Statistical mechanics of holonomic systems as a Brownian motion on smooth manifolds |
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| Authors: | Fabio Manca Pierre‐Michel Déjardin Stefano Giordano |
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| Affiliation: | 1. Institute of Electronics, Microelectronics and Nanotechnology (UMR CNRS 8520), Villeneuve d'Ascq, France;2. Laboratoire de Mathématiques et Physique (EA 4217), Université de Perpignan Via Domitia, Perpignan, France;3. International Associated Laboratory LEMAC/LICS, ECLille, Villeneuve d'Ascq, France |
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| Abstract: | The statistical mechanics of arbitrary holonomic scleronomous systems subjected to arbitrary external forces is described by specializing the Lagrange and Hamilton equations of motion to those of the Brownian motion on a manifold. In this context, the Klein‐Kramers and Smoluchowski equations are derived in covariant form, and it is demonstrated that these equations have equilibrium solutions corresponding to the Gibbs distribution, in agreement with standard thermodynamics. At last, the Langevin dynamics corresponding to the Smoluchowski limit is found to exactly correspond to the Brownian motion on a smooth manifold. These results find significant applications in the study of several statistical properties of constrained molecular assemblies (e.g. polymers) of interest in chemistry, physics and biology. |
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| Keywords: | Non‐equilibrium statistical mechanics Langevin and Fokker‐Planck equations Smoluchowski equation differential absolute calculus smooth manifolds |
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