Is there a connection between the halting problem and thermodynamic entropy?
I am not at all an expert in this area, but I believe you will be interested in reversible computing.  This involves, among other things, the study of the relationship between processes that are physically reversible and processes that are logically reversible.  I think it would be fair to say that the "founders" of the field were/are Ralph Landauer and Charles H Bennett (both of IBM research, I think.)  

It touches on quantum computing and quantum information theory, but also examines questions like "what are the limits of computation in terms of time, space and energy?"  It is known, (if I remember correctly) that you can make the energy required to perform a reversible calculation arbitrarily small by making it take an arbitrarily long time.  That is, energy $\times$ time (=action) required to perform a reversible computation can be made a constant.  This is not the case for non-reversible computations.

Many of the people studying in this area are also working on quantum computing and digitial physics (the idea that the universe is a big quantum cellular automata).  The researchers names that come to mind are Ed Fredkin, Tommaso Toffoli and Norm Margolus.

These questions are absolutely on topic for computer science.  Not just for the theory (which includes cool math as well as cool physics) but for engineers who want to know the ultimate limits of computation.  Is there a minimum volume or energy required to store a bit of information?  The action required to perform a reversible computation may be constant, but are there limits on what that constant is?  These are critical knowledge for engineers trying to push the boundaries of what is possible.