# Work needed to drive a thermodynamic system between two distributions

## TL;DR

This paper investigates the minimum work required to transition a thermodynamic system between two states within a specified time, highlighting the energy costs associated with changing energy, entropy, and position.

## Contribution

It introduces a framework to quantify the minimal work and time needed for state transitions in thermodynamic systems, including explicit solutions for specific cases.

## Key findings

- Minimum work relates to increasing internal energy, decreasing entropy, and shifting mean position.
- Explicit solutions demonstrate the trade-offs between work, time, and dissipation.
- Results provide insights into energy efficiency in thermodynamic processes.

## Abstract

In this study, the minimum amount of work needed to drive a thermodynamic system from one initial distribution to another in a given time duration is discussed. Equivalently, for given amount of work, the minimum time duration required to complete such a transition is obtained. Results show that the minimum amount of work is used to achieve the following three objectives, to increase the internal energy of the system, to decrease the system entropy, to change the mean position of the system, and with other nonzero part dissipated into environment. To illustrate the results, an example with explicit solutions is presented.