Entropy formula of N-body system

Abstract

We prove a proposition that the entropy of the system composed of finite $N$ molecules of ideal gas is the $q$-entropy or Havrda-Charv\'at-Tsallis entropy, which is also known as Tsallis entropy, with the entropic index $q=\frac{D(N-1)-4}{D(N-1)-2}$ in $D$-dimensional space. The indispensable infinity assumption used by Boltzmann and others in their derivation of entropy formulae is not involved in our derivation, therefore our derived formula is exact. The analogy of the $N$-body system brings us to obtain the entropic index of a combined system $q_C$ formed from subsystems having different entropic indexes $q_A$ and $q_B$ as $\frac{1}{1-q_C}=\frac{1}{1-q_A}+\frac{1}{1-q_B}+\frac{D+2}{2}$. It is possible to use the number $N$ for the physical measure of deviation from Boltzmann entropy.