Non-markovian dynamics-induced fictitious electric field in the presence of a time-independent magnetic field

Using the property of the solution of the Langevin dynamics with a generalized frictional memory kernel and time-dependent deterministic force field, we show that a solution method (which is very simple as well as shortcut) can be used to derive a Fokker–Planck-like equation (FPLE) for this dynamics. Using this equation, we derive a relation to find the modulation of the entropy production by a time-dependent external force. We also derive a relation between the phase space dynamics and the entropy production of the irreversible thermodynamics. Then FPLE is derived for the non-Markovian dynamics with additional force from harmonic potential, magnetic fields, and both, respectively. Thus, the method is instructive in deriving the FPLE in a shortcut way in the presence of an additional time-dependent stochastic force. Here, we have to consider that the relevant drift terms are independent of the random force.Another very important point is to be noted here. To interpret the FPLE, we recognize that the memory of the non-Markovian dynamics can induce a fictitious electric field in the presence of a time-independent magnetic field and a conservative force field. Then one may notice that like the forces from time-dependent damping, harmonic potential and magnetic field, the fictitious electric field puts its own identity to modulate the effect of the external time-dependent force in the presence of a non-Markovian thermal bath. We understand that the present study, with the recognition of the induced electric field-like quantity, will bring strong attention to different areas of non-equilibrium statistical mechanics, such as the physical tuning of conductivity of ions in solid electrolytes and stochastic thermodynamics for non-Markovian dynamics, etc. Graphical abstract