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Statistical Mechanics of a Simplified Bipartite Matching Problem: An Analytical Treatment

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Abstract

We perform an analytical study of a simplified bipartite matching problem in which there exists a constant matching energy, and both heterosexual and homosexual pairings are allowed. We obtain the partition function in a closed analytical form and we calculate the corresponding thermodynamic functions of this model. We conclude that the model is favored at high temperatures, for which the probabilities of heterosexual and homosexual pairs tend to become equal. In the limits of low and high temperatures, the system is extensive, however this property is lost in the general case. There exists a relation between the matching energies for which the system becomes more stable under external (thermal) perturbations. As the difference of energies between the two possible matches increases the system becomes more ordered, while the maximum of entropy is achieved when these energies are equal. In this limit, there is a first order phase transition between two phases with constant entropy.

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Acknowledgements

I thank M.M. Reynoso, G.G. Izús, S.E. Mangioni, and P.A. Sánchez their help and useful discussions.

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Authors and Affiliations

  1. Instituto de Investigaciones en Física de Mar del Plata, IFIMAR (CONICET-UNMdP), Funes 3350, B7602AYL, Mar del Plata, Argentina

    Matías Germán dell’Erba

  2. Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350, B7602AYL, Mar del Plata, Argentina

    Matías Germán dell’Erba

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  1. Matías Germán dell’Erba
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Correspondence to Matías Germán dell’Erba.

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dell’Erba, M.G. Statistical Mechanics of a Simplified Bipartite Matching Problem: An Analytical Treatment. J Stat Phys 146, 1263–1273 (2012). https://doi.org/10.1007/s10955-012-0447-4

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  • DOI: https://doi.org/10.1007/s10955-012-0447-4

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