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	KTH Computational Science and Engineering Centre

Title	Speaker
Stochastic Thermostats, Model Reduction and Dynamical Approximation	Ben Leimkuhler

Location	Time and Date
PDC seminar room, Teknikringen 14, level 3	14.30-15.30, 2011-02-23 (Wed)

Abstract
Based on the thermodynamic concept of a reservoir, I will discuss a computational model for interaction with unresolved degrees of freedom (a thermal bath). The starting point is the assumption that a finite restricted system can be modelled by a generalized canonical ensemble, described by a density which is a smooth function of the energy of the restricted system (an equilibrium state). A stochastic-dynamic thermostat [1] provides a restricted resolved dynamics embedded within the larger energetic bath, while leaving the desired target distribution invariant. Under certain assumptions, these methods can be proven to be ergodic, meaning that almost every extended dynamics trajectory samples the equilibrium measure; I will illustrate this with an example [2]. I will apply these techniques in the novel setting of a simplified point vortex flow on a disc, in which a modified Gibbs distribution (modelling a finite, rather than infinite, bath of weak vortices) provides a regularizing formulation for restricted system dynamics [3]. Although our method does not provide a proper dynamical closure, it is very straightforward to implement in a wide range of situations and can provide realistic averages. Numerical experiments, effectively replacing many vortices by a few artificial degrees of freedom, are in excellent agreement with the two-scale simulations that have appeared in the literature [4]. I will also comment on the issue of approximation of dynamical quantities (such as velocity autocorrelation functions) when using thermostats [5]. References: [1] Leimkuhler, B., Generalized Bulgac-Kusnezov methods for sampling of the Gibbs-Boltzmann measure, Physical Review E 026703, 2010. [2] Leimkuhler, B., Noorizadeh, E. and Theil, F., A gentle stochastic thermostat for molecular dynamics, J. Stat. Phys. 135: 261-277, 2009. [3] Dubinkina, S., Frank, J. and Leimkuhler, B., Simplified modelling of energetic interactions with a thermal bath, with application to a fluid vortex system, SIAM Multiscale Modelling and Simulation 8, 1882-1901, 2010. [4] B.F.Chler, O., Statistical mechanics of strong and weak point vortices in a cylinder, Physics of Fluids, 14, 2139-2149, 2001. [5] Leimkuhler, B., Noorizadeh, E. and Penrose, O., Comparing the efficiencies of stochastic isothermal molecular dynamics methods, preprint, 2011.

Abstract

Based on the thermodynamic concept of a reservoir, I will discuss a computational model for interaction with unresolved degrees of freedom (a thermal bath). The starting point is the assumption that a finite restricted system can be modelled by a generalized canonical ensemble, described by a density which is a smooth function of the energy of the restricted system (an equilibrium state). A stochastic-dynamic thermostat [1] provides a restricted resolved dynamics embedded within the larger energetic bath, while leaving the desired target distribution invariant. Under certain assumptions, these methods can be proven to be ergodic, meaning that almost every extended dynamics trajectory samples the equilibrium measure; I will illustrate this with an example [2].
I will apply these techniques in the novel setting of a simplified point vortex flow on a disc, in which a modified Gibbs distribution (modelling a finite, rather than infinite, bath of weak vortices) provides a regularizing formulation for restricted system dynamics [3]. Although our method does not provide a proper dynamical closure, it is very straightforward to implement in a wide range of situations and can provide realistic averages. Numerical experiments, effectively replacing many vortices by a few artificial degrees of freedom, are in excellent agreement with the two-scale simulations that have appeared in the literature [4].
I will also comment on the issue of approximation of dynamical quantities (such as velocity autocorrelation functions) when using thermostats [5].
References:
[1] Leimkuhler, B., Generalized Bulgac-Kusnezov methods for sampling of the Gibbs-Boltzmann measure, Physical Review E 026703, 2010.
[2] Leimkuhler, B., Noorizadeh, E. and Theil, F., A gentle stochastic thermostat for molecular dynamics, J. Stat. Phys. 135: 261-277, 2009.
[3] Dubinkina, S., Frank, J. and Leimkuhler, B., Simplified modelling of energetic interactions with a thermal bath, with application to a fluid vortex system, SIAM Multiscale Modelling and Simulation 8, 1882-1901, 2010.
[4] B.F.Chler, O., Statistical mechanics of strong and weak point vortices in a cylinder, Physics of Fluids, 14, 2139-2149, 2001.
[5] Leimkuhler, B., Noorizadeh, E. and Penrose, O., Comparing the efficiencies of stochastic isothermal molecular dynamics methods, preprint, 2011.

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Uppdaterad: 2007-01-06