Review on Fluctuation Theorems with Absolute Irreversibility

Abstract

In this chapter we review the notion of absolute irreversibility. While the fluctuation theorem applies to a wide range of nonequilibrium situations, it is known that it breaks down in some situations. As an example, we discuss the Jarzynski equality under free expansion with two different implementations, i.e., by removal of a partition and by shift of a partition at an infinite velocity. In the former situation the Jarzynski equality breaks down since its initial state is not in a global equilibrium but in a constrained equilibrium. In the later situation, it does not converge due to the far tail of the Maxwellian distribution. To save the integral fluctuation theorem, we introduce the notion of absolute irreversibility. Absolute irreversibility refers to the mathematical singularity of the reference probability measure with respect to the original probability measure, and physically corresponds to negatively divergent entropy production. When this singularity comes into effect, the ordinary integral fluctuation theorem can no longer be applied, and we should modify it into a form that incorporates the degree of absolute irreversibility. We show how we achieve this modification on the basis of the Lebesgue decomposition theorem. Finally, we give examples to demonstrate the validity of the fluctuation theorem with absolute irreversibility. This chapter is based on the author’s study in his master course.