Abstract
We discuss the possibility of deriving an H-theorem for the nonlinear discrete time evolution known as Ulam’s redistribution of energy problem. In this model particles are paired at random and then their total energy is redistributed between them according to some probability law. It appears useful to represent the evolution as a combination of two processes. The first is a linear transformation of two-particle distribution function due to redistribution while the second one is a kind of “reduction” which corresponds to new random pairing. Then information theory approach leads to a general inequality for the Ulam’s problem, which may be viewed as a kind of Clausius inequality. However, only for a special set of redistribution laws, given by symmetric beta distributions, this inequality results in the H-theorem. The H-functional in this case differs from the usual entropy by an additional term that vanishes only for the uniform redistribution law.
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References
Blackwell, D., Mauldin, R.D.: Ulam’s redistribution of energy problem: collision transformations. Lett. Math. Phys. 10, 149–153 (1985)
Pietruska-Pa\l uba, K.: On Ulam’s redistribution of energy problem. Lett. Math. Phys. 14, 247–252 (1987)
Holley, R., Liggett T.M.: Generalized potlatch and smoothing processes. Z. Warscheinlichkeitstheorie Verw Geb. 55, 165–195 (1981)
Durrett, R, Liggett T.M.: Fixed points of the smoothing transformation. Z. Warscheinlichkeitstheorie Verw Geb. 64 275–301 (1983)
Yakovenko, V.M., Barkley R.J. Jr.: Statistical mechanics of money, wealth, and income. Rev. Mod. Phys. 81, 1703–1725 (2009)
Patriarca, M., Heinsalu E., Chakraborti, A.: Basic kinetic wealth-exchange models: common features and open problems. Eur. Phys. J. B 73, 145–153 (2010)
Chakraborti, A., Toke, I.M., Patriarca, M., Abergel, F.: Econophysics review: II. Agent-based models. Quant. Financ., 11, 1013–1041 (2011)
Drăgulescu, A.A., Yakovenko, V.M.: Statistical mechanics of money. Eur. Phys. J. B 17, 723–729 (2000)
López-Ruiz, R., López, J.-L., Calbet, X.: Exponential wealth distribution: a new approach from functional iteration theory. ESAIM Proc. 36, 189–196 (2012)
López, J.-L., López-Ruiz, R., Calbet, X.: Exponential wealth distribution in a random market. A rigorous explanation. J. Math. Anal. Appl. 386, 19–5 (2012)
Apenko, S.M.: Monotonic entropy growth for a nonlinear model of random exchanges. Phys. Rev. E 87, 02410–1 (2013)
Lallouache, M., Jedidi, A., Chakraborti, A.: Wealth distribution: to be or not to be a Gamma? Sci. Cult. (Kolkata, India) 76, 47–8 (2010)
Cover, T.M., Thomas, J.A.: Elements of Information Theory, 2nd ed., Wiley, Hoboken (2006)
Bassetti, F., Toscani, G.: Explicit equilibria in a kinetic model of gambling. Phys. Rev. E 81, 06611–5 (2010)
Marconi, U.M.B., Puglisi, A., Vulpiani, A.: About an H-theorem for systems with non-conservative interactions. J. Stat. Mech. 8, 2 (P08003) (2013)
Attard, P.: Is the information entropy the same as the statistical mechanical entropy? arXiv:1209.5500
Maynar, P., Trizac, E.: Entropy of continuous mixtures and the measure problem. Phys. Rev. Lett. 106, 16060–3 (2011)
Resibois, P.: H-theorem for the (modified) nonlinear Enskog equation. J. Stat. Phys. 19, 593–609 (1978)
Garrido, P., Goldstein, S., Lebowitz, J.L.: Boltzmann entropy for dense fluids not in local equilibrium. Phys. Rev. Lett. 92, 05060–2 (2003)
Acknowledgements
I am very grateful to J. Gaite for pointing out Ref. [1], to R. López-Ruiz for kind hospitality in Zaragoza during NOMA'13, to A. Puglisi and E. Trizac for stimulating correspondence, and to A. Chakraborty for many interesting discussions. The work was supported in part by RFBR Grants No. 12-02-00520, 13-02-00457.
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Apenko, S. (2015). In Search of H-theorem for Ulam’s Redistribution of Energy Problem. In: López-Ruiz, R., Fournier-Prunaret, D., Nishio, Y., Grácio, C. (eds) Nonlinear Maps and their Applications. Springer Proceedings in Mathematics & Statistics, vol 112. Springer, Cham. https://doi.org/10.1007/978-3-319-12328-8_12
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DOI: https://doi.org/10.1007/978-3-319-12328-8_12
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