Ambient space formulations and statistical mechanics of holonomically constrained Langevin systems
- First Online:
- 100 Downloads
The most classic approach to the dynamics of an n-dimensional mechanical system constrained by d independent holonomic constraints is to pick explicitly a new set of (n − d) curvilinear coordinatesparametrizingthe manifold of configurations satisfying the constraints, and to compute the Lagrangian generating the unconstrained dynamics in these (n − d) configuration coordinates. Starting from this Lagrangian an unconstrained Hamiltonian H(q,p) on 2(n−d) dimensional phase space can then typically be defined in the standard way via a Legendre transform. Furthermore, if the system is in contact with a heat bath, the associated Langevin and Fokker-Planck equations can be introduced. Provided that an appropriate fluctuation-dissipation condition is satisfied, there will be a canonical equilibrium distribution of the Gibbs form exp(−βH) with respect to the flat measure dqdp in these 2(n − d) dimensional curvilinear phase space coordinates. The existence of (n − d) coordinates satisfying the constraints is often guaranteed locally by an implicit function theorem. Nevertheless in many examples these coordinates cannot be constructed in any tractable form, even locally, so that other approaches are of interest. In ambient space formulations the dynamics are defined in the full original n-dimensional configuration space, and associated 2n-dimensional phase space, with some version of Lagrange multipliers introduced so that the 2(n − d) dimensional sub-manifold of phase space implied by the holonomic constraints and their time derivative, is invariant under the dynamics. In this article we review ambient space formulations, and explain that for constrained dynamics there is in fact considerable freedom in how a Hamiltonian form of the dynamics can be constructed. We then discuss and contrast the Langevin and Fokker-Planck equations and their equilibrium distributions for the different forms of ambient space dynamics.
Unable to display preview. Download preview PDF.
- 7.W. E, E. Vanden-Eijnden, in Multiscale, Modelling, and Simulation, edited by S. Attinger, P. Koumoutsakos (Springer, Berlin, 2004), p. 35Google Scholar
- 8.L. Evans, R. Gariepy, Measure Theory and Fine Properties of Functions (CRC Press, 1992)Google Scholar
- 9.H. Federer, Geometric Measure Theory (Springer, 1969)Google Scholar
- 11.C. Hartmann, Ph.D. thesis, Free University Berlin, 2007Google Scholar
- 13.C. Hartmann, J. Latorre, C. Schütte, Proc. Comp. Sci. 1, 1591 (2010)Google Scholar
- 20.T. Lelièvre, M. Rousset, G. Stoltz., Free Energy Computations: A Mathematical Perspective (Imperial College Press, 2010)Google Scholar
- 21.T. Lelièvre, M. Rousset, G. Stoltz, Math. Comp. (to appear) (2011)Google Scholar
- 23.J. Marsden, T. Ratiu, Introduction to Mechanics and Symmetry (Springer, New York, 1999)Google Scholar
- 28.B. Øksendal, Stochastic Differential Equations, 5th edn. (Springer, Berlin, Heidelberg, 1998)Google Scholar
- 33.P. Wedin, in Matrix Pencils, Vol. 973 of Lecture Notes in Mathematics, edited by B.Kågström, A. Ruhe (Springer, Berlin Heidelberg, 1983), p. 263Google Scholar