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.
Roughly, the fluctuation theorem relates to the probability distribution of the time-averaged irreversible entropy production, denoted
is extensive), the probability of observing an entropy production opposite to that dictated by the second law of thermodynamics decreases exponentially.
Fluctuations in the velocity were recorded that were opposite to what the second law of thermodynamics would dictate for macroscopic systems.
This is untrue, as consideration of the entropy production in a viscoelastic fluid subject to a sinusoidal time dependent shear rate shows (e.g., rogue waves).
[clarification needed][dubious – discuss] 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.
One is that small machines (such as nanomachines or even mitochondria in a cell) will spend part of their time actually running in "reverse".
What is meant by "reverse" is that it is possible to observe that these small molecular machines are able to generate work by taking heat from the environment.
The environment itself continuously drives these molecular machines away from equilibrium and the fluctuations it generates over the system are very relevant because the probability of observing an apparent violation of the second law of thermodynamics becomes significant at this scale.
This is counterintuitive because, from a macroscopic point of view, it would describe complex processes running in reverse.
For example, a jet engine running in reverse, taking in ambient heat and exhaust fumes to generate kerosene and oxygen.
, and the system volume V, divided by the absolute temperature T, of the heat reservoir times the Boltzmann constant.
Thus the dissipation function is easily recognised as the Ohmic work done on the system divided by the temperature of the reservoir.
[11] However, the fluctuation theorem applies to systems arbitrarily far from equilibrium where the definition of the spontaneous entropy production is problematic.
The problem of deriving irreversible thermodynamics from time-symmetric fundamental laws is referred to as Loschmidt's paradox.
The mathematical derivation of the fluctuation theorem and in particular the second law inequality shows that, for a nonequilibrium process, the ensemble averaged value for the dissipation function will be greater than zero.
[12] This result requires causality, i.e. that cause (the initial conditions) precede effect (the value taken on by the dissipation function).
This is clearly demonstrated in section 6 of that paper, where it is shown how one could use the same laws of mechanics to extrapolate backwards from a later state to an earlier state, and in this case the fluctuation theorem would lead us to predict the ensemble average dissipation function to be negative, an anti-second law.
When we solve a problem we set the initial conditions and then let the laws of mechanics evolve the system forward in time, we don't solve problems by setting the final conditions and letting the laws of mechanics run backwards in time.
When combined with the central limit theorem, the FT also implies the Green-Kubo relations for linear transport coefficients, close to equilibrium.
There are many examples known where the distribution of time averaged dissipation is non-Gaussian and yet the FT (of course) still correctly describes the probability ratios.
Lastly the theoretical constructs used to prove the FT can be applied to nonequilibrium transitions between two different equilibrium states.
This equality shows how equilibrium free energy differences can be computed or measured (in the laboratory[13]), from nonequilibrium path integrals.
In quantum mechanical systems, however, the weak nuclear force is not invariant under T-symmetry alone; if weak interactions are present reversible dynamics are still possible, but only if the operator π also reverses the signs of all the charges and the parity of the spatial co-ordinates (C-symmetry and P-symmetry).