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The Constraint Based Statistics - Beyond the Entropy Based Statistical Mechanics

  1. Apr 24, 2010 #1
    The Constraint Based Statistics --- Beyond the Entropy Based Statistical Mechanics

    The Constraint Based Statistics --- Beyond Tsallis Entropy and Boltzmann Entropy Based Statistical Mechanics

    This post is a summary about a brand new work in the field of Nonextensive Statistical Mechanics consisting of 3 papers by a single author [1][2][3]. All of the 3 papers have been formally cited by international professional research institutes related with the Nonextensive Statistical Mechanics. Among the 3 papers, 2 of them are included in
    edited by Prof. Constantino Tsallis.

    The key ideas include but are not limited to the points described as follows.

    (1) It is demonstrated clearly that for the same classical generalized system the Tsallis power-laws with both the q > 1 and the q < 1 can be induced by the constraint of the constant harmonic mean for the so-called reciprocal energies Er and at the same time the Boltzmann distribution or the negative exponential probability distribution can be generated with the constraint of the constant arithmetic mean for the generalized energies E .

    Is the generalized system an extensive system or a nonextensive system ?

    The author thus argues that there might be no definite "extensive system" or "classical system" and there are only "classical physical parameters" and "classical constraints". For any physical system or generalized system, it is the non-natural constraints which determine both the forms of the entropies and the nonuniform equilibrium distributions.

    (2) We can only get a uniform distribution if there is no non-natural constraint and the uniform distribution can be obtained with the maximizing of both the Tsallis entropy with q<>1 and the Boltzmann-Gibbs-Shannon entropy as a special case of the Tsallis entropy with q--->1. Therefore the form of the entropy will not be exclusive if there is no non-natural constraint [3].

    (3) "How can I get a Tsallis power-law from a physical quantity while its reciprocal follows a Boltzmann distribution ?" --- You may naturally challenge the results of these papers. The results of the simulation are somehow far away from common sense just like the tunnel effects in quantum mechanics. The key is that the individuals are located at all of the energy levels with a probability at the same time, you cannot use the relationship between a single value of the energy Ei and the value of its reciprocal energy Eri to guess the probability distribution of the reciprocal energies . The decisive factor to determine the probability distribution is the constraint about all of the energy levels of the reciprocal energies.

    (4) The energies E and the reciprocal energies Er are of equal position because they are all derived from an array of random numbers with the sample size of 10,000,000. The inertia of the Boltzmann statistics may prevent one from accepting the repeatable numerical facts easily. it is recommended that as a serious scientific explorer, you should get the MATLAB code from the author and you will be convinced by the facts of the numerical experiments.

    (5) Both the simulation results and the theoretical prediction indicate a brand new phenomenon: For any system, when there is a Boltzmann Distribution generated by the constraint of the constant arithmetic mean, there may be a Tsallis power-law induced by the constraint of the constant harmonic mean, and vice versa.

    (6) Nature is far more complicated than what the Tsallis q-parameter can exclusively determine. A Constraint-Based Statistics is necessary and has been basically established [1][2][3] .

    (7) A virgin land has been discovered in the field of Nonextensive Statistical Mechanics. For example, a unified mathematical expression about the constraints has been presented [2][3], which determines in a general way the forms of the entropy and the equilibrium distributions.

    [1] X.Feng, WCFSGS, Vol.6, S1, April 2010, ISSN 1936-7260.
    [2 ]X. Feng, arXiv:cond-mat.stat-mech/1002.4254 v2 24 Feb 2010.
    [3] X. Feng, arXiv:cond-mat.stat-mech/0705.1332 v4 14 May 2007.


    All of the 3 papers have been included in Google Scholar

    A Special Edition on the Constraint-Based Statistical Mechanics has been published under the well-known name of Nonextensive Statistical Mechanics. All of the 3 papers can be found on-line.


    Last edited: Apr 24, 2010
  2. jcsd
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