Editing Loschmidt's paradox (section)

== Fluctuation theorem ==
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{{main|Fluctuation theorem}}

One approach to handling Loschmidt's paradox is the [[fluctuation theorem]], derived heuristically by [[Denis Evans]] and [[Debra Searles]], which gives a numerical estimate of the probability that a system away from equilibrium will have a certain value for the dissipation function (often an entropy like property) over a certain amount of time.<ref>D. J. Evans and D. J. Searles, Adv. Phys. '''51''', 1529 (2002).</ref> The result is obtained with the exact time reversible dynamical equations of motion and the [[universal causation]] proposition. The fluctuation theorem is obtained using the fact that dynamics is time reversible.{{Citation needed|date=April 2022}} Quantitative predictions of this theorem have been confirmed in laboratory experiments at the [[Australian National University]] conducted by [[Edith M. Sevick]] et al. using [[optical tweezers]] apparatus.<ref>{{Cite web |last=Sevick |first=Edith |title=2002 RSC Annual Report - Polymers and Soft Condensed Matter |url=https://rsc.anu.edu.au/AnnualReport/Report2002/CB-report.html |access-date=2022-04-01 |website=Research School of Chemistry. Australian National University.}}</ref> This theorem is applicable for transient systems, which may initially be in equilibrium and then driven away (as was the case for the first experiment by Sevick et al.) or some other arbitrary initial state, including relaxation towards equilibrium. There is also an asymptotic result for systems which are in a nonequilibrium steady state at all times.

There is a crucial point in the fluctuation theorem, that differs from how Loschmidt framed the paradox. Loschmidt considered the probability of observing a single trajectory, which is analogous to enquiring about the probability of observing a single point in phase space. In both of these cases the probability is always zero. To be able to effectively address this you must consider the probability density for a set of points in a small region of phase space, or a set of trajectories. The fluctuation theorem considers the probability density for all of the trajectories that are initially in an infinitesimally small region of phase space. This leads directly to the probability of finding a trajectory, in either the forward or the reverse trajectory sets, depending upon the initial probability distribution as well as the dissipation which is done as the system evolves. It is this crucial difference in approach that allows the fluctuation theorem to correctly solve the paradox.