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![[parent]](https://images.physicslibrary.org/images/uparrow.png) Viewing Message
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| ``Idea about entropy?''
by woodgrain on 2008-01-25 22:40:16 |
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| I had an idea about entropy a while ago. Remembering Boltzmann's equation S = k ln(W) where S is the entropy of the system in question, k is Boltzmann's constant (1.38 times 10^-23 J/K) and W is "the number of microstates in the macrostate". Suppose that the microstates are all mathematically symmetrical to an observer outside the system. Then there would be a finite permutation group G acting transitively on the set of microstates. Assume the simplest, that G acts regularly on the microstates, so that the order of G is W. Then W would not only be a number defining the entropy of the system, but it would also the the order of a finite group G specific to the system. Then the structure of G may say something about the system.
I thought of a prediction of this idea: When something irreversible happens to any closed system, the group G would progress from a smaller group G_1 to a larger one G_2. If it is reasonable to say that each microstate divides into an equal number of microstates in this process, and the group G_1 was a "simplification" of the later group G_2, then G_1 would be a quotient group of G_2 by a normal subgroup N, G_1 = G_2/N, with delta S = k ln(|N|). If the step from G_1 to G_2 was a minimal irreversible step then N would have to be a minimal normal subgroup of G_1. Minimal normal subgroups of finite groups are always direct products of isomorphic finite simple groups, N = H^n where H is a finite simple group (and n a natural number). Therefore the change in entropy during a minimal irreversible step would always be k n ln(|H|) where H is a finite simple group. Most often finite simple groups are cyclic groups of prime order so in this case |H| would be a prime number, but sometimes H could be a nonabelian simple group and |H| would be the size of a nonabelian simple group such as 60, 168, 360, 504, 660, ... . In other words, entropy would be quantized and the sizes of the quanta would be k n ln(|H|) where n is any natural number and H is any finite simple group. |
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