Abstract
The relevance of probability theory is obvious in a subject called “Statistical Mechanics” (SM). On the other hand, SM arose as a microscopic description of single objects made of many (invisible) parts, thus justifying from an atomistic point of view the laws of thermodynamics. As a matter of fact, experimental measurements of thermodynamic quantities are conducted on a single system of interest, hence a fundamental problem arises in connecting probabilistic computations, e.g. the averages over ensembles of identical objects, with experiments. One of the most evident aspects of macroscopic phenomena is that they are characterized by a clear trend in time, that cannot be reverted. On the other hand, our understanding of microscopic dynamics is that they are reversible in time. With the aid of analytical computations on stochastic systems, and of numerical simulations of deterministic Hamiltonian systems, we illustrate basic features of macroscopic irreversibility, thus of the microscopic foundations of the second principle of thermodynamics, along the lines of Boltzmann’s kinetic theory. It will be evidenced that in systems characterized by a very large number of degrees of freedom, irreversibility concerns single realizations of the evolution processes, in the sense of the vast majority of the far-from-equilibrium initial conditions. That the vast majority out of a collection of realizations of a given process shares certain properties is often referred to as typicality.
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Notes
- 1.
This inspired a whole branch of mathematics, known as ergodic theory, which represents one way of introducing probabilities in the analysis of the otherwise rigidly deterministic dynamical systems.
- 2.
The use of the symbol h should not lead to believe that quantum effects are taken into consideration. In the present picture, quantum mechanics plays no role.
- 3.
Hsinchu is a chinese city on the Pacific ocean.
- 4.
Most notably the opposite of the H-functional taken by Boltzmann to mirror the entropy of an isolated system.
- 5.
When dealing with non-macroscopic systems, thermodynamics does not strictly apply. This is the case, for instance, of Brownian particles immersed in a liquid. In this case, only a probabilistic, ensemble, description appears interesting and feasible.
- 6.
For instance, they have no extension and do not interact, while molecules occupy a certain volume and interact with each other. Moreover, one phase point represents a whole N-particles system, which something totally different from one of the N particles.
- 7.
The content of the paradox is the following. Given that microscopic dynamics is reversible in time, if we were able to reverse time, the dynamics should trace back its trajectory, and therefore also \(S_B\) should decrease.
References
Auletta, G., Rondoni, L., & Vulpiani, A. (2017). On the relevance of the maximum entropy principle in non-equilibrium statistical mechanics The European Physical Journal Special Topics, 226, 2327.
Baldovin, M., Caprini, L., & Vulpiani, A. (2019). Irreversibility and typicality: A simple analytical result for the Ehrenfest model. Physica A, 524, 422.
Batterman, R. W. (2002). The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction, and Emergence. Oxford University Press.
Berry, M. V. (1994). Asymptotics, singularities and the reduction of theories (p. 597). Methodology and Philosophy of Science, IX: Logic.
Bohr, T., Jensen, H. M., Paladin, G., & Vulpiani, A. (1998). Dynamical systems approach to turbulence. Cambridge University Press.
Bricmont, J. (1996). Science of chaos or chaos in Science? Annual New York Academy Science, 775, 131.
Castiglione, P., Falcioni, M., Lesne, A., & Vulpiani, A. (2008). Chaos and coarse graining in statistical mechanics. Cambridge University Press.
Cercignani, C. (1998). Ludwig Boltzmann: The man who trusted atoms. Oxford University Press.
Cerino, L., Cecconi, F., Cencini, M., & Vulpiani, A. (2016). The role of the number of degrees of freedom and chaos in macroscopic irreversibility. Physica A, 442, 486.
Chibbaro, S., Rondoni, L., Vulpiani, A. (2014). Reductionism, emergence and levels of reality. Springer.
Chibbaro, S., Rondoni, L., & Vulpiani, A. (2014). On the foundations of statistical mechanics: Ergodicity, many degrees of freedom and inference. Communications in Theoretical Physics, 62, 469.
