What is the probability of a thermodynamical transition?

Abstract

If the second law of thermodynamics forbids a transition from one state to another, then it is still possible to make the transition happen by using a sufficient amount of work. But if we do not have access to this amount of work, can the transition happen probabilistically? In the thermodynamic limit, this probability tends to zero, but here we find that for finite-sized systems, it can be finite. We compute the maximum probability of a transition or a thermodynamical fluctuation from any initial state to any final state, and show that this maximum can be achieved for any final state which is block-diagonal in the energy eigenbasis. We also find upper and lower bounds on this transition probability, in terms of the work of transition. As a bi-product, we introduce a finite set of thermodynamical monotones related to the thermo-majorization criteria which governs state transitions, and compute the work of transition in terms of them. The trade-off between the probability of a transition, and any partial work added to aid in that transition is also considered. Our results have applications in entanglement theory, and we find the amount of entanglement required (or gained) when transforming one pure entangled state into any other.

15+6 pages, 7+1 figures V3: Added discussion on heralded probability and relation to fluctuation theorems. V2: Emphasized that X can be any state and that the achievability of our result in the full thermodynamics case, holds only when the target state is block-diagonal in the energy eigenbasis