One goal here is to introduce the science of statistical inference (Gregory2005) as a segway to thermal physics e.g. from application of Newton's laws to individual objects in a Mechanics & Heat course. Bayesian inference from observed-averages leads to many aspects of thermodynamics before any physics at all is discussed.
Following Reif's classical book on the fundamentals of statistical and thermal physics (Reif1965), we make use of the monatomic ideal-gas as as an illustration of this math-first approach. Following this, we then imagine a particle-in-box model to illustrate how physics adds meat onto the bones of these equations with state models and a small number of physical axioms, in the process illustrating the assumptions behind important state-relationships like equipartition, the ideal gas law, and mass-action.
In other words we first illustrate statistical inference of various 0th, 1st, 2nd and 3rd law equations with a math-only version of the entropy-first approach already used in most senior undergrad and graduate books on thermal physics (cf.. arXiv:1106.4698). Secondly, we explore adding physical assumptions to give rise to the physical implications of those laws e.g. for energy transfer, entropy increase with time, equipartition and the ideal gas equation of state.
A second goal of this mansucript (or a followup) is to generalize this math-only but nature-inspired model of statistical inference, in context of its present-day and potential-future applications in wide-ranging fields. In that context we begin with choice/matchup multiplicities and their logarithms plus their applications, and then discuss how strategies developed in thermodynamics might be applied in these areas more generally.
Related references:
Phil C. Gregory (2005) Bayesian logical data analysis for the physical sciences: A comparative approach with Mathematica support (Cambridge U. Press, Cambridge UK) preview.
F. Reif (1965) Fundamentals of statistical and thermal physics (McGraw-Hill, NY).