In the study of stochastic partial differential equation one may wonder what is the noise? Often the stochasticity modelled in partial differential equations has its origin in thermal fluctuations. Starting from a quantum formulation of a molecular system coupled to a heat bath, I will show that ab initio Langevin dynamics, with a certain rank one friction matrix determined by the coupling, approximates the quantum system more accurately than any Hamiltonian system, for large mass ratio between the system and heat bath nuclei. I will also give an example of course-graining a stochastic molecular dynamics equation to obtain a continuum stochastic partial differential equation for phase transitions.
This is joint work with Håkon Hoel (Oslo) and Erik von Schwerin (KAUST).