# Thermodynamic uncertainty relations in a linear system

## TL;DR

This paper derives thermodynamic uncertainty relations for a Brownian particle in harmonic confinement, considering both underdamped and overdamped regimes, using thermodynamic principles and correlation matrix properties.

## Contribution

It introduces a unified approach to derive thermodynamic uncertainty relations for both underdamped and overdamped harmonic systems.

## Key findings

- Derived uncertainty relations for particle position and current.
- Applicable to both underdamped and overdamped regimes.
- Utilizes second law and correlation matrix positivity.

## Abstract

We consider a Brownian particle in harmonic confinement of stiffness $k$, in one dimension in the underdamped regime. The whole setup is immersed in a heat bath at temperature $T$. The center of harmonic trap is dragged under any arbitrary protocol. The thermodynamic uncertainty relations for both position of the particle and current at time $t$ are obtained using the second law of thermodynamics as well as the positive semi-definite property of the correlation matrix of work and degrees of freedom of the system for both underdamped and overdamped cases.