Entropy of mixing exists only for classical and quantum-like theories among the regular polygon theories

TL;DR

This paper investigates the concept of thermodynamical entropy of mixing within generalized probabilistic theories, showing it exists only in classical and quantum-like theories among regular polygon theories, which are intermediate GPTs.

Contribution

It demonstrates that thermodynamical entropy of mixing is only well-defined in classical and quantum-like theories within the regular polygon GPT framework, revealing limitations of entropy in intermediate theories.

Findings

Entropy of mixing exists only in classical and quantum theories among regular polygon theories.

Regular polygon theories do not support a natural thermodynamical entropy of mixing.

The result highlights the special status of classical and quantum theories in thermodynamics.

Abstract

The thermodynamical entropy of a system which consists of different kinds of ideal gases is known to be defined successfully in the case when the differences are described by classical or quantum theory. Since these theories are special examples in the framework of generalized probabilistic theories (GPTs), it is natural to generalize the notion of thermodynamical entropy to systems where the internal degrees of particles are described by other possible theories. In this paper, we consider thermodynamical entropy of mixing in a specific series of theories of GPTs called the regular polygon theories, which can be regarded from a geometrical perspective as intermediate theories between a classical trit and a quantum bit with real coefficients. We prove that the operationally natural thermodynamical entropy of mixing does not exist in those inbetween theories, that is, the existence of the natural entropy results in classical and quantum-like theories among the regular polygon theories.