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IMA Journal of Applied Mathematics Advance Access originally published online on September 29, 2007
IMA Journal of Applied Mathematics 2007 72(6):785-800; doi:10.1093/imamat/hxm039
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© The Author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Lambert function and a new non-extensive form of entropy

Fariel Shafee{dagger}

Physics Department, Princeton University, Princeton, NJ 0850, USA

{dagger} Email: f_shafee{at}yahoo.com

Received on May 8, 2006; Revision received March 24, 2007. Accepted on August 29, 2007

We propose a new way of defining entropy of a system, which gives a general form that is non-extensive like Tsallis entropy, but is linearly dependent on component entropies, like Renyi entropy, which is extensive. This entropy has a conceptually novel but simple origin and is mathematically easy to define by a very simple expression involving a derivative. It leads to a probability distribution function involving the Lambert function resulting from optimizing the entropy, which has hitherto never appeared in this context, and is somewhat more complex than the Shannon or Boltzmann form, but is nevertheless mathematically quite tractable. We have compared it numerically with the Tsallis and Shannon entropies. We have also considered constraints on the energy spectra imposed by the properties of the Lambert function, which are absent in the Shannon form. It may turn out to be a more appropriate candidate in a physical situation where the probability distribution does not suit any of the previously defined forms, especially when the probability density function sought is expected to be stiffer than that resulting from maximizing the other entropies. We consider the problem of defining free energy and other thermodynamic functions when the entropy is given as a general function of the probability distribution, including that for non-extensive forms. We then find that the free energy, which is central to the determination of all other quantities of interest in a thermodynamic context, can be obtained uniquely, at least numerically, even when it is the root of a transcendental equation. In particular, we examine the cases of the Tsallis form and the new form proposed by us. We compare the free energy, the internal energy and the specific heat of a simple system of two energy states for each of these forms and find significant departures for some quantities, while some others are less sensitive to the parametrization.

Keywords: Lambert function; non-extensive entropy; probability distribution function; free energy.


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