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Intrinsic irreversibility of Kolmogorov dynamical systems

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  • Courbage, M.

Abstract

We give the mathematical details and various extensions of the results stated in previous work of Courbage and Prigogine. “Intrinsic random systems” are deterministic and conservative dynamical systems for which we can associate two dissipative Markov processes through a one-to-one “change of representation”, the first leading to equilibrium for t→+∞ and the second for t→-∞. The microscopic formulation of the second principle of thermodynamics permits to lift the degeneracy by the exclusion of all states that do not approach equilibrium for t→+∞. The set of admitted initial conditions D+ is then characterized by a non-equilibrium entropy functional which is infinite for rejected initial states and takes finite values for admitted initial conditions. Thus, rejected initial states correspond to an infinite amount of information. To realize this selection rule we consider general probability measures on phase space that are not necessarily absolutely continuous and we extend the theory of transition to Markov processes to such measures. Owing to the non-invariance of D+ under the time inversion, the evolution of these states in the new representation can only be given by one of the two possible Markov processes.

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  • Courbage, M., 1983. "Intrinsic irreversibility of Kolmogorov dynamical systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 122(3), pages 459-482.
    • Handle: RePEc:eee:phsmap:v:122:y:1983:i:3:p:459-482
    DOI: 10.1016/0378-4371(83)90043-2
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    1. Misra, B. & Prigogine, I. & Courbage, M., 1979. "From deterministic dynamics to probabilistic descriptions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 98(1), pages 1-26.
    2. Martinez, S. & Tirapegui, E., 1983. "Entropy functionals associated to shifts dynamical systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 122(3), pages 593-601.
    3. Edith Penrose, 1981. "Divergent Oil Interests of the First and Third Worlds," The World Economy, Wiley Blackwell, vol. 4(4), pages 435-445, December.
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    2. Gialampoukidis, I. & Gustafson, K. & Antoniou, I., 2013. "Financial Time Operator for random walk markets," Chaos, Solitons & Fractals, Elsevier, vol. 57(C), pages 62-72.

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