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Thermodynamics of Small Magnetic Particles

Department of Physics, Universidad de La Frontera, Francisco Salazar 01145, Temuco, Chile
Center for the Development of Nanoscience and Nanotechnology, 9170124 Santiago, Chile
Department of Physics, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile
Departamento de Física, Instituto de Física Aplicada, Universidad Nacional de San Luis-CONICET, Ejército de Los Andes 950, San Luis D5700BWS, Argentina
Author to whom correspondence should be addressed.
Entropy 2017, 19(9), 499;
Received: 2 August 2017 / Revised: 1 September 2017 / Accepted: 13 September 2017 / Published: 15 September 2017
(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)
PDF [568 KB, uploaded 15 September 2017]


In the present paper, we discuss the interpretation of some of the results of the thermodynamics in the case of very small systems. Most of the usual statistical physics is done for systems with a huge number of elements in what is called the thermodynamic limit, but not all of the approximations done for those conditions can be extended to all properties in the case of objects with less than a thousand elements. The starting point is the Ising model in two dimensions (2D) where an analytic solution exits, which allows validating the numerical techniques used in the present article. From there on, we introduce several variations bearing in mind the small systems such as the nanoscopic or even subnanoscopic particles, which are nowadays produced for several applications. Magnetization is the main property investigated aimed for two singular possible devices. The size of the systems (number of magnetic sites) is decreased so as to appreciate the departure from the results valid in the thermodynamic limit; periodic boundary conditions are eliminated to approach the reality of small particles; 1D, 2D and 3D systems are examined to appreciate the differences established by dimensionality is this small world; upon diluting the lattices, the effect of coordination number (bonding) is also explored; since the 2D Ising model is equivalent to the clock model with q = 2 degrees of freedom, we combine previous results with the supplementary degrees of freedom coming from the variation of q up to q = 20 . Most of the previous results are numeric; however, for the case of a very small system, we obtain the exact partition function to compare with the conclusions coming from our numerical results. Conclusions can be summarized in the following way: the laws of thermodynamics remain the same, but the interpretation of the results, averages and numerical treatments need special care for systems with less than about a thousand constituents, and this might need to be adapted for different properties or devices. View Full-Text
Keywords: small systems; thermodynamics; magnetization; Ising model; Potts model; dilution small systems; thermodynamics; magnetization; Ising model; Potts model; dilution

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Vogel, E.E.; Vargas, P.; Saravia, G.; Valdes, J.; Ramirez-Pastor, A.J.; Centres, P.M. Thermodynamics of Small Magnetic Particles. Entropy 2017, 19, 499.

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