Drossel, B. (2015). On the relation between the second law of thermodynamics and classical and quantum mechanics. In B. Falkenburg & M. Morrison (Eds.), Why more is different (p. 41). Springer.
Dumas, H. S. (2014). The KAM story. World Scientific.
Falcioni, M., Palatella, L., Pigolotti, S., Rondoni, L., & Vulpiani, A. (2007). Initial growth of Boltzmann entropy and chaos in a large assembly of weakly interacting systems. Physica A, 385, 170.
Fermi, E. (1956). Thermodynamics. Dover Publications.
Gallavotti, G. (Ed.). (2007). The Fermi-Pasta-Ulam problem: A status report. Springer.
Gaspard, P. (2005). Chaos, scattering and statistical mechanics. Cambridge University Press.
Gibbs, J. W. (1906). The scientific papers of J. Willard Gibbs (Vol. 1). Longmans, Green and Company.
Gnedenko, B. V. (2018). Theory of probability. Routledge.
Goldstein, S. (2012). Typicality and notions of probability in physics. In Y. Ben-Menahem & M. Hemmo (Eds.), Probability in physics (p. 59). Springer.
Goldstein, S. (2001). Boltzmann’s approach to statistical mechanics. In J. Bricmont et al. (Eds.), Chance in physics (p. 39). Springer.
Huang, K. (2009). Introduction to statistical physics. CRC Press.
Jaynes, E. T. (1957). Information theory and statistical mechanics. Physical Review, 106, 620.
Kac, M. (1957). Probability and related topics in physical sciences. American Mathematical Soc.
Kadanoff, L. P. (013). Relating theories via renormalization. Studies in History and Philosophy of Science Part B, 44, 22.
Khinchin, A. I. (1949). Mathematical foundations of statistical mechanics. Dover.
Landau, L. D., & Lifshitz, E. M. (1980). Statistical physics. Butterworth-Heinemann.
Lanford, O. E. (1981). The hard sphere gas in the Boltzmann- Grad limit. Physica A, 106, 70.
Lebowitz, J. L. (1993). Boltzmann’s entropy and time’s arrow. Physics Today, 46, 32.
Ma, S. K. (1985). Statistical mechanics. World Scientific.
Mazur, P., & van der Linden, J. (1963). Asymptotic form of the structure function for real systems. Journal of Mathematical Physics, 4, 271.
Peliti, L. (2011). Statistical mechanics in a nutshell. Princeton University Press.
Prigogine, I., & Stengers, I. (1979). La nouvelle alliance: métamorphose de la science. Gallimard.
Shafer, G., & Vovk, V. (2001). Probability and finance. Wiley.
Uffink, J. (1995). Can the maximum entropy principle be explained as a consistency requirement? Studies in History and Philosophy of Science Part B, 26, 223.
Vulpiani, A., Cecconi, F., Cencini, M., Puglisi, A., & Vergni, D. (Eds.). (2014). Large deviations in physics. The legacy of the law of large numbers. Springer.
Zanghì, N. (2005). I fondamenti concettuali dell’approccio statistico in fisica. In V. Allori, M. Dorato, F. Laudisa & N. Zanghì (Eds.), La natura delle cose: Introduzione ai fondamenti e alla filosofia della fisica (p. 202). Carocci.
Zurek, W. H. (2018). Eliminating ensembles from equilibrium statistical physics: Maxwell’s demon, Szilard’s engine, and thermodynamics via entanglement. Physics Reports, 755, 1.
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Chibbaro, S., Rondoni, L., Vulpiani, A. (2022). Probability, Typicality and Emergence in Statistical Mechanics. In: Wuppuluri, S., Stewart, I. (eds) From Electrons to Elephants and Elections. The Frontiers Collection. Springer, Cham. https://doi.org/10.1007/978-3-030-92192-7_20
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