- #1

bobdavis

- 19

- 8

An uncontained system of particles interacting only by elastic collision is the same as a gas undergoing free expansion. If, for simplicity, the particles are assumed to be identical with equal mass, then the evolution of the system can be treated as if the particles pass through each other without colliding and can be drawn as just the straight world-lines extending from the initial velocity vectors of each particle (both backward and forward in time). There are finitely many particles, so finitely many world-lines and finitely many points of intersection, which all occur within a finite time-interval, before which some particles may be drifting toward each other and after which the particles are all simply drifting away from each other. According to thermodynamics, the expansion of the gas is an irreversible process corresponding to an increase in entropy.

Loschmidt's paradox suggests that it's paradoxical for an irreversible process to arise from time-symmetric dynamics. But in this example the dynamics are time-symmetric and the irreversible expansion appears to happen in either direction. In the normal-time picture there is a time before which there are no intersections of world-lines, and in the reversed-time picture this becomes the time after which there are no intersections and the particles are all drifting away from each other. In either case, leading up to the region of collisions it might appear as though the system is spontaneously compressing over the long-term but eventually after traversing the region of collisions the system will be spontaneously expanding again, and irreversibly in the sense that the system can never spontaneously compress again.

This simple example seems to suggest that there's nothing inherently paradoxical about an irreversible process arising from time-symmetric dynamics, excepting that time-symmetric dynamics might allow a finite duration after any particular starting time where the system can appear to be temporarily running an irreversible process in reverse, but will eventually get past it and permanently fall into an actual normal-time irreversible process.

Is this a correct understanding of applying and resolving Loschmidt's paradox for the example of free gas expansion?