Fluctuation Relations associated to an arbitrary bijection in path space

TL;DR

The paper develops a unifying framework for fluctuation relations in stochastic dynamics driven by a trajectory-space bijection $R$ that need not be an involution, defined via a vector-valued entropic functional for observables $A_T$. It shows how a covariance structure $\Sigma_T^{\mathbb{P}} = \langle w, A_T\rangle$ together with $A_T(R(X)) = \mathcal{R}A_T(X)$ yields finite-time FRs for vector observables and, under large-deviation principles, asymptotic FRs that generalize isometric/spatial relations. The authors apply the framework to canonical path probabilities and non-degenerate diffusions, presenting explicit results for time-homogeneous semi-Markov processes, Langevin dynamics with harmonic potentials and rotations in momentum space, and diffusions with time-local spatial transformations, including multiplicative noise. These results extend fluctuation relations beyond time-reversal, accommodate non-Markovian and non-isotropic settings, and provide concrete computable expressions for cumulant generating functions and rate functions, offering a broad toolkit for probing nonequilibrium statistics. The work broadens the applicability of FRs and furnishes practical recipes to derive new relations in diverse stochastic systems.

Abstract

We introduce a framework to identify Fluctuation Relations for vector-valued observables in physical systems evolving through a stochastic dynamics. These relations arise from the particular structure of a suitable entropic functional and are induced by transformations in trajectory space that are invertible but are not involutions, typical examples being spatial rotations and translations. In doing so, we recover as particular cases results known in the literature as isometric fluctuation relations or spatial fluctuation relations and moreover we provide a recipe to find new ones. We mainly discuss two case studies, namely stochastic processes described by a canonical path probability and non degenerate diffusion processes. In both cases we provide sufficient conditions for the fluctuation relation to hold, considering either finite time or asymptotically large times.

Figures (1)

• Figure 1: (i) Level curves of the function $\Lambda_0(k)$, \ref{['eq:lambdaSemiMarkov']}, for $p=0.7, q=0.3,\lambda=1$ and $n=2$. The curve on the left bottom corresponds to $\Lambda_0=-0.5$ and the subsequent ones correspond to increments of $0.3$. (ii) Level curves of the function $I_0(a)$, \ref{['eq:ratefunctionSemiMarkov']}, for $p=0.7, q=0.3,\lambda=1$ and $n=2$. The closed curve on the bottom left corresponds to $I_0=0.1$ and the other curves are related to increments of $0.4$. (iii) Rate function $I_0$ for $a_1=a_2=a$ and different values of $n$. As before we set $p=0.7,q=0.3,\lambda=1$. One can see the non-monotonic behavior with respect to $n$ for small $a$, like for instance around $a=0.2$.

Theorems & Definitions (12)

• Lemma 2.1: Mother fluctuation relation
• proof
• Theorem 4.2: Finite-time FRs
• proof
• Theorem 4.5: Asymptotic FRs
• proof
• Corollary 5.2: Finite-time FRs for Canonical Processes
• Corollary 5.3: Asymptotic FRs for Canonical Processes
• Lemma 6.2: Modified diffusion
• proof