I have been studying for quite some time now about entropic functionals, including Boltzmann-Gibbs, Renyi, Kaniadakis and Tsallis, and I am familiar with the properties that a functional has to fulfill in order for it to represent the entropy of a system.
But I am more interested to find the detailed mathematical way in which these entropic functionals are defined. I would like to know the strict mathematical definition of an entropic funtional.
For example, for Boltzmann-Gibbs, $S_{BG}=-k\sum p_i \ln p_i$, or Tsallis entropy $S_q=\left( 1-\sum p_i^q \right)/(q-1)$:
- Which functional space are they defined in?
- What are the properties of that space? (norm, metric, boundeness, completeness, etc)
Is there a detailed report/analysis answering these questions?
Finally, I have the following question. Given the foundations of classical statistical mechanics, it comes to my understanding that in order to study the microstates of the system we start with Newton's law, therefore a deterministic approach. But in the thermodynamical limit, we end up with probability density functions, therefore a stochastic approach.
What are the conditions that have to be met in order for these two, in general contradictory approaches, to coexist?
I have been quite buffled by the above.
Thank you.