Loschmidt's Paradox - Fluctuation Theorem

Fluctuation Theorem

One approach to handling Loschmidt's paradox is the fluctuation theorem, proved by Denis Evans and Debra Searles, which gives a numerical estimate of the probability that a system away from equilibrium will have a certain change in entropy over a certain amount of time. The theorem is proved with the exact time reversible dynamical equations of motion and the Axiom of Causality. The fluctuation theorem is proved using the fact that dynamics is time reversible. 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.

However, the fluctuation theorem assumes that the system is initially in a non-equilibrium state, so it can be argued that the theorem only verifies the time-asymmetry of the second law of thermodynamics based on an a priori assumption of time-asymmetric boundary conditions. If no low-entropy boundary conditions in the past are assumed, the fluctuation theorem should give exactly the same predictions in the reverse time direction as it does in the forward direction, meaning that if you observe a system in a nonequilibrium state, you should predict that its entropy was more likely to have been higher at earlier times as well as later times. This prediction would be at odds with everyday experience, since if you film a typical nonequilibrium system and play the film in reverse, you typically see the entropy steadily decreasing rather than increasing. Thus we still have no explanation for the arrow of time that is defined by the observation that the fluctuation theorem gives correct predictions in the forward direction but not the backward direction, so the fundamental paradox remains unsolved.

Note, however, that if you were looking at an isolated system which had reached equilibrium long in the past, so that any departures from equilibrium were the result of random fluctuations, then the backwards prediction would be just as accurate as the forward one, because if you happen to see the system in a nonequilibrium state it is overwhelmingly likely that you are looking at the minimum-entropy point of the random fluctuation (if it were truly random, there's no reason to expect it to continue to drop to even lower values of entropy, or to expect it had dropped to even lower levels earlier), meaning that entropy was probably higher in both the past and the future of that state. So, the fact that the time-reversed version of the fluctuation theorem does not ordinarily give accurate predictions in the real world is reason to think that the nonequilibrium state of the universe at the present moment is not simply a result of a random fluctuation, and that there must be some other explanation such as the Big Bang starting the universe off in a low-entropy state (see below).