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Non-equilibrium symmetry of cyclic first-passage times

Daniel Maria Busiello, Shiling Liang, Simone Pigolotti

TL;DR

The paper addresses cyclic first-passage times along a cycle of N>2 states in a stationary stochastic system, showing that at equilibrium the clockwise and counterclockwise cyclic times share the same distribution, while out of equilibrium they obey a detailed fluctuation theorem. The main method reveals a trajectory-level symmetry by combining time reversal with a cutting/pasting operation, yielding p_CW(τ,s)/p_CCW(τ,-s) = e^{s} and connecting cycle timing to entropy production via large-deviation theory. Key contributions include a general framework for CW/CCW cyclic FPTs on arbitrary cycles, a rigorous detailed fluctuation theorem, and explicit expressions for the mean, variance, and third cumulant of the entropy-production rate; these are validated in an enzymatic-cycle model. The results introduce a novel nonequilibrium symmetry in stochastic systems and offer a practical route to infer dissipation from partial trajectory data, with potential broad applicability to biological cyclic processes and experimental diagnostics.

Abstract

We study the sum of first passage times along an arbitrary cycle made up of N>2 states of a small physical system. We show that, if the system is at thermodynamic equilibrium, this sum follows the same probability distribution regardless of whether the cycle is explored clockwise or counterclockwise. Out of equilibrium, the distributions of clockwise and counterclockwise cyclic first passage times are related by a detailed fluctuation theorem. This result descends from a symmetry of clockwise and counterclockwise trajectories, which combines time reversal with swapping portions of the trajectories. We then relate the entropy produced along the cycle with the entropy production of the whole system using large deviation theory. Our results reveal a novel symmetry in stochastic systems, of potential broad applicability in non-equilibrium physics.

Non-equilibrium symmetry of cyclic first-passage times

TL;DR

The paper addresses cyclic first-passage times along a cycle of N>2 states in a stationary stochastic system, showing that at equilibrium the clockwise and counterclockwise cyclic times share the same distribution, while out of equilibrium they obey a detailed fluctuation theorem. The main method reveals a trajectory-level symmetry by combining time reversal with a cutting/pasting operation, yielding p_CW(τ,s)/p_CCW(τ,-s) = e^{s} and connecting cycle timing to entropy production via large-deviation theory. Key contributions include a general framework for CW/CCW cyclic FPTs on arbitrary cycles, a rigorous detailed fluctuation theorem, and explicit expressions for the mean, variance, and third cumulant of the entropy-production rate; these are validated in an enzymatic-cycle model. The results introduce a novel nonequilibrium symmetry in stochastic systems and offer a practical route to infer dissipation from partial trajectory data, with potential broad applicability to biological cyclic processes and experimental diagnostics.

Abstract

We study the sum of first passage times along an arbitrary cycle made up of N>2 states of a small physical system. We show that, if the system is at thermodynamic equilibrium, this sum follows the same probability distribution regardless of whether the cycle is explored clockwise or counterclockwise. Out of equilibrium, the distributions of clockwise and counterclockwise cyclic first passage times are related by a detailed fluctuation theorem. This result descends from a symmetry of clockwise and counterclockwise trajectories, which combines time reversal with swapping portions of the trajectories. We then relate the entropy produced along the cycle with the entropy production of the whole system using large deviation theory. Our results reveal a novel symmetry in stochastic systems, of potential broad applicability in non-equilibrium physics.
Paper Structure (7 sections, 40 equations, 3 figures)

This paper contains 7 sections, 40 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Detailed fluctuation theorem for cyclic first-passage times
  3. The symmetry leading to the detailed fluctuation theorem
  4. Cyclic first-passage times and entropy production rate
  5. Example: enzymatic cycle
  6. Discussion
  7. Alternative derivation of the detailed fluctuation theorem

Figures (3)

  • Figure 1: Clockwise (CW) and counterclockwise (CCW) trajectories over a set of three visible states, $x_0$, $x_1$, and $x_2$, among many hidden states in the system. In the right representation, the CW (top) and CCW (bottom) trajectories are highlighted.
  • Figure 2: Equivalence class of cyclic trajectories in six examples of increasing complexity. (a) Example of a trajectory including a single minimal cycle. The two $x_0$s whose cuts lead to a ${\rm CW}$ and a $\overline{\rm CCW}$ trajectory are marked with arrows. The five trajectories associated with all possible cuts are shown in the right, alongside the position of the minimal cycle. The function $\phi(x)$ associated with the equivalence class is also shown. (b) Two overlapping minimal cycles are present and two different cuts lead to a CW and a $\overline{\rm CCW}$ trajectory. (c-d) Two cases in which none of the cuts lead to either ${\rm CW}$ or $\overline{\rm CCW}$ trajectories, due to the presence of multiple non-overlapping minimal cycles. (e-f) Examples in which multiple ${\rm CW}$ and $\overline{\rm CCW}$ trajectories are present in the same equivalence class. In example (f), one of the trajectories belongs to both ${\rm CW}$ and $\overline{\rm CCW}$. In all examples, the number of ${\rm CW}$ trajectories in each equivalence class is equal to the number of $\overline{\rm CCW}$ trajectories.
  • Figure 3: (a) Enzymatic cycle in which two substrates, $S_1$ and $S_2$, are converted into a product, $P$. Substrates and product concentrations are chemostatted. Only substrate-bound states are visible (not shaded) and the system is externally driven to cycle clockwise (red arrows). (b) Distribution of the first passage time associated with clockwise ($ES_1-ES_1S_2-ES_2-ES_1$) and counterclockwise ($ES_1-ES_2-ES_1S_2-ES_1$) visible cycles. (c) Numerical validation of the fluctuation theorem in Eq. \ref{['eq:detailed']}. The clockwise distribution is recovered from the counterclockwise statistics upon integration over all possible values of the entropy production. (d) Mean and variance of the entropy production rate estimated from the large-deviation argument in Eq. \ref{['eq:res1']} (applied to clockwise and counterclockwise cycles) coincide with their corresponding theoretical values. Here we denote by $\langle \cdot \rangle_{\rm CW}$ an average performed over the distribution of clockwise cycles (and similarly for $\langle \cdot \rangle_{\rm CCW}$), while $\langle \cdot \rangle_{\rm th}$ indicates the corresponding theoretical expression. (e) The same comparison is shown for the third central moment. Numerical values have been obtained from $20$ different realizations for each cycle direction (clockwise and counterclockwise), all reported in the figure. Energies ($\epsilon_i$) and energetic barriers ($B_{ij}$) are: $\epsilon_E = 0, \epsilon_{ES_1} = 0.023, \epsilon_{ES_1S_2} = 0.20, \epsilon_{ES_2} = 0.023, \epsilon_{EP} = 0.19, B_{E,ES_1} = 0.24, B_{E,ES_2} = 0.28, B_{E,EP} = 0.27, B_{ES_1,ES_1S_2} = 0.28, B_{ES_2,ES_1S_2} = 0.23, B_{ES_1S_2,EP} = 0.26$. We take $k_B T = 1$, $[S_1] = [S_2] = [P] = 1$, and the nonequilibrium driving equal to $\Delta\mu = 2$ in panels (a-c).