Optimal work associated with off-centered harmonic Brownian motion at any friction damping

Work and heat distributions of an inertial Brownian particle

The work and heat distribution densities of a classical Brownian particle immersed in a heat bath and interacting with an external field have been extensively analyzed from different approaches. In this article, a previous method based on basic principles of stochastic dynamics is extended to derive these functions. The starting point is the inertial Langevin equation and the external field is an off-center harmonic potential driven by an external protocol. Unlike previous works where the driving is arbitrary, the so-called optimal protocol that minimizes the mechanical work is used instead. The corresponding work and heat distributions are derived through a procedure based on getting a generic Fokker-Planck equation indistinctly of the variable under consideration. The work distribution is calculated for different initial conditions and values of the friction coefficient of the thermal fluid ranging from the periodic or very low-underdamped mode up to the overdamped regime. It is a Gaussian as that of previous experiments of a particle trapped in an optical tweezers moved at constant velocity. Some aspects about the heat distribution is analyzed in terms of the statistical features of the non-Gaussian noise accompanying its dynamics to give an account of experimental results. It is concluded that the easiness in calculating the work distribution cannot be applied to heat. It requires numerical calculations.