In the following phrase, we are debating if a possessive genitive form should be used or not:
"Such mapping enabled the use of the classical statistical mechanics' tools"
Should "classical statistical mechanics" be modified with the apostrophe to express a genitive form?
No. In this case (barring a possible reading of your sentence that John suggests but which I'm less sure about), what you have is simply a compound and there's no need for the apostrophe.
As a rough rule of thumb, compare with a comparable situation in which the word doesn't end in "s". For example, in "classical statistical chemistry tools", you'd never even think in your wildest dreams of saying "chemistry's tools", right?
Another more complex rule of thumb for spotting a 'compound' is that the individual elements making up the compound can't readily be "co-indexed" (="refer to the same thing or person as") with a pronoun. So consider the case of: "Mary's dog bit her", where "Mary" and "her" can refer to the same person. Then compare this with "mechanics tools help us to understand it": here, "it" cannot refer to "mechanics". Or in other words, the case of a possessive and the case of a compound are not the same type of structure and the 's is only used for the genuine possessive case (the "Mary" case, where the individual person/thing can be referred to 'from the outside' by a pronoun).
The correct phrasing would be
... use of tools from classical statistical mechanics
After all, use relates to tools and defering the later too much is incorrect.
As much as you can, avoid using more than two modifiers: 'classical statistical mechanics' already has two adjectives and a noun. Using this as an object and deriving its possessive form to qualify 'tools' is stretching it too far.
As a consideration for the reader, it is better to rephrase and improve readability.
Yes, it can be used, but the definite article (the) should be dropped in that case:
Such mapping enabled the use of classical statistical mechanics' tools.
Using the phrase classical statistical mechanics without a definite article makes it a name for the theory (I've seen it capitalized), and since mechanics ends in -s and wouldn't be pronounced with an extra syllable /-əz/ in the possessive, mechanics' gets a solo apostrophe, according to orthodox apostrophic theories.
That's if and only if you want to make it possessive in writing, note. In speech
mechanics'can't be distinguished frommechanicsanyway, since apostrophes are always silent. And yet somehow we all manage to understand either version in speech and only notice a problem in writing. Amazing.
Whereas leaving the definite article in makes the classical statistical mechanics just one of several statistical mechanicses, or of several other kinds of mechanics, all unspecified, which is precisely what you want to avoid when you invoke a classical theory to explain something. Especially if you are interested in the tools and not the theories.
