Loschmidt's paradox

Arrow of time

Any process that happens regularly in the forward direction of time but rarely or never in the opposite direction, such as entropy increasing in an isolated system, defines what physicists call an thermodynamic arrow of time given time-symmetric fundamental laws, since time-symmetry implies that for any process compatible with these fundamental laws, a reversed version that looked exactly like a film of the first process played backwards would be equally compatible with the same fundamental laws, and would even be equally probable if one were to pick the system's initial state randomly from the phase space of all possible states for that system.

Although most of the arrows of time described by physicists are thought to be special cases of the thermodynamic arrow, there are a few that are believed to be unconnected, like the cosmological arrow of time based on the fact that the universe is expanding rather than contracting, and the fact that there are a few processes in particle physics actually violate time-symmetry, although they respect a related symmetry known as CPT symmetry. In the case of the cosmological arrow, most physicists believe that entropy would continue to increase even if the universe began to contract (although the physicist Thomas Gold once proposed a model in which the thermodynamic arrow would reverse in this phase). In the case of the violations of time-symmetry in particle physics, the situations in which they occur are rare and are only known to involve a few types of meson particles. Furthermore, due to CPT symmetry reversal of time direction is equivalent to renaming particles as antiparticles and vice versa. Therefore this cannot explain Loschmidt's paradox.

Fluctuation theorem

One approach to handling Loschmidt's paradox is the 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).

The Big Bang

Another way of dealing with Loschmidt's paradox is to see the second law as an expression of a set of boundary conditions, in which our universe's time coordinate has a low-entropy endpoint: the Big Bang. From this point of view, the arrow of time is determined entirely by the direction that leads to the Big Bang, and a hypothetical universe with a maximum-entropy Big Bang would have no arrow of time. The theory of cosmic inflation tries to give reason why the early universe had such a low entropy.

See also

• Maximum entropy thermodynamics for one particular perspective on entropy, reversibility and the Second Law
• Poincaré recurrence theorem
• Reversibility
• Statistical mechanics

References

• J. Loschmidt, Sitzungsber. Kais. Akad. Wiss. Wien, Math. Naturwiss. Classe 73, 128–142 (1876)