Editing Fluctuation theorem (section)

== Relation to Loschmidt's paradox ==

The [[second law of thermodynamics]], which predicts that the entropy of an isolated system out of equilibrium should tend to increase rather than decrease or stay constant, stands in apparent contradiction with the [[T-symmetry|time-reversible]] equations of motion for classical and quantum systems. The time reversal symmetry of the equations of motion show that if one films a given time dependent physical process, then playing the movie of that process backwards does not violate the laws of mechanics. It is often argued that for every forward trajectory in which entropy increases, there exists a time reversed anti trajectory where entropy decreases, thus if one picks an initial state randomly from the system's [[phase space]] and evolves it forward according to the laws governing the system, decreasing entropy should be just as likely as increasing entropy. It might seem that this is incompatible with the [[second law of thermodynamics]] which predicts that entropy tends to increase. 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.<ref>{{Cite journal |last1=Evans |first1=Denis J. |last2=Searles |first2=Debra J. |date=2002 |title=The Fluctuation Theorem |url=http://www.tandfonline.com/doi/abs/10.1080/00018730210155133 |journal=Advances in Physics |language=en |volume=51 |issue=7 |pages=1529–1585 |doi=10.1080/00018730210155133 |bibcode=2002AdPhy..51.1529E |s2cid=10308868 |issn=0001-8732}}</ref> 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. This second prediction, which is inconsistent with the real world, is obtained using an anti-causal assumption. That is to say that effect (the value taken on by the dissipation function) precedes the cause (here the later state has been incorrectly used for the initial conditions). The fluctuation theorem shows how the second law is a consequence of the assumption of causality. 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.