Thermodynamic work statistics for Ornstein-Uhlenbeck-type heat baths

In this work we explore the validity of the transient work fluctuation theorem as well as the Jarzynski equality. In the case of a Brownian particle dragged through a fluid by an optical trap, the fluid plays the role of a non-Markovian heat bath characterized by an specific Ornstein-Uhlenbeck friction memory kernel. To achieve our goal, we used an alternative method which transformed the generalized Langevin equation into an equivalent third-order non-Markovian Langevin equation. By means of the exact analytical solution of this Langevin equation, we calculated the statistics of thermodynamic work which established the Crooks relation, and then the Jarzynski equality came immediately. The fluctuation theorems are applicable to small systems where energy fluctuations with respect to the average behavior are big enough to be measured. The examples of such a class of systems are found inside living organisms.