Regular Article

Density distribution in two Ising systems with particle exchange

¹ Laboratoire de Physique et Chimie Théoriques, CNRS UMR 7019, Université de Lorraine, 54506 Vandoeuvre-lès-Nancy, France
² Department of Physics and Center for Theoretical Physics, Seoul National University, Seoul 151-747, Korea
³ Center for Computational Science and Social and Economic Engineering Initiative, Korea Institute of Science and Technology, Seoul 136-791, Korea
⁴ Department of Economics and Institute for Research in Finance & Economics, Seoul National University, Seoul 151-742, Korea

^a e-mail: jean-yves.fortin@univ-lorraine.fr
^b e-mail: mychoi@snu.ac.kr

Received: 26 January 2018
Received in final form: 29 August 2018
Published online: 10 December 2018

Abstract

Various physical and social systems are subject to exchanges of their constituent particles, in addition to usual energy exchanges or fluctuations. In this paper, we consider a system consisting of two Ising systems, a one-dimensional lattice (solid) and a fully connected system (gas) or reservoir (with constant fugacity), and exchanging particles between the two, and study the exact distribution of particles as a function of the internal couplings, temperature, and external field. Particles (with spins) in the gas can be adsorbed onto the one-dimensional lattice (corresponding to condensation) or desorbed back into the reservoir (evaporation). The distribution of the number of particles on the lattice is computed exactly and the thermodynamic limit is studied by means of the saddle-point analysis. It is found that the probability follows a cumulative Gumbel distribution, with the argument proportional to the free energy cost of removing one site.