# Thermodynamic Cost for Classical Counterdiabatic Driving

## TL;DR

This paper investigates the thermodynamic cost of classical counterdiabatic driving, establishing a universal trade-off between speed and cost, and introduces new cost definitions relevant when excess work vanishes, with applications to harmonic oscillators.

## Contribution

It provides a new complementary definition of thermodynamic cost for classical systems and demonstrates the classical counterpart of quantum results on counterdiabatic driving.

## Key findings

- A universal speed-cost trade-off for classical counterdiabatic driving.
- New cost measure relevant when average excess work is zero.
- Application to a time-dependent harmonic oscillator.

## Abstract

Motivated by the recent growing interest about the thermodynamic cost of Shortcuts to Adiabaticity (STA), we consider the cost of driving a classical system by the so-called Counterdiabatic Driving (CD). To do so, we proceed in three steps: first we review a general definition recently put forward in the literature for the thermodynamic cost of driving a Hamiltonian system; then we provide a new complementary definition of cost, which is of particular relevance for cases where the average excess work vanishes; finally, we apply our general framework to the case of CD. Interestingly, we find that in such case our results are the exact classical counterparts of those reported in Funo et al. PRL, 118(10):100602, 2017. In particular we show that a universal trade-off between speed and cost for CD also exists in the classical case. To illustrate our points we consider the example of a time-dependent harmonic oscillator subject to different strategies of adiabatic control.