DIFFERENT TYPES OF THE DEFORMED EXPONENTIAL FUNCTIONS IN THE STATISTICAL MECHANICS

AUTHOR(S) / АУТОР(И): Predrag M. Rajković , Sladjana D. Marinković , Miomir S. Stanković

Download Full Pdf   

DOI: 10.46793/ICSSM25.228R

ABSTRACT / САЖЕТАК:

In recent developments in various sciences, the need is noted to define and use deformed versions of the exponential function. In this paper, the consideration of such functions has two purposes: to have the exponential function as their special case, and, even more, to acquit their inauguration from a mathematical point of view. Starting from well-known Tsallis and Kaniadakis versions, we proposed our own deformed versions and connect them with others. It leads to definition of the deformed numbers and operators. We find their series expansions and derived differential and difference properties. They are important in forming and explaining continuous and discrete models of numerous phenomena in statistical mechanics, thermostatics, information theory, cybernetics, control theory, etc. We illustrate it by analyzing of the different versions of Malthus model in population dynamics. Also, we look back on the well-known law of composed interest in the economy by the deformed exponential function.

KEYWORDS / КЉУЧНЕ РЕЧИ:

Exponential function, differential operator, difference operator, Malthus model

ACKNOWLEDGEMENT / ПРОЈЕКАТ:

This work was supported by the Ministry of Science, Technological Development and Innovation of the Republic of Serbia [grants No. 451-03-65/2024- 03/200102 and No. 451- 03-66/2024-03/200109].

REFERENCES / ЛИТЕРАТУРА:

• Bernoulli,J., Opera omnia, Vol. 3, 1697.
• Tsallis, C., Possible generalization of Boltzmann-Gibbs statistics, Journal of Statistical Physics, Vol. 52, 479-487,1988.
• Abe, S., Nonextensive statistical mechanics of q–bosons based on the q–deformed entropy, Physics Letters A, Vol. 224, 229-236, 1998.
• Kaniadakis, G., Non–linear kinetics underlying generalized statistics, Physica A, Vol. 296, 405-425, 2001.
• Stanković, M.S., Marinković, S.D., Rajković, P.M., The deformed and modified Mittag– Leffler polynomials, Mathematical and Computer Modelling, Vol. 54, 721-728, 2011.
• Stanković, M.S., Marinković, S.D., Rajković, P.M., The deformed exponential functions of two variables in the context of various statistical mechanics, Applied Mathematics and Computation, Vol. 218, 2439-2448, 2011.
• Borges, E., A possible deformed algebra and calculus inspired in nonextensive thermostatics, Physica A, Vol. 340, 95-101, 2004.
• Kaniadakis, G., Maximum entropy principle and power-law tailed distributions, The European Physical Journal B, Vol. 70, No 3, 3-13, 2009.
• Andreis, S. de, Ricci, P.E., Modelling Population Growth via Laguerre-Type Exponentials, Mathematical and Computer Modelling, Vol. 42, 1421-1428, 2005.
• Bretti, G., Ricci, P.E., Laguerre-type special functions and population dynamics, Applied Mathematics and Computation, Vol. 187, 89-100, 2007.
• Malthus, T.R., Principle of Population, Penguin, Harmondsworth, England, 1798.
• Murray, J., Mathematical Biology, Springer-Verlag, Berlin, 1989.