Nonlinear Fokker-Planck Equation for an Overdamped System with Drag Depending on Direction
We investigate a one-dimensional, many-body system consisting of particles interacting via repulsive, short-range forces, and moving in an overdamped regime under the effect of a drag force that depends on direction. That is, particles moving to the right do not experience the same drag as those mov...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-09-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/9/1621 |
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| author | Angel Ricardo Plastino Roseli S. Wedemann Constantino Tsallis |
| author_facet | Angel Ricardo Plastino Roseli S. Wedemann Constantino Tsallis |
| author_sort | Angel Ricardo Plastino |
| collection | DOAJ |
| description | We investigate a one-dimensional, many-body system consisting of particles interacting via repulsive, short-range forces, and moving in an overdamped regime under the effect of a drag force that depends on direction. That is, particles moving to the right do not experience the same drag as those moving to the left. The dynamics of the system, effectively described by a non-linear, Fokker–Planck equation, exhibits peculiar features related to the way in which the drag force depends on velocity. The evolution equation satisfies an <i>H</i>-theorem involving the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>q</mi></msub></semantics></math></inline-formula> nonadditive entropy, and admits particular, exact, time-dependent solutions closely related, but not identical, to the <i>q</i>-Gaussian densities. The departure from the canonical, <i>q</i>-Gaussian shape is related to the fact that in one spatial dimension, in contrast to what occurs in two or more spatial dimensions, the drag’s dependence on direction entails that its dependence on velocity is necessarily (and severely) non-linear. The results reported here provide further evidence of the deep connections between overdamped, many-body systems, non-linear Fokker–Planck equations, and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>q</mi></msub></semantics></math></inline-formula>-thermostatistics. |
| first_indexed | 2024-03-10T07:10:45Z |
| format | Article |
| id | doaj.art-96113f8c2e574424ad3c9656fac913cb |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T07:10:45Z |
| publishDate | 2021-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-96113f8c2e574424ad3c9656fac913cb2023-11-22T15:27:32ZengMDPI AGSymmetry2073-89942021-09-01139162110.3390/sym13091621Nonlinear Fokker-Planck Equation for an Overdamped System with Drag Depending on DirectionAngel Ricardo Plastino0Roseli S. Wedemann1Constantino Tsallis2CeBio y Departamento de Ciencias Básicas, Universidad Nacional del Noroeste de la Província de Buenos Aires, UNNOBA-Conicet, Roque Saenz Peña, Junin 456, ArgentinaInstituto de Matemática e Estatística, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier 524, Rio de Janeiro 20550-900, RJ, BrazilCentro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, RJ, BrazilWe investigate a one-dimensional, many-body system consisting of particles interacting via repulsive, short-range forces, and moving in an overdamped regime under the effect of a drag force that depends on direction. That is, particles moving to the right do not experience the same drag as those moving to the left. The dynamics of the system, effectively described by a non-linear, Fokker–Planck equation, exhibits peculiar features related to the way in which the drag force depends on velocity. The evolution equation satisfies an <i>H</i>-theorem involving the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>q</mi></msub></semantics></math></inline-formula> nonadditive entropy, and admits particular, exact, time-dependent solutions closely related, but not identical, to the <i>q</i>-Gaussian densities. The departure from the canonical, <i>q</i>-Gaussian shape is related to the fact that in one spatial dimension, in contrast to what occurs in two or more spatial dimensions, the drag’s dependence on direction entails that its dependence on velocity is necessarily (and severely) non-linear. The results reported here provide further evidence of the deep connections between overdamped, many-body systems, non-linear Fokker–Planck equations, and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>q</mi></msub></semantics></math></inline-formula>-thermostatistics.https://www.mdpi.com/2073-8994/13/9/1621non-linear Fokker–Planck equationdirection-dependent drag<i>H</i>-theorem |
| spellingShingle | Angel Ricardo Plastino Roseli S. Wedemann Constantino Tsallis Nonlinear Fokker-Planck Equation for an Overdamped System with Drag Depending on Direction Symmetry non-linear Fokker–Planck equation direction-dependent drag <i>H</i>-theorem |
| title | Nonlinear Fokker-Planck Equation for an Overdamped System with Drag Depending on Direction |
| title_full | Nonlinear Fokker-Planck Equation for an Overdamped System with Drag Depending on Direction |
| title_fullStr | Nonlinear Fokker-Planck Equation for an Overdamped System with Drag Depending on Direction |
| title_full_unstemmed | Nonlinear Fokker-Planck Equation for an Overdamped System with Drag Depending on Direction |
| title_short | Nonlinear Fokker-Planck Equation for an Overdamped System with Drag Depending on Direction |
| title_sort | nonlinear fokker planck equation for an overdamped system with drag depending on direction |
| topic | non-linear Fokker–Planck equation direction-dependent drag <i>H</i>-theorem |
| url | https://www.mdpi.com/2073-8994/13/9/1621 |
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