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::::::Requested outline: A molecule or billiard ball moves elastically in a rigid enclosure. The enclosure is so shaped that the particle never exactly retraces its trajectory. The particle goes nearly everywhere in the enclosure. It often visits every finite small region of the enclosure. It traces out a possibly or nearly spacefilling trajectory. The entropy of the thermodynamic system is a property of the state of thermodynamic equilibrium, and is so defined. It is not a property of an instantaneous point on the trajectory. The nearly spacefilling trajectory, taken as a whole, defines the entropy of the thermodynamic equilibrium state. It is time invariant because it is defined by the whole trajectory. This is the way mathematicians think about the matter nowadays. The concept of probability provides an attractive mathematical procedure to calculate the entropy, as in Monte Carlo, but that is not the only way to do the calculation. The entropy is a geometric property of the trajectory, and can be calculated directly in geometrical terms, without appeal to probability.
::::::In this simple example, only space is explored, and the particle moves with constant speed except at instants of collision. In examples with several particles that can be excited or de-excited in a collision, collisions generate various speeds, so that a more complicated phase space is required. This excitation–de-excitation possibility dispels the mystery of why systems that lack it do not show the usual phenomena. See below.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 21:17, 20 December 2020 (UTC)
::::::* Do you agree that for a system with entropy fluctuations, some will constitute a decrease in entropy and will therefore be, strictly speaking, in violation of the second law?
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