The fluctuation theorem (FT), which originated from statistical mechanics, deals with the relative probability that the entropy of a system which is currently away from thermodynamic equilibrium (i.e., maximum entropy) will increase or decrease over a given amount of time. While the second law of thermodynamics predicts that the entropy of an isolated system should tend to increase until it reaches equilibrium, it became apparent after the discovery of statistical mechanics that the second law is only a statistical one, suggesting that there should always be some nonzero probability that the entropy of an isolated system might spontaneously decrease; the fluctuation theorem precisely quantifies this probability.
Statement
Roughly, the fluctuation theorem relates to the probability distribution of the time-averaged irreversible entropy production, denoted . The theorem states that, in systems away from equilibrium over a finite time t, the ratio between the probability that takes on a value A and the probability that it takes the opposite value, −A, will be exponential in At. In other words, for a finite non-equilibrium system in a finite time, the FT gives a precise mathematical expression for 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 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 expressions in non-equilibrium statistical mechanics that is valid far from equilibrium.
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 is equivalent to the Second Law of Thermodynamics.
History
The FT was first proposed and tested using computer simulations, by Denis Evans, E.G.D. Cohen and Gary Morriss in 1993. The first derivation was given by Evans and Debra Searles in 1994. Since then, much mathematical and computational work has been done to show that the FT applies to a variety of statistical ensembles. The first laboratory experiment that verified the validity of the FT was carried out in 2002. 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. In 2020, observations at high spatial and spectral resolution of the solar photosphere have shown that solar turbulent convection satisfies the symmetries predicted by the fluctuation relation at a local level.
Second law inequality
A simple consequence of the fluctuation theorem given above is that if we carry out an arbitrarily large ensemble of experiments from some initial time t=0, and perform an ensemble average of time averages of the entropy production, then an exact consequence of the FT is that the ensemble average cannot be negative for any value of the averaging time t:
This inequality is called the second law inequality. This inequality can be proved for systems with time dependent fields of arbitrary magnitude and arbitrary time dependence.
It is important to understand what the second law inequality does not imply. It does not imply that the ensemble averaged entropy production is non-negative at all times. This is untrue, as consideration of the entropy production in a viscoelastic fluid subject to a sinusoidal time dependent shear rate shows (e.g., water waves). In this example the ensemble average of the time integral of the entropy production over one cycle is however nonnegative - as expected from the second law inequality.
Nonequilibrium partition identity
Another remarkably simple and elegant consequence of the fluctuation theorem is the so-called "nonequilibrium partition identity" (NPI):
Thus in spite of the Second Law Inequality which might lead you to expect that the average would decay exponentially with time, the exponential probability ratio given by the FT exactly cancels the negative exponential in the average above leading to an average which is unity for all time.