Thermodynamic is the study of the flow of heat of the transformation of work into heat. Our under standing of Thermodynamic is largely related to equilibrium states while the most of processes that happen in nature are far from equilibrium. Non - equilibrium systems can not be described according to classic Thermodynamic. In recent years inventors and engineers have triad to build a machine with small dimensions of nanometer. Concepts of classic Thermodynamic are not suitable for such small systems. However progress in recent decades in the field of Non – equilibrium statistical mechanic has been led to pointed the accurate results foe small systems and also far from equilibrium in which can be pointed the fluctuation theorem. These theorem are including of entropy production theorem crooks theorem Bochkov and Kuzovlev theorem Jarzynski equality and etc. these theorem to be applied to systems that are arbitrarily far from equilibrium or non – equilibrium systems. Thirdly for the first time microscopic dynamics. By stating these theorems the solution of Loschmidt irreversibility paradox can be possible that for the first time was expressed in 1876 by Loschmidt. He stated this paradox in connection with the second law thermodynamic by Boltzma of law of newtons motion in this thesis the kinds of fluctuation theorems and their derivation are considered and we extended generalization of crooks theorem for a situation in which the systems in driven away from equilibrium by work performed on it and during this process the temerature of systems is changed. We also extended generalization of kawai theorem a relation that connects between the dissipation work and the measure of the distinguishability of forward density from its reverse one. We worked out for a harmonic oscillator thet its temiserature changed. In the end we investigated expectations macroscopic from related quantities of entropy.
