Stochastic Theory of Coarse-Grained Deterministic Systems: Martingales and Markov Approximations
Many works have been devoted to show that Thermodynamics and Statistical Physics can be rigorously deduced from an exact underlying classical Hamiltonian dynamics, and to resolve the related paradoxes. In particular, the concept of equilibrium state and the derivation of Master Equations should resu...
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| Autores principales: | , |
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| Formato: | Capítulo de libro |
| Lenguaje: | inglés |
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IntechOpen
2021
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| Acceso en línea: | https://www.intechopen.com/chapters/75310 |
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| _version_ | 1879164742362202112 |
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| author | Moreau, Michel Gaveau, Bernard |
| author_browse | Gaveau, Bernard Moreau, Michel |
| author_facet | Moreau, Michel Gaveau, Bernard |
| author_sort | Moreau, Michel |
| building | IntechOpen |
| collection | Acceso abierto |
| description | Many works have been devoted to show that Thermodynamics and Statistical Physics can be rigorously deduced from an exact underlying classical Hamiltonian dynamics, and to resolve the related paradoxes. In particular, the concept of equilibrium state and the derivation of Master Equations should result from purely Hamiltonian considerations. In this chapter, we reexamine this problem, following the point of view developed by Kolmogorov more than 60 years ago, in great part known from the work published by Arnold and Avez in 1967. Our setting is a discrete time dynamical system, namely the successive iterations of a measure-preserving mapping on a measure space, generalizing Hamiltonian dynamics in phase space. Using the notion of Kolmogorov entropy and martingale theory, we prove that a coarse-grained description both in space and in time leads to an approximate Master Equation satisfied by the probability distribution of partial histories of the coarse-grained state. |
| doi_str_mv | 10.5772/intechopen.95903 |
| first_indexed | 2026-10-15T21:39:10Z |
| format | Book Chapter |
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<dc:title>Stochastic Theory of Coarse-Grained Deterministic Systems: Martingales and Markov Approximations</dc:title>
<dc:creator>Michel Moreau</dc:creator>
<dc:creator>Bernard Gaveau</dc:creator>
<dc:subject>Physical Sciences, Engineering and Technology</dc:subject>
<dc:description>Many works have been devoted to show that Thermodynamics and Statistical Physics can be rigorously deduced from an exact underlying classical Hamiltonian dynamics, and to resolve the related paradoxes. In particular, the concept of equilibrium state and the derivation of Master Equations should result from purely Hamiltonian considerations. In this chapter, we reexamine this problem, following the point of view developed by Kolmogorov more than 60 years ago, in great part known from the work published by Arnold and Avez in 1967. Our setting is a discrete time dynamical system, namely the successive iterations of a measure-preserving mapping on a measure space, generalizing Hamiltonian dynamics in phase space. Using the notion of Kolmogorov entropy and martingale theory, we prove that a coarse-grained description both in space and in time leads to an approximate Master Equation satisfied by the probability distribution of partial histories of the coarse-grained state.</dc:description>
<dc:publisher>IntechOpen</dc:publisher>
<dc:date>2021-02-18</dc:date>
<dc:type>Chapter, Part Of Book</dc:type>
<dc:identifier>https://www.intechopen.com/chapters/75310</dc:identifier>
<dc:identifier>doi:10.5772/intechopen.95903</dc:identifier>
<dc:language>en</dc:language>
<dc:relation>ISBN:978-1-83969-123-2</dc:relation>
<dc:rights>https://creativecommons.org/licenses/by/3.0/</dc:rights>
<dc:source>https://www.intechopen.com/books/10471 ; Advances in Dynamical Systems Theory, Models, Algorithms and Applications</dc:source>
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| id | intech-75310 |
| institution | IntechOpen |
| isbn | 978-1-83969-123-2 |
| language | English |
| last_indexed | 2026-10-15T21:39:10Z |
| publishDate | 2021 |
| publishDateRange | 2021 |
| publishDateSort | 2021 |
| publisher | IntechOpen |
| publisherStr | IntechOpen |
| record_format | oai_dc |
| spelling | intech-753102021-02-18 Stochastic Theory of Coarse-Grained Deterministic Systems: Martingales and Markov Approximations Michel Moreau Bernard Gaveau Physical Sciences, Engineering and Technology Many works have been devoted to show that Thermodynamics and Statistical Physics can be rigorously deduced from an exact underlying classical Hamiltonian dynamics, and to resolve the related paradoxes. In particular, the concept of equilibrium state and the derivation of Master Equations should result from purely Hamiltonian considerations. In this chapter, we reexamine this problem, following the point of view developed by Kolmogorov more than 60 years ago, in great part known from the work published by Arnold and Avez in 1967. Our setting is a discrete time dynamical system, namely the successive iterations of a measure-preserving mapping on a measure space, generalizing Hamiltonian dynamics in phase space. Using the notion of Kolmogorov entropy and martingale theory, we prove that a coarse-grained description both in space and in time leads to an approximate Master Equation satisfied by the probability distribution of partial histories of the coarse-grained state. IntechOpen 2021-02-18 Chapter, Part Of Book https://www.intechopen.com/chapters/75310 doi:10.5772/intechopen.95903 en ISBN:978-1-83969-123-2 https://creativecommons.org/licenses/by/3.0/ https://www.intechopen.com/books/10471 ; Advances in Dynamical Systems Theory, Models, Algorithms and Applications |
| spellingShingle | Physical Sciences, Engineering and Technology Moreau, Michel Gaveau, Bernard Stochastic Theory of Coarse-Grained Deterministic Systems: Martingales and Markov Approximations |
| title | Stochastic Theory of Coarse-Grained Deterministic Systems: Martingales and Markov Approximations |
| title_full | Stochastic Theory of Coarse-Grained Deterministic Systems: Martingales and Markov Approximations |
| title_fullStr | Stochastic Theory of Coarse-Grained Deterministic Systems: Martingales and Markov Approximations |
| title_full_unstemmed | Stochastic Theory of Coarse-Grained Deterministic Systems: Martingales and Markov Approximations |
| title_short | Stochastic Theory of Coarse-Grained Deterministic Systems: Martingales and Markov Approximations |
| title_sort | stochastic theory of coarse grained deterministic systems martingales and markov approximations |
| topic | Physical Sciences, Engineering and Technology |
| url | https://www.intechopen.com/chapters/75310 |
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