I came across a stochastic dynamical system which is modeled with a conservative Hamiltonian component and an Ornstein-Uhlenbeck component. It is meant to represent small perturbations around a Hamiltonian system, and broadly I'm trying to understand what this means. Very introductory background references would be helpful.
More specifically, the SDE is represented as: $$ \mathrm{d} x = J \nabla H(x)\, \mathrm{d} t - \gamma S \nabla H (x)\, \mathrm{d} t + \sqrt{\gamma \tau} S^{1/2}\, \mathrm{d} W_t $$ where $H(x)$ is the Hamiltonian, $J$ is skew-symmetric, $S$ is symmetric matrices, $W_t$ is a Wiener process, $\gamma$ is a damping parameter, and $\tau$ is the temperature. If $\gamma = 0$, then we have Hamiltonian dynamics. If $\tau = 0$, then we have steepest descent to equilibrium.
A few questions:
- What is the purpose / interpretation of $J$ and $S$?
- In the references I've looked through, $\gamma$ is commonly referred to as "friction" and doesn't have a $\nabla H(x)$ term in front. Is $\mathrm{d} x = - \gamma x\, \mathrm{d} t + \sqrt{\gamma \tau} S^{1/2}\, \mathrm{d} W_t$ still the OU part? Does the value of $\gamma$ lead to over- and under-damped equations?
- Is "Langevin dynamics" a broad term for an SDE with a conservative Hamiltonian part and an OU part?