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Fluctuation theorem

The fluctuation theorem (FT) is a mathematical expression for the probability ratio that time-averaged irreversible entropy production, takes on a value, A, to the opposite value, −A, in systems away from equilibrium. In other words, for a finite non-equilibrium system in a finite time, the FT gives the probability that entropy will flow in a direction opposite to that dictated by the second law of thermodynamics.

Mathematically, the FT is expressed as:

This means that as the time or system size increases (since Σ is extensive), the probability of observing an entropy production opposite to that dictated by the second law of thermodynamics decreases exponentially. The FT is one of the few expression in non-equililbrium statistical mechanics that is valid far from equilibrium.

The FT was first introduced by Evans, Cohen and Morriss in 1993 in the journal Physical Review Letters. Since then, much mathematical and computational work has been done to show that the FT applies to a variety of statistical ensembles. Recently, the first laboratory experiment was performed that verified the validity of the FT. In this experiment, a plastic bead was pulled through a solution by a laser. Fluctuations in the velocity were recorded that were opposite to what the second law of thermodynamics would dictate for macroscopic systems.

Note that the FT does not state that the second law of thermodynamics is wrong or invalid. The second law of thermodynamics is a statement about macroscopic systems. The FT is more general. It can be applied to both microscopic and macroscopic systems. When applied to macroscopic systems, the FT verifies the second law of thermodynamics.

There are many important implications from the FT. One is that small machines (such as nanomachines or even mitochondria in a cell) will spend part of their time actually running in "reverse". By "reverse", it is meant that they function so as to run in a way opposite to that to which they are presumably designed. As an example, consider a jet engine. If a jet engine were to run in "reverse" in this context, it would take in exhaust fumes and create kerosene and oxygen.