Cycle counts and affinities in stochastic models of nonequilibrium systems

Patrick Pietzonka, Jules Guioth, and Robert L. Jack

Phys. Rev. E 104, 064137 – Published 27 December 2021

Abstract

For nonequilibrium systems described by finite Markov processes, we consider the number of times that a system traverses a cyclic sequence of states (a cycle). The joint distribution of the number of forward and backward instances of any given cycle is described by universal formulas which depend on the cycle affinity, but are otherwise independent of system details. We discuss the similarities and differences of this result to fluctuation theorems, and generalize the result to families of cycles, relevant under coarse graining. Finally, we describe the application of large deviation theory to this cycle-counting problem.

Received 2 August 2021
Accepted 8 December 2021

DOI:https://doi.org/10.1103/PhysRevE.104.064137

Physics Subject Headings (PhySH)

Fluctuation theorems Stochastic processes

Large deviation & rare event statistics

Statistical Physics & Thermodynamics

Authors & Affiliations

Patrick Pietzonka ¹, Jules Guioth ^1,2, and Robert L. Jack ^1,3

¹Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
²Univ. Lyon, ÉNS de Lyon, Univ. Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France
³Yusuf Hamied Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom

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Vol. 104, Iss. 6 — December 2021

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Images

Figure 1
(a) Diagram showing a simple system of four states, with transitions indicated by straight arrows. The cycle $C = A B C A$ is also indicated. (b) Example trajectory for a time period $[0, τ]$ , with jumps between states at times $t_{X} = (t_{1}, t_{2}, t_{3}, t_{4})$ . The sequence of states is $x_{X} = (D, A, B, C, A)$ so $n_{C} (X) = 1$ and $n_{C}^{R} (X) = 0$ .
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Figure 2
(Top) A trajectory $X$ of the four-state system in Fig. 1, and the corresponding sequence of completions of the cycle $C$ or $C^{R}$ . We keep track of the departure and arrival times for state $A$ that enclose instances of $C$ or $C^{R}$ . The second such instance is highlighted in red. (Bottom) Applying time reversal to the highlighted part of the trajectory leads to the trajectory ${\tilde{X}}_{2}$ , transforming the cycle $C^{R}$ into $C$ .
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Figure 3
A network where $B$ and $B^{'}$ form a pair of substates. If the physical driving mechanism does not distinguish between these substates, variations of the cycle $C$ visiting either $B$ or $B^{'}$ all have the same affinity and can be lumped together in a single family of cycles.
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Figure 4
SCGF $Ψ (s, λ)$ (color coded) for the current and traffic of the cycle $C = A B C A$ in Fig. 1. Panels (a) and (b) differ in the choice of rates, but the affinity $A_{C} = 3$ is fixed. Lines of constant value of $s + g$ are shown in solid white. They prescribe the overall shape of the SCGF, including the symmetry with respect to $λ = - A_{C} / 2$ (dashed white line) corresponding to the fluctuation symmetry (27). Parameters: (a) $w (A \to B) = w (B \to C) = w (C \to A) = 0.5, w (C \to A) = w (D \to A) = 2, w (B \to A) = w (A \to C) = 0.5 e^{- 2}, w (C \to B) = 0.5 e, w (D \to C) = w (A \to D) = 2 e^{2}$ ; (b) $w (A \to B) = w (B \to C) = w (C \to A) = 1, w (C \to D) = w (D \to A) = w (A \to D) = 0.1, w (B \to A) = w (C \to B) = w (A \to C) = e^{- 1}, w (D \to C) = 0.1 e^{2}$ .
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Figure 5
A trajectory $X$ of the four-state system in Fig. 1, and the corresponding reduced trajectory $Y$ . The reduced trajectory keeps track of the arrival and departure times for state $A$ , and on whether excursions from $A$ are instances of $C$ or $C^{R}$ or some other cycle (indicated as $O$ ). The initial and final parts of the trajectory consist of generic words (indicated by $W$ ). This reduced trajectory has $S_{Y} = (C, O, C^{R}, C)$ .
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Figure 6
Transitions in the extended state space for counting completions of the cycles $C$ and $C^{R}$ for the network of Fig. 1. Transitions within the original state space $Γ$ (circled states) are shown in black. Additional states (boxed) are the partial completions of cycles $C$ (solid) and $C^{R}$ (dashed). Yellow arrows indicate attempted completions of cycles. These attempts may fail (via the transitions shown in red) or lead to a successful completion of a cycle (via the transitions shown in green).
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