# Fluctuation Relations Associated to an Arbitrary Bijection in Path Space

## TL;DR

This paper introduces a new framework for identifying fluctuation relations in physical systems using invertible transformations in trajectory space.

## Contribution

The paper provides a general method to derive fluctuation relations from arbitrary invertible transformations, extending beyond involutions.

## Key findings

- Fluctuation relations can be derived from invertible transformations in trajectory space, such as spatial rotations and translations.
- The framework recovers known isometric fluctuation relations and offers a recipe to discover new ones.
- Sufficient conditions for fluctuation relations are established for both finite and asymptotically large times.

## 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 relations to hold, considering either finite time or asymptotically large times.