# Unifying Thermodynamic Uncertainty Relations

## TL;DR

This paper presents a new geometric approach to derive and strengthen thermodynamic uncertainty relations, broadening their applicability and achieving optimal bounds without complex probabilistic techniques.

## Contribution

It introduces a unifying geometric method to generalize and improve TURs, including a new optimal bound based on entropy production and a novel bound for stationary Markov processes.

## Key findings

- Derived a generalized TUR using Euclidean geometry
- Established a new optimal TUR based on entropy production
- Proved bounds for stationary Markov processes surpassing previous results

## Abstract

We introduce a new technique to bound the fluctuations exhibited by a physical system, based on the Euclidean geometry of the space of observables. Through a simple unifying argument, we derive a sweeping generalization of so-called Thermodynamic Uncertainty Relations (TURs). We not only strengthen the bounds but extend their realm of applicability and in many cases prove their optimality, without resorting to large deviation theory or information-theoretic techniques. In particular, we find the best TUR based on entropy production alone and also derive a novel bound for stationary Markov processes, which surpasses previous known bounds. Our results derive from the non-invariance of the system under a symmetry which can be other than time reversal and thus open a wide new spectrum of applications.