Statistical System based on $p$-adic numbers

TL;DR

This paper introduces a novel statistical system based on $p$-adic numbers, demonstrating that phase transition phenomena can occur even with a single degree of freedom due to the $p$-adic structure.

Contribution

It presents a new framework for statistical mechanics using $p$-adic numbers, showing phase transitions in simple systems with finite degrees of freedom.

Findings

Phase transition observed in a $p$-adic based system with one degree of freedom.

Thermodynamical quantities like free energy and entropy are computed.

$p$-adic structure enables phase transitions without infinite degrees of freedom.

Abstract

We propose statistical systems based on $p$-adic numbers. In the systems, the Hamiltonian is a standard real number which is given by a map from the $p$-adic numbers. Therefore we can introduce the temperature as a real number and calculate the thermodynamical quantities like free energy, thermodynamical energy, entropy, specific heat, etc. Although we consider a very simple system, which corresponds to a free particle moving in one dimensional space, we find that there appear the behaviors like phase transition in the system. Usually in order that a phase transition occurs, we need a system with an infinite number of degrees of freedom but in the system where the dynamical variable is given by $p$-adic number, even if the degree of the freedom is unity, there might occur the phase transition.