# Perturbative Expansion of Irreversible Work in Fokker-Planck Equation a la Quantum Mechanics

## TL;DR

This paper introduces a perturbative method inspired by quantum mechanics to systematically expand and calculate irreversible work in Fokker-Planck systems, simplifying computations and enabling optimization of control protocols.

## Contribution

It develops a novel perturbative expansion approach for the Fokker-Planck equation, analogous to quantum adiabatic calculations, to efficiently compute irreversible work and optimize control protocols.

## Key findings

- First order expansion yields exact results for harmonic potentials.
- Method simplifies calculations by avoiding coupled differential equations.
- Optimized protocols minimize irreversible work effectively.

## Abstract

We discuss the systematic expansion of the solution of the Fokker-Planck equation with the help of the eigenfunctions of the time-dependent Fokker-Planck operator. The expansion parameter is the time derivative of the external parameter which controls the form of an external potential. Our expansion corresponds to the perturbative calculation of the adiabatic motion in quantum mechanics. With this method, we derive a new formula to calculate the irreversible work order by order, which is expressed as the expectation value with a pseudo density matrix. Applying this method to the case of the harmonic potential, we show that the first order term of the expansion gives the exact result. Because we do not need to solve the coupled differential equations of moments, our method simplifies the calculations of various functions such as the fluctuation of the irreversible work per unit time. We further investigate the exact optimized protocol to minimize the irreversible work by calculating its variation with respect to the control parameter itself.