# zbMATH — the first resource for mathematics

Disequilibrium, thermodynamic relations, and Rényi’s entropy. (English) Zbl 1372.94373
Summary: The disequilibrium concept $$(D)$$ was introduced by López-Ruiz, Mancini, and Calbet 20 years ago together with their successful notion of statistical complexity. In this note, we show that, in a classical, canonical-ensemble environment, $$D$$ displays interesting thermodynamic properties and is able to replace the partition function. Also, we show that for a generalized statistical complexity-family that involves Rényi’s entropy of order $$q$$, the maximal value of these new complexities is attained in the case $$q = 1$$.

##### MSC:
 94A17 Measures of information, entropy 82B05 Classical equilibrium statistical mechanics (general)
##### Keywords:
disequilibrium; canonical ensemble; Rényi’s entropy
Full Text:
##### References:
 [1] López-Ruiz, R.; Mancini, H. L.; Calbet, X., A statistical measure of complexity, Phys. Lett. A, 209, 321-326, (1995) [2] Crutchfield, J. P., The calculi of emergence: computation, dynamics and induction, Physica D, 75, 11-54, (1994) · Zbl 0860.68046 [3] Feldman, D. P.; Crutchfield, J. P., Measures of statistical complexity: why?, Phys. Lett. A, 238, 244-252, (1998) · Zbl 1026.82505 [4] Martin, M. T.; Plastino, A.; Rosso, O. A., Statistical complexity and disequilibrium, Phys. Lett. A, 311, 126-132, (2003) · Zbl 1060.60500 [5] Rudnicki, L.; Toranzo, I. V.; Sánchez-Moreno, P.; Dehesa, J. S., Monotone measures of statistical complexity, Phys. Lett. A, 380, 377-380, (2016) · Zbl 1349.81065 [6] López-Ruiz, R., A statistical measure of complexity, (Kowalski, A.; Rossignoli, R.; Curado, E. M.C., Concepts and Recent Advances in Generalized Information Measures and Statistics, (2013), Bentham Science Books New York), 147-168 [7] (Sen, K. D., Statistical Complexity, Applications in Electronic Structure, (2011), Springer Berlin) [8] Mitchell, M., Complexity: A guided tour, (2009), Oxford University Press Oxford, England · Zbl 1215.00003 [9] Martin, M. T.; Plastino, A.; Rosso, O. A., Generalized statistical complexity measures: geometrical and analytical properties, Physica A, 369, 439-462, (2006) [10] López-Ruiz, R., Complexity in some physical systems, Int. J. Bifurc. Chaos, 11, 2669-2673, (2001) [11] Pathria, R. K., Statistical mechanics, (1996), Butterworth-Heinemann Oxford, UK · Zbl 0862.00007 [12] Sañudo, J.; López-Ruiz, R., Calculation of statistical entropic measures in a model of solids, Phys. Lett. A, 376, 2288-2291, (2012) [13] Catalán, R. G.; Garay, J.; López-Ruiz, R., Features of the extension of a statistical measure of complexity to continuous systems, Phys. Rev. E, 66, (2002) [14] Lima, J. A.S.; Plastino, A. R., On the classical energy equipartition theorem, Braz. J. Phys., 30, 1, 176-180, (2000) [15] Tolman, R. C., The principles of statistical mechanics, 95, (2010), University Press Oxford, Great Britain [16] Jaynes, E. T., Information theory and statistical mechanics, Phys. Rev. E, 106, (1957) · Zbl 0084.43701 [17] Baez, J. C., Rényi entropy and free energy, (2011) [18] Lenzi, E. K.; Mendes, R. S.; da Silva, L. R., Statistical mechanics based on Rényi entropy, Physica A, 280, 337-345, (2000) [19] Jizba, P.; Arimitsu, T., The world according to Rényi: thermodynamics of multifractal systems, Ann. Phys., 312, 17-59, (2004) · Zbl 1044.82001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.