Thermodynamic bounds and symmetries in first-passage problems of fluctuating currents

TL;DR

This work develops a unified framework for thermodynamic bounds on first-passage problems of fluctuating currents in Markov chains with two absorbing boundaries, linking dissipation to time-averaged and time-reversed statistics through a coarse-grained KL-divergence approach. A central advance is extending the effective affinity concept to discrete-time chains and deriving a refined bound that incorporates the full distribution of first-passage times, via the negative-threshold statistics $\mathcal{I}_-(1/\overline{j})$. The paper also reveals symmetry properties: optimal currents satisfy a speed symmetry between positive and negative thresholds, and a generalised symmetry holds for generic currents using a dual, time-reversed Doob-transformed process, with concrete demonstrations on low-dimensional Markov models. Overall, these results enable thermodynamic inference from first-passage data and provide tools for coarse-grained descriptions of nonequilibrium systems such as molecular motors.

Abstract

We develop a method for deriving thermodynamic bounds for first-passage problems of currents with two boundaries in Markov chains. Using this method, we derive a thermodynamic bound on the rate of dissipation in terms of the splitting probability and the first-passage time statistics of a fluctuating current, which is a refinement of a previously derived inequality. We also show that the concept of effective affinity, originally developed for continuous-time Markov chains, naturally extends to discrete-time Markov chains. Furthermore, we analyse symmetries in first-passage problems of fluctuating currents with two boundaries. We show that optimal currents -- those for which the effective affinity fully accounts for the dissipation -- satisfy a symmetry property: the current's average speed to reach the positive threshold equals the current's speed to reach the negative threshold. The developed approach uses a coarse-graining procedure for the average entropy production at random times and uses martingale methods to perform time-reversal of first-passage quantities.

Figures (8)

• Figure 1: Graphical illustration of the first passage problem Eq. \ref{['eq:T']}. Multiple trajectories of a fluctuating current $J(t)$ are plotted as a function of time until the first time $T$ when $J$ exits the interval $(-\ell_-,\ell_+)$ (middle panel). Trajectories terminating at $\ell_+$ are coloured in blue and those terminating at $\ell_-$ are coloured in red. One example trajectory for each case is highlighted in black. The thresholds $\ell_\pm$ are marked as brown dashed lines. The large deviation rate functions $\mathcal{I}_+(t/\ell_+)$ and $\mathcal{I}_-(t/\ell_-)$ of the scaled first passage time conditioned on terminating at $\ell_+$ or $-\ell_-$, respectively, are plotted above and below the plot (yellow, solid lines); note that the y-axis is inverted in the bottom panel. Furthermore, the corresponding conditioned first-passage time distributions $p_T(t|+)$ and $p_T(t|-)$ at both boundaries are shown (blue, solid lines). Data is from the two-dimensional random walker model described in Sec. \ref{['sec:2DRandomd']}, with parameters $\Delta=0.6$, $\rho=1$, $\nu=\ln(4/3)$, and thresholds $\ell_+=100$ and $\ell_-=27$.
• Figure 2: Illustrated application of the first-passage problem (\ref{['eq:T']}) for the case of a two-headed motor protein bounded to a biofilament. The motion of the motor is biased towards the filament's plus end. The fluctuating current $J(t)$ denotes the position of the center of mass of the motor along the filament. In this example two natural first-passage problems $T$ are for $\ell_-=\ell_+=a$, in which case $T$ is the motor dwell time, and $-\ell_-$ and $\ell_+$ set equal to the end points of the filament, in which case $T$ is the total duration of the motor's motion bound to the filament.
• Figure 3: A graphical illustration of the refined bound Eq. \ref{['eq:newboundIJform']} for the rate of dissipation $\dot{s}$. The scaled cumulant generating function, $\lambda_J(a)$, is plotted as a function of $a$ (blue solid line), and the tangents of $\lambda_J(a)$ at $a=0$ and $a=\tilde{a}$ are indicated (red solid lines). The right-hand side of Eq. \ref{['eq:newboundIJform']} is $\mathcal{I}_J(-\overline{j})$, which is equal to the Legendre transform of $\lambda_J(a)$ evaluated at $a=\tilde{a}$ where $\lambda_J'(\tilde{a}) = \overline{j}$. Instead, the right-hand side of the coarser bound \ref{['eq:sjas']} is given by $a^\ast \overline{j}$ with $\lambda_J(a^\ast)=0$. Note also that $\lambda_J(0)=0$ and $\lambda_J'(0)=-\overline{j}$. The two bounds are equivalent when $\tilde{a} = a^\ast$, which is equivalent with the symmetry \ref{['eq:weaksymm']}.
• Figure 4: Classification of currents according to symmetries and optimality. Relationships between sets of currents with different properties are shown (see Sec. \ref{['sec:symmetry']} for definitions). A star indicates a set for which example currents are known to exist, while a question mark denotes a set whose non-emptiness remains unknown.
• Figure 5: Diagram illustrating the effects of the Doob transform \ref{['eq:qhat']} and the time-reversal transformation \ref{['eq:qTimeRev']} on the scaled cumulant generating function $\lambda_J$ of the current $J$. (a) Plot of the scaled cumulant generating function $\lambda_J(a)$ in a Markov process described by the matrix $\mathbf{q}$. (b) Plot of the scaled cumulant generating function $\lambda^\dagger_J(a)$ in the time-reversed process described by the matrix $\mathbf{q}^\dagger$ (Eq.\ref{['eq:qTimeRev']}). (c) Plot of the scaled cumulant generating function $\hat{\lambda}_J(a)$ in the dual process described by the matrix $\mathbf{\hat{q}}$, which is the Doob-transform of $\mathbf{q}$ with tilting $a^\ast$ (Eq.\ref{['eq:qhat']}). (d) Plot of the scaled cumulant generating function $\hat{\lambda}^\dagger_J(a)$ in the conjugate process described by the matrix $\mathbf{\hat{q}^\dagger}$, which is the time-reversal of the dual process (Eq.\ref{['eq:qhatdagger']}). Note that the diagram is commutable, as one can arrive at the conjugate process (corresponding to the plot in panel (d)) from the original process (corresponding to the plot in panel (a)) by traversing the arrows clockwise or counter-clockwise. The values of the effective affinities $a^\ast$ and $a^{\ast,\dagger}$ in each case are also marked.