We derive a Crooks-Jarzynski-type identity for computing free energy differences between metastable states that is based on nonequilibrium diffusion processes. Furthermore we outline a brief derivation of an infinite-dimensional stochastic partial differential equation that can be used to efficiently generate the ensemble of trajectories connecting the metastable states.
We propose a robust and efficient numerical discretization scheme for the infinitesimal generator of a diffusion process based on a finite volume approximation. The resulting discrete-space operator can be interpreted as a jump process on the mesh whose invariant measure is precisely the cell approximation of the Boltzmann distribution of the original process. Moreover the resulting jump process preserves the detailed balance property of the original stochastic process.
We revisit the problem of the linear response of a constrained mechanical system. In doing so we show that the standard expressions of Green and Kubo carry over to the constrained case without any alteration. The argument is based on the appropriate definition of constrained expectations by means of which Liouville’s theorem and the Green-Kubo relations naturally follow.
We propose a nonequilibrium sampling method for computing free energy profiles along a given reaction coordinate. The method consists of two parts: a controlled Langevin sampler that generates nonequilibrium bridge paths conditioned by the reaction coordinate, and Jarzynski’s formula for reweighting the paths. Our derivation of the equa- tions of motion of the sampler is based on stochastic perturbation of a controlled dissipative Hamiltonian system, for which we prove Jarzynski’s identity as a special case of the Feynman-Kac formula. We illustrate our method by means of a suitable numerical example and briefly discuss issues of optimally choosing the control protocol for the reaction coordinate.
We study balanced model reduction of partially-observed linear stochastic differential equa- tions of Langevin type. Balancing the equations of motion gives rise to a singularly perturbed system of equations with slow and fast degrees of freedom, and we prove that in the limit of the fast variables becoming infinitely fast, the solutions converge to the solution of a reduced-order Langevin equation. We illustrate the method with several numerical examples and discuss the relation to model reduction of deterministic control systems that have an underlying Hamiltonian structure.
We study balanced model reduction for stable bilinear systems in the limit of partly vanishing Hankel singular values. We show that the dynamics admit a splitting into fast and slow subspaces and prove an averaging principle for the slow dynamics. We illustrate our method with an example from stochastic control (density evolution of a dragged Brownian particle) and discuss issues of structure preservation and positivity.