Symmetry in Statistical Mechanics

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I have of late been reflecting on something.

Generally as a rough approximation we may divide physics into classical mechanics, quantum mechanics, classical field theory (like E/M, fluid mechanics...), quantum field theory, and then statistical mechanics.

All the classical and quantum mechanics and field theories seem to me to possess a certain elegance in that they are fundamentally based on symmetry. Think about it - Lorentz covariance and gauge invariance give rise to all the fundamental fields and particles, we can understand momentum, angular momentum, energy and all that as just consequences of a symmetry group with appropriate generators, etc... It feels very natural. Apart from some basic postulates (like those of quantum theory, or the principle of least action), symmetry and the elegance that results from it seem to pervade these theories.

Statistical mechanics seems to be a bit less so. In a sense, it has its own elegance, in that from the assumption of equal a priori probabilities we can derive so much. But I don't immediately see any group theory or symmetry principles that pervade the rest of physics. In a way, this bothers me. Admittedly I have only had up to undergraduate level physics, but I wonder if anyone could illuminate for me how statistical mechanics, phase transitions, critical phenomena, etc... can be seen as more 'natural' and more in sync with the rest of physics, which seems to fall in to place much more smoothly.

Thanks.
 

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Andy Resnick
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Interesting question- there are models incorporating symmetry (Ising-type models and the order parameter, for example), and there is the renormalization group (scaling), which I don't understand all that well.
 

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