Indeed, statistical mechanics is applicable to the systems with many degrees of freedom (typically of the order of the Avogadro constant, $N_A\sim 10^{23}$). Gases, liquids, and solids routinely fill this definition, as they consist of huge numbers of atoms or molecules, each of which has at least three degrees of freedom (for molecules we also need incorporate their rotation and elastic movement).
There are two ways how statistical mechanics can be applied to proteins (see, e.g., the discussion in the book by Doi):
- one may consider a solution containing a macroscopic (i.e. $\sim N_A$) number of polymer molecules
- one may consider a single polymer molecule as a statistical ensemble, because a polimer with many links can adopt huge number of configurations and therefore satisfies most of the criteria necessary for applying statistical phsyics. This is specially true for polymers with $10^5-10^6$ links, but provides useful guidelines for relatively short protein molecules, with typical lengths of a few hundreds od amino-acids.
As an example we may consider protein folding (see, e.g., this question): a protein would fold spontaneously even in a vacuum, and even if the forces between its parts were non-existent. The reason is entropy: a chain of molecules can adopt billions of configurations in space, of which the stretched configurations would form only a small fraction. Statistical physics tells us that all these configurations are equiprobable, and we therefore are likely (what practically amounts to certainty) to find protein in a folded configuration. (This equiprobable state is referred to as maximal entropy or maximum of an appropriate thermodynamic potential, if we are not dealing with a closed system.)
Adding interactions between protein links and between the protein and the solvent molecules makes the problem even more amenable to statistical physics analysis (although now one may have to deal with folding-unfolding phase transition).
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