Abstract
Although the formula engraved on Boltzmann’s tombstone is Eq. (8.1), connecting the entropy of a system with the measure associated with its macrostate, his name is at least as intimately associated with the Boltzmann equation and the H-theorem, describing, in a more quantitative manner, convergence to equilibrium for a low-density gas. This H-theorem is of great interest in light of our previous discussion, because it can be seen as a concrete implementation of the general scheme that we introduced as the “typicality account.”
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There is no issue here as to whether we let the clock run “forward” or “backward”—the problem is symmetric with respect to the time evolution in both directions.
References
Davies, P. C. W. (1977). The physics of time asymmetry. Berkeley and Los Angeles: University of California Press.
Fournier, N. & Guillin, A. (2014). On the rate of convergence in Wasserstein distance of the empirical measure. Probability Theory and Related Fields, 162, 1–32.
Goldstein, S. (2001). Boltzmann’s approach to statistical mechanics. In J. Bricmont, D. Dürr, M. C. Galavotti, G. Ghirardi, F. Petruccione, & N. Zanghì (eds.). Chance in physics: Foundations and perspectives (pp. 39–54). Berlin: Springer.
Goldstein, S. & Lebowitz, J. L. (2004). On the (Boltzmann) entropy of non-equilibrium systems. Physica D: Nonlinear Phenomena, 193(1), 53–66.
Hauray, M. & Jabin, P.-E. (2015). Particle approximation of Vlasov equations with singular forces: Propagation of chaos. Annales scientifiques de l’École normale supérieure, 48(4), 891–940.
Jeans, J. H. (1915). On the theory of star-streaming and the structure of the universe. Monthly Notices of the Royal Astronomical Society, 76(2), 70–84.
Lanford, O. E. I. (1975). Time evolution of large classical systems. In J. Moser (ed.). Dynamical systems, theory and applications. Lecture Notes in Physics (Vol. 38, pp. 1–111). Berlin: Springer.
Lazarovici, D. & Pickl, P. (2017). A mean field limit for the Vlasov–Poisson system. Archive for Rational Mechanics and Analysis, 225(3), 1201–1231.
Price, H. (2002). Burbury’s last case: The mystery of the entropic arrow. In C. Callender (Ed.). Time, reality & experience (pp. 19–56). Cambridge: Cambridge University Press.
Spohn, H. (1991). Large scale dynamics of interacting particles. Berlin: Springer.
Uffink, J. (2007). Compendium of the foundations of classical statistical physics. In Philosophy of physics (pp. 923–1074). Elsevier.
Uffink, J. (2017). Boltzmann’s work in statistical physics. In Zalta, E. N., (Ed.). The Stanford encyclopedia of philosophy (spring 2017 ed.). Metaphysics Research Lab, Stanford University
Villani, C. (2002). A review of mathematical topics in collisional kinetic theory. In S. Friedlander & D. Serre (Eds.). Handbook of mathematical fluid dynamics (Vol. 1, pp. 71–74). North-Holland: Elsevier Science.
Vlasov, A. A. (1938). On vibration properties of electron gas. Journal of Experimental and Theoretical Physics, 8(3), 291.
Vlasov, A. A. (1968). The vibrational properties of an electron gas. Soviet Physics Uspekhi, 10(6), 721.
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Lazarovici, D. (2023). Boltzmann Equation and the H-Theorem. In: Typicality Reasoning in Probability, Physics, and Metaphysics. New Directions in the Philosophy of Science. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-031-33448-1_10
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