# Loschmidt's paradox

**Loschmidt's paradox**, also known as the **reversibility paradox**, **irreversibility paradox** or * Umkehreinwand*, is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics. This puts the time reversal symmetry of (almost) all known low-level fundamental physical processes at odds with any attempt to infer from them the second law of thermodynamics which describes the behaviour of macroscopic systems. Both of these are well-accepted principles in physics, with sound observational and theoretical support, yet they seem to be in conflict; hence the paradox.

## Origin

Johann Loschmidt's criticism was provoked by the H-theorem of Boltzmann, which employed kinetic theory to explain the increase of entropy in an ideal gas from a non-equilibrium state, when the molecules of the gas are allowed to collide. In 1876, Loschmidt pointed out that if there is a motion of a system from time t_{0} to time t_{1} to time t_{2} that leads to a steady decrease of *H* (increase of entropy) with time, then there is another allowed state of motion of the system at t_{1}, found by reversing all the velocities, in which *H* must increase. This revealed that one of Boltzmann's key assumptions, molecular chaos, or, the *Stosszahlansatz*, that all particle velocities were completely uncorrelated, did not follow from Newtonian dynamics. One can assert that possible correlations are uninteresting, and therefore decide to ignore them; but if one does so, one has changed the conceptual system, injecting an element of time-asymmetry by that very action.

Reversible laws of motion cannot explain why we experience our world to be in such a comparatively low state of entropy at the moment (compared to the equilibrium entropy of universal heat death); and to have been at even lower entropy in the past.

### Before Loschmidt

In 1874, two years before the Loschmidt paper, William Thomson defended the second law against the time reversal objection.^{[1]}

## 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 arrow of time in nature. This term only refers to an observation of an asymmetry in time, it is not meant to suggest an explanation for such asymmetries. Loschmidt's paradox is equivalent to the question of how it is possible that there could be a 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 a few processes in particle physics actually violate time-symmetry, while 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.

## Dynamical systems

Current research in dynamical systems offers one possible mechanism for obtaining irreversibility from reversible systems. The central argument is based on the claim that the correct way to study the dynamics of macroscopic systems is to study the transfer operator corresponding to the microscopic equations of motion. It is then argued that the transfer operator is not unitary (*i.e.* is not reversible) but has eigenvalues whose magnitude is strictly less than one; these eigenvalues corresponding to decaying physical states. This approach is fraught with various difficulties; it works well for only a handful of exactly solvable models.^{[2]}

Abstract mathematical tools used in the study of dissipative systems include definitions of mixing, wandering sets, and ergodic theory in general.

## 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 appears at odds with everyday experience in systems that are not closed, 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 starting point: the Big Bang. From this point of view, the arrow of time is determined entirely by the direction that leads away from 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

- ↑ Thomson, W. (Lord Kelvin) (1874/1875). The kinetic theory of the dissipation of energy,
*Nature*, Vol. IX, 1874-04-09, 441–444. - ↑ Dean J. Driebe,
*Fully Chaotic Maps and Broken Time Symmetry*, (1999) Kluwer Academic ISBN 0-7923-5564-4

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

## External links

- Reversible laws of motion and the arrow of time by Mark Tuckerman
- A toy system with time-reversible discrete dynamics showing entropy increase