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Statistical mechanics provides a sort of Monte Carlo procedure to estimate that integral, using ergodic theorems.
Non-equilibrium is characterized by some sequence of measurements drifting a significant 'distance' through phase space. The drift may involve repeated distinct visits of the whole-system instantaneous microstate to some region of phase space, but it must be evident that they are repeated distinct and separate visits, not just little excursions in a permanent and persistent hovering pattern. In general, for a non-equilibrium trajectory through the phase space of whole-system instantaneous microstates, over some long observation time interval <math>(t_{\mathrm{initial}},t_{\mathrm{final}})</math>, the trajectory will drift from some region <math>R_{\mathrm {initial}} \subset R_0</math> to some other region <math>R_{\mathrm {final}} \subset R_0</math>, with negligible overlap <math>p_{\mathrm A} \in R_{\mathrm A}</math> {initial}} \cap R_{\mathrm{final}}</math>. Thermodynamic entropy does not apply here. Other so-called 'entropies' may be defined ''ad lib'', but they refer to some kind of 'time rate of entropy production'.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 20:09, 19 December 2020 (UTC)
: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)
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::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]]) 23:35, 19 December 2020 (UTC)
::I have to revise the above.
::That the overlap <math>R_{\mathrm {initial}} \cap R_{\mathrm{final}}</math> should be negligible certainly gives a non-equilibrium trajectory. But such is well and truly and thoroughly non-equilibrium. More tightly, non-equilibrium just needs <math>R_{\mathrm {initial}} \ne R_{\mathrm{final}}</math>, though that doesn't exactly settle things, because I haven't tightly said what I mean by the trajectory being in a region <math>R</math> at a time <math>t</math>. What sort of region is <math>R\,</math>?
::For thermodynamic equilibrium, the condition <math>R_{\mathrm {initial}} = R_{\mathrm{final}}</math> is necessary and sufficient if at least one of {<math>t_{\mathrm{initial}} \to - \infty \,</math>, <math>t_{\mathrm{final}} \to + \infty</math> } holds.
::We may consider a thermodynamic process that starts when two equilibrium systems <math>\mathrm A</math> and <math>\mathrm B \,</math>, that separately occupy regions <math>R_{\mathrm A}</math> and <math>R_{\mathrm B} \,</math>, are exposed to each other, and ends when a thermodynamic process isolates the final joint system, so that its initial instantaneous microstate obeys the conditions <math>p_{\mathrm A} \in R_{\mathrm A}</math> and <math>p_{\mathrm B} \in R_{\mathrm B}</math> and <math>p_{\mathrm{initial}} = (p_{\mathrm A},p_{\mathrm B})</math> with <math>\,p_{\mathrm {initial}} \in R_{\mathrm {joint}}</math> in an obvious notation for the final thermodynamic equilibrium. The second law requires something such as <math>R_{\mathrm {joint}} \sub R_{\mathrm A} \times R_{\mathrm B}</math>, in a suitable notation, with <math>\sub</math> denoting a proper subset relation. The second law requires more, making a strict statement about entropies.
::A non-equilibrium process is not so simple to define microscopically in general terms. But surely it requires at least definite initial and final conditions? And that they belong to different regions, <math>R_{\mathrm {initial}} \ne R_{\mathrm{final}}</math> in some sense. But it doesn't require such strict separation as makes negligible the overlap <math>R_{\mathrm {initial}} \cap R_{\mathrm{final}} \,</math>.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 03:15, 20 December 2020 (UTC)
== Outstanding questions ==
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