Editing Fluctuation theorem (section)

== Statement ==

{{Rewrite section|1=Entropy production is not formally defined here, which could lead to confusion. In the standard treatment of the Evans-Searles Theorem in the literature, “entropy production” is typically not normalized by the Boltzmann constant (unlike in other areas of thermodynamics literature). As a result, it does not appear in the exponential. However, if we choose not to explicitly introduce entropy production, readers might assume it is normalized by the Boltzmann constant, in which case we would need to include a factor of its reciprocal in the exponential. --[[User:Maximilian Janisch|Maximilian Janisch]] ([[User talk:Maximilian Janisch|talk]]) 23:07, 1 January 2025 (UTC)|date=January 2025}}

Roughly, the fluctuation theorem relates to the probability distribution of the time-averaged irreversible [[entropy production]], denoted <math>\overline{\Sigma}_t</math>. The theorem states that, in systems away from equilibrium over a finite time ''t'', the ratio between the probability that <math>\overline{\Sigma}_t</math> takes on a value ''A'' and the probability that it takes the opposite value, &minus;''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:
: <math> \frac{\Pr(\overline{\Sigma}_{t}=A)}{\Pr(\overline{\Sigma}_{t}=-A)}=e^{At}.</math>

This means that as the time or system size increases (since <math>\Sigma</math> is [[extensive variable|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.