Generic signatures of non-Markovian dynamics in the presence of a magnetic field.

In this paper, we have presented the three-dimensional dynamics in the velocity space of a Brownian particle coupled to a non-Markovian thermal bath in the presence of a magnetic field (MF), as well as external deterministic and stochastic force fields. The directions of the MF and the deterministic force are taken to be arbitrary. Furthermore, there are no restrictions on the nature of correlation or the strength of the components of the Gaussian thermal and external noises. We solve the coupled Langevin equations of motion and demonstrate that, in the steady state, a nontrivial symmetry breaking emerges in the motion, manifested in the components of the average velocity resulting from the combined effects of the MF and the deterministic force. The average velocity in this state may be independent of the nature of the thermal bath. We further identify that, in the absence of the external stochastic force (ESF), the relevant variances do not depend on the magnetic field for Markovian dynamics (MD). However, for the corresponding non-Markovian dynamics (NMD), these quantities do depend on the MF, serving as a generic signature of non-Markovianity. Although the equations of motion are coupled, the equal-time cross-correlations vanish for MD in the absence of ESF, whereas they may become nonzero for NMD provided the magnetic field has at least two nonzero components. This can be regarded as another generic signature of non-Markovian dynamics. The consequences of these signatures are discussed in terms of the information entropy, the nonequilibrium kinetic temperature, and the entropic temperature. Writing a Fokker-Planck-like equation (FPLE) in the velocity space for a Brownian particle coupled to a Markovian thermal bath in the presence of nonwhite Gaussian external noise remains an open problem. It is noteworthy that we formulate the FPLE corresponding to the coupled Langevin equations of motion, which appears to be an even more complex and open issue. Using this equation, we compare the relaxation processes in Markovian and non-Markovian dynamics in terms of the total entropy production, the entropy production of irreversible thermodynamics, and their respective bounds.