# Loschmidt's paradox in Bohmian Mechanics

In Bohmian mechanics, the position of the particles must have a random distribution given by $\rho = |\Psi|^2$, where $\Psi$ is the wave function, in order to be compatible with Born rule in standard Quantum Mechanics. If I understand correctly, this is referred to as the quantum equilibrium hypothesis (QEH). There is a priori no reason not to consider non-quantum distributions but then an equivalent of Boltzmann H-theorem is claimed to guarantee that such a non-quantum distribution would very quickly relax to $\rho = |\Psi|^2$, and would therefore never be observed.

But then how would one evade Loschmidt's argument in this context? The equations of motions for the particles, and for the wave function, are definitively deterministic. The situation seems actually worse than in classical Newtonian mechanics since the particle do actually not have a real dynamic: the equations of motion boils down to the speed of the particle being directly determined by the wave function.

• I'm not sure that Loschmidt's paradox has been resolved in any context? So it seems like you are asking one of the unsolved questions of physics. (first one on this list en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics ) – JMLCarter Nov 22 '17 at 23:42
• @JMLCarter: I would tend to disagree with that. Loschmidt's paradox was already very convincingly answered by Boltzmann. Note that this paradox is really about the apparent incompatibility between reversible microscopic equations of motion and irreversible macroscopic behavior, and I would contend that this aspect is rather well understood. The general problem of the arrow of time is of course more complex. – Yvan Velenik Nov 23 '17 at 11:14
• @JMLCarter: two nice references (among many others) that address this point of view on the irreversibility "paradox" are this paper by Joel Lebowitz and this paper by Jean Bricmont. – Yvan Velenik Nov 23 '17 at 13:35
• Actually, a discussion of the question raised by the OP can be found in the papers contained in this book. (It is not difficult to find copies of the book on the internet if you cannot access it directly.) – Yvan Velenik Nov 23 '17 at 13:47
• @Velenik. Thanks for the links. I don't want to hijack this thread with need for clarification special to me. I don't understand the distinction between this paradox and the problem of the arrow of time (yet). – JMLCarter Nov 23 '17 at 14:06