A derivation (and quantification) of the third law of thermodynamics

TL;DR

This paper provides the first general derivation and quantification of the third law of thermodynamics, establishing fundamental limits on cooling processes and the unattainability of absolute zero temperature.

Contribution

It introduces a universal derivation of the unattainability principle applicable to quantum and classical systems, linking resource requirements to cooling limits.

Findings

Cooling temperature scales as an inverse power of time.

Positive heat capacity of the bath is essential for unattainability.

Finite-time perfect cooling is possible if heat capacity is negative.

Abstract

The third law of thermodynamics has a controversial past and a number of formulations due to Planck, Einstein, and Nernst. It's most accepted version, the unattainability principle, states that "any thermodynamic process cannot reach the temperature of absolute zero by a finite number of steps and within a finite time." Although formulated in 1912, there has been no general proof of the principle, and the only evidence we have for it is that particular cooling methods become less efficient as the temperature decreases. Here we provide the first derivation of a general unattainability principle, which 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 particular temperature, and translate these resources into a minimal time or number of steps by considering the notion of a Thermal Machine which obeys similar restrictions to universal computers. We generally find that the obtainable temperature can scale as an inverse power of the cooling time. Our argument relies on the heat capacity of the bath being positive, and we show that if this is not the case then perfect cooling in finite time is in principle possible. 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.