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Macroscopic laws, microscopic dynamics, time's arrow and Boltzmann's entropy

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  • Lebowitz, Joel L.

Abstract

I discuss Boltzmann's resolution of the apparent paradox: microscopic dynamics are time-symmetric but the behavior of macroscopic objects, composed of microscopic constituents, is time-asymmetric. Noting the great disparity between macroscales and microscales Boltzmann developed a statistical approach which explains the observed macroscopic behavior. In particular it predicts the increase with time of the “Boltzmann entropy”, SB(X), for “almost all” microscopic states X, of a nonequilibrium macroscopic system. The quantitative description of the macroscopic evolution, and ipso facto the compatibility between the macroscopic descriptions and microscopic descriptions, is illustrated by an example: the rigorous derivation of a diffusion equation for the typical macroscopic density profile of a Lorentz gas of independent electrons moving according to Hamiltonian dynamics. The role of low entropy “initial states” is emphasized.

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  • Lebowitz, Joel L., 1993. "Macroscopic laws, microscopic dynamics, time's arrow and Boltzmann's entropy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 194(1), pages 1-27.
    • Handle: RePEc:eee:phsmap:v:194:y:1993:i:1:p:1-27
    DOI: 10.1016/0378-4371(93)90336-3
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    Cited by:

    1. Michel, Denis, 2018. "A probabilistic rate theory connecting kinetics to thermodynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 503(C), pages 26-44.
    2. Pérez-Cárdenas, Fernando C. & Resca, Lorenzo & Pegg, Ian L., 2016. "Microscopic reversibility and macroscopic irreversibility: A lattice gas model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 457(C), pages 82-92.
    3. Kalogeropoulos, Nikolaos, 2022. "Coarse-graining and symplectic non-squeezing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 589(C).
    4. Ali, S.A. & Cafaro, C. & Kim, D.-H. & Mancini, S., 2010. "The effect of microscopic correlations on the information geometric complexity of Gaussian statistical models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(16), pages 3117-3127.

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