Testing ground for fluctuation theorems: The one-dimensional Ising model

C. G. O. Lemos, M. Santos, A. L. Ferreira, and W. Figueiredo
Phys. Rev. E 97, 042121 – Published 13 April 2018

Abstract

In this paper we determine the nonequilibrium magnetic work performed on a Ising model and relate it to the fluctuation theorem derived some years ago by Jarzynski. The basic idea behind this theorem is the relationship connecting the free energy difference between two thermodynamic states of a system and the average work performed by an external agent, in a finite time, through nonequilibrium paths between the same thermodynamic states. We test the validity of this theorem by considering the one-dimensional Ising model where the free energy is exactly determined as a function of temperature and magnetic field. We have found that the Jarzynski theorem remains valid for all the values of the rate of variation of the magnetic field applied to the system. We have also determined the probability distribution function for the work performed on the system for the forward and reverse processes and verified that predictions based on the Crooks relation are equally correct. We also propose a method to calculate the lag between the current state of the system and that of the equilibrium based on macroscopic variables. We have shown that the lag increases with the sweeping rate of the field at its final value for the reverse process, while it decreases in the case of the forward process. The lag increases linearly with the size of the chain and with a slope decreasing with the inverse of the rate of variation of the field.

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  • Received 5 February 2018

DOI:https://doi.org/10.1103/PhysRevE.97.042121

©2018 American Physical Society

Physics Subject Headings (PhySH)

Statistical Physics & Thermodynamics

Authors & Affiliations

C. G. O. Lemos* and M. Santos

  • Departamento de Física, Universidade Federal de Santa Catarina, Campus Universitário, Trindade, 88040-900, Florianópolis, Santa Catarina, Brazil

A. L. Ferreira

  • Departamento de Física, I3N, Universidade de Aveiro, 3810-193, Aveiro, Portugal

W. Figueiredo§

  • Departamento de Física, Universidade Federal de Santa Catarina, Campus Universitário, Trindade, 88040-900, Florianópolis, Santa Catarina, Brazil and Instituto de Física, Universidade de São Paulo, Rua do Matão, 1371, 05508-090 São Paulo, São Paulo, Brazil

  • *carlos.gentil@posgrad.ufsc.br
  • marcio.santos@ufsc.br
  • alf@ua.pt
  • §wagfig@gmail.com

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Issue

Vol. 97, Iss. 4 — April 2018

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