Counting observables in stochastic excursions

Abstract

Understanding fluctuations of observables across stochastic trajectories is essential for various fields of research, from quantum thermal machines to biological motors. We introduce a framework to analyze the statistics of counting observables in sub-trajectories$-$dubbed as stochastic excursions$-$of processes out of equilibrium. Given a partition of the state space into two sets $A$ and $B$, an excursion is defined as the segment of the trajectory that starts with a transition from $A$ to $B$ and ends upon the first return from $B$ to $A$. Our approach offers analytical expressions for the full distribution of counting observables (such as currents, heat, work, entropy production, and dynamical activity) and the excursion duration, capturing their correlations and finite-time fluctuations. As our main result, we uncover a nontrivial fundamental relation between fluctuations of counting observables at the single-excursion level and the steady state noise obtained from full counting statistics, offering a tool to inspect noise sources. We also show the existence of a fluctuation theorem and a thermodynamic uncertainty relation at the level of individual excursions. We discuss examples from distinct fields in which the excursion framework naturally addresses relevant questions, and explore in more detail how analyzing excursions yields additional insights into the operation of the three-qubit absorption refrigerator.

Figures (2)

• Figure 1: The general setup of stochastic excursions: (a) A system undergoes stochastic dynamics on a finite set of states that are split into two regions, $A$ (gray) and $B$ (green). An excursion starts with a transition $A\to B$ and ends with the first transition $B\to A$ (red arrows). The states in $A$ do not have to be connected or clustered, they can be spread out across the state space. Examples of excursions in relevant problems: (b) The salesman performs excursions leaving home and visiting cities. While questions related to time are interesting, a comprehensive characterization of the process requires counting observables such as the amount of money earned or distance traveled per excursion and their interplay with excursion times. (c) Quantum thermodynamic models such as heat engines are often be modeled as a classical Markov jump process. Excursions around the ground state provide information of cyclic heat transfer and work extraction through counting observables. (d) Cellular sensing is often modeled as a stochastic process where some states represent unbound receptors while others represent bound. Subsequent occurrences of unbound receptors mark excursions through bound states. Key quantities such as the environmental concentration of ligands are learned from the properties of such excursions. (e) If states in $B$ are hidden, trajectories within it form excursions; if duration and counting observables are empirically available at the end of excursions, their statistics contains information for inferring properties of $B$.
• Figure 2: (a) Energy diagram of the three qubit absorption refrigerator. Excursions from the ground state $|000\rangle$ can be classified as: success, where the cold current is $+1$; fail, where the cold current is zero; or disaster, there the cold current is $-1$. The different arrows show one example of each type. (b) Covariance between cold current and excursion duration as a function of $n_c$, for different values of $\Gamma_c$ with fixed $g = 10$. (c), (d) Diffusion coefficient and its decomposition for (c) entropy production, and (d) dynamical activity as a function of $g$, with $n_c = 0.1$ and $\Gamma_c = 0.1$. For all plots, we fixed $\Gamma_h = 50, \ \Gamma_w = 0.5, \ n_h = 0.005,\ n_w = 0.5$. Parameters are well in accordance with the parameter regime of aamirThermallyDrivenQuantum2025 and the classical approximation.