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:::::::::I think of it this way: It is an *assumption* that every trajectory will visit any neighborhood in phase space with a probability proportional to the "volume" of that neighborhood. This is just another way of saying that each microstate is equally probable. Phase space may be divided up into a large number of macrostates, each with their own information entropy. For systems with a large number of particles, the microstates corresponding to the equilibrium macrostate hugely outnumber the volume of the nonequilibrium microstates combined. It follows that, starting from a nonequilibrium microstate, the trajectory will wander into the equilibrium macrostate region and practically never leave. Observationally, that is the signature of an equilibrium state - the macrostate is unchanging. Since the information entropy of a macrostate (and, by Boltzmann's equation, the thermodynamic entropy) is proportional to the log of the phase space volume occupied by that macrostate, the information entropy of the equilibrium macrostate is the largest. A trajectory from a non-equilibrium microstate does not "drift" in any particular direction any more than a trajectory from an equilibrium microstate does. A sort of random walk from any point in phase space will almost certainly walk you into an equilibrium microstate, and almost certainly not walk you into a non-equilibrium microstate, no matter what kind of state you started from. In phase space, trajectories do not "hover" around equilibrium microstates. The macrostate variables do "hover" around their means, however. [[User:PAR|PAR]] ([[User talk:PAR|talk]]) 21:17, 19 December 2020 (UTC)
::::::::::We are putting our cards on the table. I have some problems with your just above comment.
::::::::::Your point of view is that of statistical mechanics. Statistical mechanics is a clever, indeed brilliant and even masterly, and handy mathematical procedure for a sort of Monte Carlo integration, using a concept of random walking, relying on ergodic assumptions. Statistical mechanics is a highly sophisticated topic, taught after several years of advanced education in physics. I don't see it as obvious that it is suitable for novices who are uneducated in physics.
::::::::::The notions of 'an equilibrium microstate' and of 'a non-equilibrium microstate' belong specifically to statistical mechanics.
::::::::::A physical trajectory as conceived by Boltzmann is not a random walk, but is generated by Newton's laws of motion. Mathematicians today try to deal with such trajectories as such. Every point on an equilibrium trajectory has an equal status, as, if you like, 'an equilibrium microstate'. No point on an equilibrium trajectory is 'a non-equilibrium microstate'. Thermodynamic equilibrium and non-equilibrium are characterized by trajectories, not by isolated points. Boltzmann continued the work of Maxwell and others, using the statistical mechanical procedure, but that does not actually make a Newtonian trajectory into an actual random walk.
::::::::::It might be said that the switch from Newtonian trajectory to random walk is made by a mind projection fallacy. It is not obvious that we should impose that fallacy on novices who are not expected to be trained in academic physics.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 22:04, 19 December 2020 (UTC)
== Outstanding questions ==
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