Ergodicity and the Numerical Simulation of Hamiltonian Systems.

Publication Type:

Journal Article

Authors:

Tupper, P.F.

Source:

SIAM Journal on Applied Dynamical Systems, Volume 4, Issue 3, p.563-587 (2005)

Abstract:

We discuss the long-time numerical simulation of Hamiltonian systems of ordinary differential equations. Our goal is to explain the ability of symplectic integration schemes such as Stormer-Verlet to compute accurate long-time averages for these systems in the context of molecular dynamics. This paper introduces a weakened version of ergodicity that allows us to study this problem. First, we demonstrate the utility of the weakened ergodicity definition by showing that it is a property of Hamiltonian systems robust to perturbations. Second, we study what the weakened ergodicity of a Hamiltonian system implies about numerical simulations of the system. In the case where a numerical method is volume-conserving and approximately energyconserving, we show that long-time averages are approximated well for suffciently small step lengths.

Last edited by on Sun, 08/26/2007 - 02:13

Continuous and Tractable models for the Variation of Evolutionary Rates

Publication Type:

Journal Article

Source:

Mathematical Biosciences, Volume 199, Issue 2, p.216-233 (2006)

Keywords:

Evolutionary rate; Molecular clocks; CIR process; Diffusionprocesses; Covarion; Phylogenetics.

Abstract:

We propose a continuous model for evolutionary rate variation across sites and over the tree and derive exact transition probabilities under this model. Changes in rate are modelled using the CIR process, a diffusion widely used in financial applications. The model directly extends the standard gamma distributed rates across site model, with one additional parameter governing changes in rate down the tree. The parameters of the model can be estimated directly from two well-known statistics: the index of dispersion and the gamma shape parameter of the rates across sites model. The CIR model can be readily incor- porated into probabilistic models for sequence evolution. We provide here an exact formula for the likelihood of a three taxa tree. Larger trees can be evaluated using Monte-Carlo methods.

Last edited by on Mon, 08/27/2007 - 15:50