Numerical methods for extended Hamiltonian systems with applications in statistical mechanics

We present numerical methods for integrating the dynamics of extended Hamiltonian systems arising from applications in statistical mechanics. Classical Hamiltonian systems are augmented with extended variables to produce configurations from a particular equilibrium distribution. We start by considering the Nose Hamiltonian, and show how discretizations can be developed for constant temperature dynamics which preserve symplectic and time-reversible properties in extended phase space. Backward error analysis is used to estimate the error in the statistical averages generated by constant temperature methods. The validity of these error estimates is investigated in a series of numerical experiments involving a Lennard-Jones gas.; A second class of extended Hamiltonian systems arises from the approximation of Feynman-Kac path integrals for quantum statistical mechanics. We consider the discretization of this class of infinite dimensional path integrals by reduction to a finite-dimensional approximating subspace. We prove that in its primitive form, the error induced by this approximation technique is of order 1/d, where d is the dimension of the subspace. However, this order result can be drastically improved when the density matrix is applied with a modified reference potential. In numerical experiments, several discretizations based on subspace approximations are applied to one-dimensional model problems. It is shown that a new method using Hermite cubic splines is the most efficient for calculating the average energy.