Is it possible to infer on the thermodynamics of two problems if a reduction from $B$ to $A$ exists? Peter Shor commented on this post:

years of experience in theoretical computer science says that the thermodynamic behavior of two NP complete problems are in general not similar.



What can we say about the distribution/thermodynamics of $A$ w.r.t. $B$ if we know that $A \le_M B$ (there is a reduction from $B$ to $A$), or if we know that $A, B \in C$ where $C$ is some complexity class ($P$, $NP$, co-$NP$, etc.)?

What can we say on the thermodynamics if there isn't a reduction from $A$ to $B$, or given their complexity classes?



I am not sure if it is a wicked idea to use complexity theory whose main concern is the worst-case scenarios while discussing thermodynamics, or statistical physics, whose general aim is to characterize phase transitions and describe the average case (principle of maximum entropy). Here is somehow a related work that states &quot;NP problems&quot; in terms of an Ising model.
With thermodynamics you have to be careful with the kind of reductions you allow, or (as Peter Shor pointed out) there can be essentially no thermodynamic relationship implied by a reduction. For example, if we consider not just the complexity of languages, but also the complexity of functions, every language is equivalent (under pretty simple reductions) to a permutation of $\Sigma^*$, and permutations can in principle be computed with zero thermodynamic cost.