Symmetry in Statistical Mechanics

[tiyusufaly]

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 possesses 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 into place much more smoothly.

Thanks.

 

[Andy Resnick]

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.

 

[astrokreng]

I can understand your thoughts on the role of symmetry in statistical mechanics. It is true that classical and quantum mechanics and field theories are based on symmetry principles, such as Lorentz covariance and gauge invariance. These symmetries play a fundamental role in explaining the behavior of particles and fields.

However, in statistical mechanics, the concept of symmetry is not as straightforward. Unlike the other branches of physics, statistical mechanics deals with the behavior of many particles in a system, rather than individual particles. This introduces a level of complexity that cannot be fully explained by symmetry principles alone.

That being said, there are still symmetries that exist in statistical mechanics, but they may not be as obvious as in other areas of physics. For example, the concept of entropy, which is central to statistical mechanics, can be seen as a measure of symmetry breaking in a system. Phase transitions and critical phenomena can also be understood in terms of symmetry breaking.

Furthermore, the elegant mathematical framework of statistical mechanics, particularly the use of probability distributions and partition functions, can be seen as a manifestation of symmetry. These tools allow us to make predictions about the behavior of a system based on its symmetries.

In addition, the idea of equal a priori probabilities, which you mentioned, can also be seen as a form of symmetry. It assumes that all microstates of a system are equally probable, which is a form of symmetry between the different possible configurations of the system.

Overall, while the role of symmetry in statistical mechanics may not be as obvious as in other areas of physics, it is still a fundamental concept that plays a crucial role in understanding the behavior of complex systems. I hope this helps to illuminate the connection between statistical mechanics and the rest of physics.