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Probability distributions of Hamiltonian changes in linear magnetic systems under discontinuous perturbations

H. Híjar, J. Quintana, G. Sutmann.

Journal of Statistical Mechanics: Theory and Experiment (JSTAT), 2008, P05009, (2008)

Abstract
A model for the stochastic evolution of a linear paramagnetic system in contact with a thermal bath and subjected to variations in time of an external magnetic field, H, is presented. Changes in the Hamiltonian of this system, HH, defined through the relation W = ∫ dH, (∂HH/∂H), are considered for the special case in which the external field is varied from an initial to a final value in discontinuous successive steps. Distribution functions of W, PF(W) and PR{W} , corresponding to switching on and switching off processes, respectively, are explicitly calculated. To study the relaxation process, it is assumed that in between successive variations of the external field, the system follows a linear Langevin dynamics in the presence of a constant field. The dependences of PF(W) and PR{W} on the number of steps in which the external field changes from initial to final values as well as on the time rate of the processes are presented. These distributions are used to estimate free energy differences between two equilibrium states via the relations of Jarzynski and Crooks, and it is verified that these relations yield correct estimations of the free energy difference. Results of the model are illustrated firstly, for a system of independent two-level spins in the thermodynamic limit. Finally, a comparison is also performed with distributions of W in the two-dimensional Ising model obtained independently from numerical simulations, carried out in the paramagnetic phase and restricted to perturbations that produce a linear response. A good agreement is found between simulations and the present calculations.


Keyword(s): exact results; probability theory
DOI: 10.1088/1742-5468/2008/05/P05009
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