The Generalized Stochastic Smoluchowski Equation
We study the dynamics of a system of overdamped Brownian particles governed by the generalized stochastic Smoluchowski equation associated with a generalized form of entropy and involving a long-range potential of interaction [P.H. Chavanis, Entropy <b>17</b>, 3205 (2015)]. We first negl...
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-10-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/21/10/1006 |
| _version_ | 1828146076681502720 |
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| author | Pierre-Henri Chavanis |
| author_facet | Pierre-Henri Chavanis |
| author_sort | Pierre-Henri Chavanis |
| collection | DOAJ |
| description | We study the dynamics of a system of overdamped Brownian particles governed by the generalized stochastic Smoluchowski equation associated with a generalized form of entropy and involving a long-range potential of interaction [P.H. Chavanis, Entropy <b>17</b>, 3205 (2015)]. We first neglect fluctuations and provide a macroscopic description of the system based on the deterministic mean field Smoluchowski equation. We then take fluctuations into account and provide a mesoscopic description of the system based on the stochastic mean field Smoluchowski equation. We establish the main properties of this equation and derive the Kramers escape rate formula, giving the lifetime of a metastable state, from the theory of instantons. We relate the properties of the generalized stochastic Smoluchowski equation to a principle of maximum dissipation of free energy. We also discuss the connection with the dynamical density functional theory of simple liquids. |
| first_indexed | 2024-04-11T20:42:42Z |
| format | Article |
| id | doaj.art-9563a89e17574bb6960c9fb1e29bb4e6 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-11T20:42:42Z |
| publishDate | 2019-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-9563a89e17574bb6960c9fb1e29bb4e62022-12-22T04:04:10ZengMDPI AGEntropy1099-43002019-10-012110100610.3390/e21101006e21101006The Generalized Stochastic Smoluchowski EquationPierre-Henri Chavanis0Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, 31000 Toulouse, FranceWe study the dynamics of a system of overdamped Brownian particles governed by the generalized stochastic Smoluchowski equation associated with a generalized form of entropy and involving a long-range potential of interaction [P.H. Chavanis, Entropy <b>17</b>, 3205 (2015)]. We first neglect fluctuations and provide a macroscopic description of the system based on the deterministic mean field Smoluchowski equation. We then take fluctuations into account and provide a mesoscopic description of the system based on the stochastic mean field Smoluchowski equation. We establish the main properties of this equation and derive the Kramers escape rate formula, giving the lifetime of a metastable state, from the theory of instantons. We relate the properties of the generalized stochastic Smoluchowski equation to a principle of maximum dissipation of free energy. We also discuss the connection with the dynamical density functional theory of simple liquids.https://www.mdpi.com/1099-4300/21/10/1006fokker-planck equationsfree energygeneralized thermodynamicsinstantonsdynamical density functional theory |
| spellingShingle | Pierre-Henri Chavanis The Generalized Stochastic Smoluchowski Equation Entropy fokker-planck equations free energy generalized thermodynamics instantons dynamical density functional theory |
| title | The Generalized Stochastic Smoluchowski Equation |
| title_full | The Generalized Stochastic Smoluchowski Equation |
| title_fullStr | The Generalized Stochastic Smoluchowski Equation |
| title_full_unstemmed | The Generalized Stochastic Smoluchowski Equation |
| title_short | The Generalized Stochastic Smoluchowski Equation |
| title_sort | generalized stochastic smoluchowski equation |
| topic | fokker-planck equations free energy generalized thermodynamics instantons dynamical density functional theory |
| url | https://www.mdpi.com/1099-4300/21/10/1006 |
| work_keys_str_mv | AT pierrehenrichavanis thegeneralizedstochasticsmoluchowskiequation AT pierrehenrichavanis generalizedstochasticsmoluchowskiequation |