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Derivation of some new distributions in statistical mechanics using maximum entropy approach

The maximum entropy principle has been earlier used to derive the Bose Einstein(B.E.), Fermi Dirac(F.D.) & Intermediate Statistics(I.S.) distribution of statistical mechanics. The central idea of these distributions is to predict the distribution of the microstates, which are the particle of the system, on the basis of the knowledge of some macroscopic data. The latter information is specified in the form of some simple moment constraints. | Yugoslav Journal of Operations Research 24 (2014) Number 1, 145-155 DOI: 10.2298/YJOR120912031R DERIVATION OF SOME NEW DISTRIBUTIONS IN STATISTICAL MECHANICS USING MAXIMUM ENTROPY APPROACH Amritansu RAY Department of Mathematics, Rajyadharpur Deshbandhu Vidyapith, Serampore, Hooghly – 712203, West Bengal, India. amritansu_ray06@yahoo.co.in S.K. MAJUMDER Department of Mathematics, Bengal Engineering and Science University (BESU), Shibpur, Howrah – 711103, West Bengal, India. majumder_sk@yahoo.co.in Received: September 2012 / Аccepted: June 2013 Abstract: The maximum entropy principle has been earlier used to derive the Bose Einstein(B.E.), Fermi Dirac(F.D.) & Intermediate Statistics(I.S.) distribution of statistical mechanics. The central idea of these distributions is to predict the distribution of the microstates, which are the particle of the system, on the basis of the knowledge of some macroscopic data. The latter information is specified in the form of some simple moment constraints. One distribution differs from the other in the way in which the constraints are specified. In the present paper, we have derived some new distributions similar to B.E., F.D. distributions of statistical mechanics by using maximum entropy principle. Some proofs of B.E. & F.D. distributions are shown, and at the end some new results are discussed. Keywords: Bose-Einstein (B.E.) distribution, Fermi-Dirac (F.D.) distribution, Lagrange’s multiplier. Shannons’ measure, Jaynes principle. MSC: 94A17, 82B10. A.Ray, S.K. Majumder/ Derivation of Some New Distributions 146 1. INTRODUCTION The term “entropy” was introduced by Clausius in nineteenth-century thermodynamics, and is the subject of second law of thermodynamics, which states that in an isolated thermodynamic system, entropy will either remain constant or increase towards its maximum but cannot decrease. As we know, an isolated system is the one which is closed to inputs of both matter and energy; so in an isolated system, the