinity that captures the thermodynamic driving on that channel. This leads to a general lower bound for the stationary entropy production rate that extends the well-known Thermodynamic Uncertainty Relation (TUR). Such an augmented TUR, which incorporates the statistics of the recurrences, proves to be tighter than the standard one far from equilibrium, and potentially offers new opportunities for the optimization and design of biological and chemical out-of-equilibrium systems at the nanoscale.

Figures (3)

• Figure 1: (a) An example of Markov jump process where only bidirectional transitions along the channel connecting states $\alpha$ and $\beta$ are observable. The shaded part thus represents the hidden part of the network. In this example, without loss of generality, only one channel $\alpha\to\beta$ is present; however, multiple channels connecting $\alpha$ and $\beta$ may exist, and they can be incorporated in the hidden part of the network, or can be lumped together in a single observable channel. Additionally, we assume that each hidden channel is bidirectional, so that the entropy production rate $\sigma^{\rm ss}$ is well defined schnakenberg1976network and our results can be compared with the standard TUR Eq. \ref{['eq:TUR']}. (b) Network of the effective process in the space of transitions. The sequence of transitions $\ell \in \lbrace \uparrow,\downarrow\rbrace$, with $\uparrow:\alpha \to \beta$ and $\downarrow:\beta \to \alpha$, is generated from the discrete-time process with transition matrix $\boldsymbol{\rm P}$ [see Eq. \ref{['eq:trans-transition']}]. (c) A typical realization of the process $\mathcal{N}_t$ where only $\uparrow$ and $\downarrow$ are observed. Specifically, a transition $\uparrow$ increases the integrated current $\mathcal{N}_t$ by 1, whereas a transition $\downarrow$ decreases the current by 1. Here, we also indicate the inter-transition times $\tau_{\ell'\ell}$ between successive occurrences of transitions $\ell'$ and $\ell$, and the recurrence times $\tau_{\ell}$ of transitions $\ell \in\lbrace \uparrow,\downarrow\rbrace$, which are blind to consecutive occurrences of the inverse transition $\bar{\ell}$.
• Figure 2: Plot of $J$ against $\Delta \mathcal{P}^\tau$ for $10^6$ randomly generated instances of the network in Fig. \ref{['fig:setup']}(a). The rate constants were drawn as $k=10^X$, where $X$ is uniformly distributed in the interval $[-3,0)$. Notably, there exist non-vanishing values of $\Delta \mathcal{P}^\tau$ for $J = 0$, which is associated to dissipation occurring in the hidden part of the network. As an example (in the inset), we generated several instances ($10^3$) of the same network where the $\alpha$ and $\beta$ are now disconnected, with uniformly distributed rates between 0.1 and 1, so that $\boldsymbol{\rm p}^{\rm stall}$ is the stationary (stalling) probability of the process. $k_{\alpha\beta}$ and $k_{\alpha\beta}$ are then re-introduced from $k_{\alpha\beta} = k_{\beta\alpha} {\rm p}_\beta^{\rm stall}/{\rm p}_\alpha^{\rm stall}$polettini2017effectivepolettini2019effective, with $k_{\beta\alpha}$ uniformly distributed between 0.1 and 1, thereby ensuring that $J=0$.
• Figure 3: Comparison between lower bounds for the entropy production rate $\sigma^{\rm ss}$ obtained from the long-time precision coefficient of a single integrated current: standard TUR [Eq. \ref{['eq:TUR']}] (circles) vs. asymptotic bound [Eq. \ref{['eq:asymptotic-bound']}] (crosses). The points are obtained from the same instances as in Fig. \ref{['fig:deltaP']}. The values of $\sigma^{\rm ss}$ are divided into equally sized intervals. Then, for each interval, we pick the minimum value of the difference $\sigma^{\rm ss} - \hat{\sigma}$, for $\hat{\sigma} = \sigma^{\rm TUR},\sigma^\ell$, with $\sigma^\ell$ corresponding to the right-hand side of Eq. \ref{['eq:asymptotic-bound']}. Each panel represents a different choice of observable edge for the same network, highlighted by the bidirectional arrow in the insets. This plot shows that while the TUR does not reach saturation as $\sigma^{\rm ss}$ grows, the bound Eq. \ref{['eq:asymptotic-bound']} consistently saturates in a wide interval. The irregularities at very high $\sigma^{\rm ss}$ are due to statistical errors given by the limited amount of data falling in that region (less than 10 networks per interval, shaded region).