Entropy-production fluctuation theorem for a generalized Langevin particle in crossed electric and magnetic fields

We study fluctuations of entropy production for a charged Brownian particle confined in a harmonic trap and driven out of equilibrium by crossed electric and magnetic fields. The magnetic field is constant and perpendicular to the plane of motion, while the electric field is time dependent and provides the driving. The non-Markovian dynamics is modeled by a generalized Langevin equation with memory and Gaussian noise. This setting represents a charged Brownian degree of freedom in a structured bath, where delayed friction modifies relaxation while the magnetic field couples the transverse coordinates. Using the exact solution of this linear dynamics, we obtain the time-dependent Gaussian phase-space probability density and from it compute the trajectory-dependent total entropy production. For two solvable driving protocols -- direct forcing by a prescribed time-dependent electric force and dragging of the harmonic trap center -- we prove analytically that the total entropy production obeys a detailed fluctuation theorem.