I'm looking for materials (books, articles, etc) which focus ONLY on the mathematics of statistical mechanics (as I have no background in physics). The materials may have some simple explanations or intuitions behind the important concepts in statistical mechanics.
My personal favorite is "Mathematical Foundations Of Statistical Mechanics" by A. I. Khinchin (a mathematician) and G. Gamow. The content remains mathematically rigorous throughout, but nonetheless very readable. In chapter two, both the Liouville and Birkhoff theorems are derived, followed up by a long discussion on metric decomposability of phase space and its relations to ergodicity. Chapter three covers a beautiful intro to the Ergodic problem. Near the end the Ideal monatomic gas is closely treated, followed by a chapter on Foundations of classical thermodynamics. The book is about 170 pages, yet covering all the major topics, with all the important proofs, just goes to show how succinctly this book is written.
Another classic, which doesn't lack in rigor, is Landau & Lifshitz Vol. 5 Statistical Physics, which is also accessible online. One that shouldn't go unmentioned, is Statistical Mechanics: A set of lectures by R. P. Feynman, covering also Quantum Statistical Mechanics, maybe less readable for a non-physicist compared to Khinchin's, but definitely worth considering.
A very good book along the lines you seem to want is Gallavotti's Statistical Mechanics - A Short Treatise, which can be downloaded from here. He covers many of the classical topics, with a detailed discussion of foundational issues, the role of ergodicity/mixing, etc.
From a very different point of view, with a colleague, we have just finished writing a mathematically rigorous introductory book on the equilibrium statistical mechanics of lattice systems. The final version, as it was sent to the publisher (Cambridge University Press), can be downloaded here; it should more or less coincide (up to changes that may happen when correcting the galley proofs) with the version that will be published around mid 2017. We have tried hard to make the book as readable and student-friendly as possible.
Simon's book The Statistical Mechanics of Lattice Gases is somehow similar to our book, but covers less material (but more in depth; a second volume was planned, but will most likely never exist). It also covers quantum models.
Another book similar to ours, but much harder, is Georgii's famous Gibbs Measures and Phase Transitions. (In a sense, our book can be considered as an introduction to this book, although we cover some important topics that are not discussed the latter.)
There are of course other books aimed at mathematicians/mathematical physicists on more specific topics (e.g., disordered systems, integrable models, relations with large deviations theory or dynamical systems, etc.).