Statistical mechanics, the physics of many-particle systems, is a profound intellectual achievement. A statistical approach to systems with many degrees of freedom makes perfect sense. It's ridiculous to think about solving Newton's laws (or the Schroedinger equation, for that matter) for all the gas molecules in this room. Apart from being computationally intractable, it would be silly for the vast majority of issues we care about, since the macroscopic properties of the air in the room are approximately the same *now* as they were when you began reading this sentence. Instead of worrying about every molecule and their interactions, we characterize the macroscopic properties of the air by a small number of parameters (the pressure, temperature, and density). The remarkable achievement of statistical physics is that it places this on a firm footing, showing how one can go from the microscopic degrees of freedom, through a statistical analysis, and out the other side with the macroscopic parameters.

However, there are still some tricky bits, and even equilibrium statistical mechanics continues to be an active topic of research. For example, one key foundation of statistical mechanics is the idea of replacing time-averages with energy-averages. When we teach stat mech to undergrads, we usually say something like, a system explores all of the microscopic states available to it (energetically, and within other constraints) with equal probability. That is, if there are five ways to arrange the system's microscopic degrees of freedom, and all five of those ways have the same energy, then the system in equilibrium is equally likely to be found in any of those five "microstates". How does this actually work, though? Should we think of the system somehow bopping around (making transitions) between these microstates? What guarantees that it would spend, on average, equal time in each? These sorts of issues are the topic of papers like this one, from the University of Tokyo. Study of these issues in classical statistical mechanics led to the development of ergodic theory (which I have always found opaque, even when described by the best in the business). In the quantum limit, when one must worry about physics like entanglement, this is still a topic of active research, which is pretty cool.
## 1 comment:

Consider jaynes Maximum entropy approach.. It is consistent, makes sense, and mathematically simpler..

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