Statistical system based on p-adic numbers

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 degree of freedom is unity, there might occur the phase transition.