By "physical means", I mean, for example, using water pouring down tubes, or combining chemicals, etc. Basically, using some experiment in the physical world to perform some computation. I'll call this physical computation.
As the simplest example, if I want to add the numbers 3 and 5, I could just pour 3mL into a cup and 5mL into the same cup, and measure how many mL I end up with.
I know that physical computation will probably not be completely precise, but ignore that, and assume that all experiments and measurements are carried out at their theoretical perfect precision. Another way to think about it is that we have a perfect physics simulator that's efficient up to real time.
Let the class of problems that can be solved efficiently (in a polynomial number of steps) through physical computation be called Phys-P.
My question is: what is Phys-P's relation with P? Are there problems that physical compututation can solve faster than traditional computation? Does leveraging nature and physics give you an exponential speed-up? I'd like to think yes, but I can't think of a particular NP-hard problem that physical computation would solve efficiently. Maybe something like how the structure of a crystal that grows (given some initial conditions) will solve TSP, or something like that.
If possible, I'd like to keep BQP (e.g. quantum computation) out of the discussion, and focus on running physical/chemical experiments.