Effect of time-dependent temperature and field-bath interaction on the diffusion of generalized Brownian motion

The dynamics of a Brownian particle immersed in a bath with variable temperature where both subsystems interact with an external parabolic field are analyzed under the framework of the generalized Langevin equation. This analysis is based on adopting a heuristic method applied to the classical Zwanzig version formulated in 1990 by Brey and Casado. These authors obtained a consistent heat dissipation as a time-dependent temperature external protocol. Including the field-bath interaction, the new dynamic equation retains, in general, the same functional characteristics, with the difference that now the Brownian particle interacts with the external field through an effective frequency modulated by the parameters that characterize the thermal bath. As a consequence, the memory kernel and colored noise are modified by the original field frequency. A method to solve the resulting equation of motion of the Brownian particle is proposed, and the reduced version of its associated Fokker-Planck equation shows that the system would diffuse anomalously, in which the diffusion is a function of time. The solution to the new generalized Langevin equation is analyzed for the average position, and the noise correlation due to the velocity of the Brownian particle for a dedicated heat dissipation model becomes analytic. It is calculated to partially investigate the viability of the equations derived from the theory. From the ensemble-averaged mean-square displacement of the central particle, it is concluded that the process is partially subdiffusive in the time considered in the calculation.