The reason the Poincaré recurrence theorem (also called Zermelo-Poincaré recurrence) posed a problem is that Boltzmann constructed his entire theory with the assumption that time had a direction. To be precise, he defined his now famous H-theorem such that time increased in the "correct" direction. That is, Boltzmann's equation describes a time-irreversible system which is directly in contradiction with Zermelo-Poincaré recurrence. The interesting part is that Boltzmann made a few more assumptions that we now know to be incorrect but we do not fully understand why the failure of these assumptions does not render the predicted results of his equation useless. Those assumptions are:
- that the velocities of any two particles in the system described by the particle velocity distribution functions are uncorrelated (this fails the moment two particles collide); and
- that the collisions result in a time-irreversible evolution of the equations.
We know that 1 fails the moment two particles collide, which kind of defeats the purpose of having a collision operator (Note that this assumption is now called the propagation of one-sided chaos).
We know that Boltzmann's collision operator assumed that particles collided like billiard balls undergoing elastic collisions, a process that is time-reversible. This originally led to what is called Loschmidt’s paradox – How can an irreversible process result from lots of reversible processes (perhaps a poor paraphrasing?)? This was eventually resolved physicists realized that the reversible microdynamics involved in collisions were not contradictory to irreversible macrodynamics (i.e. thermodynamics) if one uses the right amount of probability to interpret the macroscopic model. One must realize that the initial choice of the distribution function in the derivation of the Boltzmann equation introduces the necessary and sufficient probabilities. Unfortunately, there is no mathematically rigorous argument against Loschmidt’s paradox.
Poincaré's recurrence theorem remained as far as I know, unproven until 1919 when Caratheodóry proved it. Why then did it represent an issue to Boltzmann?
It was an issue because Boltzmann's system was ultimately made of reversible processes yet it predicted irreversible dynamics. If the system is irreversible, then one cannot return to the original state. It turns out that the irreversibility is somewhat artificially introduced in multiple ways, one of which is assuming a continuous distribution function to describe a discrete number of particles, i.e., by assuming a continuous distribution function one loses the ability to follow any individual particle in time, thus one loses information. One can also render a system irreversible through the choice of initial and/or boundary conditions.
Boltzmann died in 1906, did he not know about this? I believe he did, because his ergodic hypothesis is designed exactly to combat this assertion of the poincaré's theorem.
Yes, Boltzmann was aware of Poincaré's theorem. In one of the references below (I believe, it's been several years since I dug through them) one of the authors comments that someone asked Boltzmann why he could assign a sign to his H-theorem, i.e., how could he force time to go in one direction? Boltzmann apparently quipped a remark along the lines of "Show me an example where it does the opposite." I do not recall the exact quote/exchange, but the point is that Boltzmann was aware of this issue.
- Evans, D.J. "On the entropy of nonequilibrium states," J. Statistical Phys. 57, pp. 745–758, doi:10.1007/BF01022830, 1989.
- Evans, D.J., and G. Morriss Statistical Mechanics of Nonequilibrium Liquids, 1st Edition, Academic Press, London, 1990.
- Evans, D.J., and D. J. Searles "Equilibrium microstates which generate second law violating steady states," Phys. Rev. E 50, pp. 1645–1648, doi:10.1103/PhysRevE.50.1645, 1994.
- Evans, D.J., E.G.D. Cohen, and G.P. Morriss "Viscosity of a simple fluid from its maximal Lyapunov exponents," Phys. Rev. A 42, pp. 5990–5997, doi:10.1103/PhysRevA.42.5990, 1990.
- Gressman, P.T., and R.M. Strain "Global classical solutions of the Boltzmann equation with long-range interactions," Proc. Nat. Acad. Sci. USA 107, pp 5744–5749, doi:10.1073/pnas.1001185107, 2010.
- Hoover, W. (Ed.) Molecular Dynamics, Lecture Notes in Physics, Berlin Springer Verlag, 258, 1986.
- Villani, C., Chapter 2, A review of mathematical topics in collisional kinetic theory, pp. 71–74, North-Holland, Washington, D.C., doi:10.1016/S1874-5792(02)80004-0, 2002.
- Villani, C. "Entropy production and convergence to equilibrium for the Boltzmann equation," in XIVTH International Congress on Mathematical Physics, Edited by J.-C. Zambrini, pp. 130–144, doi:10.1142/9789812704016_0011, 2006.