Numerical study of the Holstein model in different thermostats

Based on the Holstein Hamiltonian, the dynamics of the charge introduced into the molecular chain of sites was modeled at different temperatures. In the calculation, the temperature of the chain is set by the initial data — random Gaussian distributions of velocities and site displacements. Various options for the initial charge density distribution are considered. Long-term calculations show that the system moves to fluctuations near a new equilibrium state. For the same initial velocities and displacements, the average kinetic energy, and, accordingly, the temperature of the $T$ chain, varies depending on the initial distribution of the charge density: it decreases when a polaron is introduced into the chain, or increases if at the initial moment the electronic part of the energy is maximum. A comparison is made with the results obtained previously in the model with a Langevin thermostat. In both cases, the existence of a polaron is determined by the thermal energy of the entire chain.
According to the simulation results, the transition from the polaron mode to the delocalized state occurs in the same range of thermal energy values of a chain of $N$ sites $\sim NT$ for both thermostat options, with an additional adjustment: for the Hamiltonian system the temperature does not correspond to the initially set one, but is determined after long-term calculations from the average kinetic energy of the chain.
In the polaron region, the use of different methods for simulating temperature leads to a number of significant differences in the dynamics of the system. In the region of the delocalized state of charge, for high temperatures, the results averaged over a set of trajectories in a system with a random force and the results averaged over time for a Hamiltonian system are close, which does not contradict the ergodic hypothesis. From a practical point of view, for large temperatures $T\approx 300$K, when simulating charge transfer in homogeneous chains, any of these options for setting the thermostat can be used.