A general derivation and quantification of the third law of thermodynamics

Lluís Masanes,Jonathan Oppenheim

doi:10.1038/ncomms14538

Abstract

The most accepted version of the third law of thermodynamics, the unattainability principle, states that any process cannot reach absolute zero temperature in a finite number of steps and within a finite time. Here, we provide a derivation of the principle that applies to arbitrary cooling processes, even those exploiting the laws of quantum mechanics or involving an infinite-dimensional reservoir. We quantify the resources needed to cool a system to any temperature, and translate these resources into the minimal time or number of steps, by considering the notion of a thermal machine that obeys similar restrictions to universal computers. We generally find that the obtainable temperature can scale as an inverse power of the cooling time. Our results also clarify the connection between two versions of the third law (the unattainability principle and the heat theorem), and place ultimate bounds on the speed at which information can be erased.

Highlights

The most accepted version of the third law of thermodynamics, the unattainability principle, states that any process cannot reach absolute zero temperature in a finite number of steps and within a finite time
Walther Nernst’s first formulation of the third law of thermodynamics[1], called the heat theorem, was the subject of intense discussion[2]. Nernst claimed that he could prove his heat theorem using thermodynamical arguments while Einstein, who refuted several versions of Nernst’s attempted derivation, was convinced that classical thermodynamics was not sufficient for a proof, and that quantum theory had to be taken into account
One can potentially cool at a faster rate in systems violating the heat theorem, we show that the unattainability principle still holds

Summary

Introduction

The most accepted version of the third law of thermodynamics, the unattainability principle, states that any process cannot reach absolute zero temperature in a finite number of steps and within a finite time. Our results clarify the connection between two versions of the third law (the unattainability principle and the heat theorem), and place ultimate bounds on the speed at which information can be erased. Since we understand the entropy at zero temperature to be the logarithm of the ground-state degeneracy, the validity of the heat theorem is contingent on whether the degeneracy changes for different parameters of the Hamiltonian. The bound we obtain quantifies the extent to which a change in entropy at T 1⁄4 0 affects the cooling rate Of this debate, the validity of the unattainability principle has remained open. Strict unattainability in the sense of Nernst is not really a physically meaningful

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