Relationship between entropy and the number of symmetries

Question

We count symmetries of a system by counting the number of transformations/operations under which a feature of the system remains unchanged.
Entropy is a measure of the number of microstates that correspond to the observed macroscopic state (which remains unchanged when the change of microstates remain in a certain set, similar to the way symmetries work).

Is this a mere analogy or would it be possible to define entropy in terms of the concept of symmetry?

OzzieO · Answer

@ali I'll take a feeble stab at this.
First, this is my guess at what you mean.
Here is a "system" microstate 00110. Here is another "system" microstate 11000. Distinct microstates and the operation was roughly interchange places 3,4 and 1,2.
A system macrostate property is the sum over "places" for a given microstate. In this case both microstates have the macrostate property of 2. Conceptually, total entropy for this system is the number of ways I can get macrostate 2 out of rearrangements of the two 1's and 3 0's.
If we accept the definition provided for symmetry, then the symmetric "feature" is 2 and the symmetry "transformations" are the rearrangements of 1's and 0's. Within the context of this simple example, your question is justifiable but I sure don't know how far it goes. It only takes one counter-example to disprove any equivalence type of statement.

Relationship between entropy and the number of symmetries

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