Reasonable and robust Hamiltonians violating the Third Law

TL;DR

This paper characterizes a class of classical Hamiltonians in one dimension that violate the third law of thermodynamics, providing a geometric description of their couplings and demonstrating the physical reasonableness of a specific example.

Contribution

It defines the subvolume of such Hamiltonians using graph properties and presents a physically reasonable Hamiltonian within this class.

Findings

Identifies a wedge in parameter space where Hamiltonians violate the third law.

Provides a simple expression for a specific Hamiltonian H* within this wedge.

Shows that the coupling constants of H* decrease smoothly with interaction range.

Abstract

It has recently been shown that the third law of thermodynamics is violated by an entire class of classical Hamiltonians in one dimension, over a finite volume of coupling-constant space, assuming only that certain elementary symmetries are exact, and that the interactions are finite-ranged. However, until now, only the existence of such Hamiltonians was known, while almost nothing was known of the nature of the couplings. Here we show how to define the subvolume of these Hamiltonians---a `wedge' W in a d-dimensional space---in terms of simple properties of a directed graph. We then give a simple expression for a specific Hamiltonian H* in this wedge, and show that H* is a physically reasonable Hamiltonian, in the sense that its coupling constants lie within an envelope which decreases smoothly, as a function of the range l, to zero at l=r+1, where r is the range of the interaction.