In this paper, the stochastic diffusion process of a charged classical harmonic oscillator in a constant magnetic field is exactly described through the analytical solution of the associated Langevin equation. Due to the presence of the magnetic field, stochastic diffusion takes place across and along the magnetic field. Along the magnetic field, the Brownian motion is exactly the same as that of the ordinary one-dimensional classical harmonic oscillator, which was very well described in Chandrasekhar's celebrated paper [Rev. Mod. Phys. 15, 1 (1943)]. Across the magnetic field, the stochastic process takes place on a plane, perpendicular to the magnetic field. For internally Gaussian white noise, this planar-diffusion process is exactly described through the first two moments of the positions and velocities and their corresponding cross correlations. In the absence of the magnetic field, our analytical results are the same as those calculated by Chandrasekhar for the ordinary harmonic oscillator. The stochastic planar diffusion is also well characterized in the overdamped approximation, through the solutions of the Langevin equation.

The theoretical study about the transient and stationary fluctuation theorems is extended to include the effects of electromagnetic fields on a charged Brownian particle. In particular, we consider a harmonic trapped Brownian particle under the action of a constant magnetic field pointing perpendicular to a plane and a time-dependent electric field acting on this plane. The electric field is seen to be responsible for the motion of the center in the harmonic trap, giving as a result a time-dependent dragging. Our study is focused on the solution of the Smoluchowski equation associated with the over-damped Langevin equation and also considers two particular cases for the motion of the harmonic trap minimum. The first one is produced by a linear time-dependent electric field and, in the second case an oscillating electric field produces a circular motion. In this last case we have found resonant behavior in the mean work when the electric field is tuned with Larmor's frequency. Some comparisons are made with other works in the absence of the magnetic field.

In this work we give an alternative method to calculate the transition probability densities (TPD) for the velocity space, phase space, and Smoluchowsky configuration space of a Brownian gas of charged particles in the presence of a constant magnetic field. Our proposal consists in transforming, by means of a rotation matrix, the Langevin equation of a charged particle in the velocity space into another velocity space where the behavior is quite similar to that of ordinary Brownian motion. A similar strategy is also applied to the phase-space. In consequence, in the transformed space both the Fokker-Planck and Fokker-Planck-Kramers equations are solved following Chandrasekhar's methodology. Our results are compared with those obtained by Czopnik and Garbaczewski [Phys. Rev. E 63, 021105 (2001)].

The nonlinear relaxation time and the statistics of the first passage time distribution in connection with the quasideterministic approach are used to detect weak signals in the decay process of the unstable state of a Brownian particle embedded in memory thermal baths. The study is performed in the overdamped approximation of a generalized Langevin equation characterized by an exponential decay in the friction memory kernel. A detection criterion for each time scale is studied: The first one is referred to as the receiver output, which is given as a function of the nonlinear relaxation time, and the second one is related to the statistics of the first passage time distribution.

The detection of weak signals through nonlinear relaxation times for a Brownian particle in an electromagnetic field is studied in the dynamical relaxation of the unstable state, characterized by a two-dimensional bistable potential. The detection process depends on a dimensionless quantity referred to as the receiver output, calculated as a function of the nonlinear relaxation time and being a characteristic time scale of our system. The latter characterizes the complete dynamical relaxation of the Brownian particle as it relaxes from the initial unstable state of the bistable potential to its corresponding steady state. The one-dimensional problem is also studied to complement the description.

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In this work, it is shown that the detailed fluctuation theorem for the total entropy production of a charged particle in a two-dimensional harmonic trap under the action of an electromagnetic field is valid in two physical situations. The proof of the theorem is achieved if the particle is initially distributed with a canonical distribution at equilibrium with the thermal bath. The two examined cases are the following: in the first case, the charged particle in the harmonic trap is subjected to an arbitrary time-dependent electric field; in the second one, the minimum of the harmonic trap is arbitrarily dragged by such an electric field. The theoretical framework is developed within the context of stochastic thermodynamics and the Langevin dynamics for the charged particle.

We study the statistical properties of the total work associated with the Langevin equation for an electrically charged Brownian particle in a two-dimensional harmonic trap and in the presence of a uniform magnetic field. The calculations are performed under the overdamped approximation. The center of the harmonic trap is dragged in an arbitrary time-dependent way. As a result we have found the relation of the averaged work and the variance in the work distribution in the presence of the magnetic field. In addition, the Jarzynski equality (JE) is considered when the potential associated with the working force contains a time-dependent term, giving a way to calculate the change in the free energy. The particular cases in Jayannavar and Sahoo's work [Phys. Rev. E 75, 032102 (2007)] and their use of the JE are also discussed.