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Assuming each of the N microstates for a given macrostate are equally possible with probability p=1/N, the Shannon Entropy is -Σp.log(p) = -N.p.log(p)=-1.log(1/
by enugu 2y ago
Assuming each of the N microstates for a given macrostate are equally possible with probability p=1/N, the Shannon Entropy is -Σp.log(p) = -N.p.log(p)=-1.log(1/N)=log(N), which is the physics interpretation.
In the continuous version, you would get log(V) where V is the volume in phase space occupied by the microstates for a given macrostate.
Liouville's theorem that the volume is conserved in phase space implies that any macroscopic process can only move all the microstates from a macrostate A into a macrostate B only if the volume of B is bigger than the volume of A. This implies that the entropy of B should be bigger than the entropy of A which is the Second Law.
- cubefox 2y agoThe second law of thermodynamics is time-asymmetric, but the fundamental physical laws are time-symmetric, so from them you can only predict that the entropy of B should be bigger than the entropy of A irrespective of whether B is in the future or the past of A. You need the additional assumption (Past Hypothesis) that the universe started in a low entropy state in order to get the second law of thermodynamics. > If our goal is to predict the future, it suffices to choose a distribution that is uniform in the Liouville measure given to us by classical mechanics (or its quantum analogue). If we want to reconstruct the past, in contrast, we need to conditionalize over trajectories that also started in a low-entropy past state — that the “Past Hypothesis” that is required to get stat mech off the ground in a world governed by time-symmetric fundamental laws. https://www.preposterousuniverse.com/blog/2013/07/09/cosmology-and-the-past-hypothesis/ https://www.preposterousuniverse.com/blog/2013/07/09/cosmolo...
- kgwgk 2y agoThe second law of thermodynamics is about systems that are well described by a small set of macroscopic variables. The evolution of an initial macrostate prepared by an experimenter who can control only the macrovariables is reproducible. When a thermodynamical system is prepared in such a reproducible way the preparation is happening in the past, by definition. The second law is about how part of the information that we had about a system - constrained to be in a macrostate - is “lost” when we “forget” the previous state and describe it using just the current macrostate. We know more precisely the past than the future - the previous state is in the past by definition.