Introduction

For a system under thermal conditions in a heat bath with temperature T, the dynamics of each of the system particles is influenced by interactions with the heat-bath particles. If quantum effects are negligible (what we will assume in the following), the classical motion of any system particle looks erratic; the particle follows a stochastic path. The system can “gain” energy from the heat bath by these collisions (which are typically more generally called “thermal fluctuations”) or “lose” energy by friction effects (dissipation). The total energy of the coupled system of heat bath and particles is a conserved quantity, i.e., fluctuation and dissipation refer to the energetic exchange between heat bath and system particles only. Consequently, the coupled system is represented by a microcanonical ensemble, whereas the particle system is in this case represented by a canonical ensemble: The energy of the particle system is not a constant of motion. Provided heat bath and system are in thermal equilibrium, i.e., heat-bath and system temperature are identical, fluctuations and dissipation balance each other. This is the essence of the celebrated fluctuation-dissipation theorem [74]. In equilibrium, only the statistical mean of the particle system energy is constant in time.

This canonical behavior of the system particles is not accounted for by standard Newtonian dynamics (where the system energy is considered to be a constant of motion). In order to perform molecular dynamics (MD) simulations of the system under the influence of thermal fluctuations, the coupling of the system to the heat bath is required. This is provided by a thermostat, i.e., by extending the equations of motion by additional heatbath coupling degrees of freedom [75].