Entropy formula of N-body system

Jae Wan Shim

Published 2018-12-13Version 1

Tsallis entropy is a generalisation of Boltzmann entropy. The generalisation is characterised by a single parameter known as the entropic index $q$ and Boltzmann entropy is retrieved when $q$ approaches one. Boltzmann derived his entropy formula by considering a physical system composed of infinite number of ideal gases where the infinity was an indispensable constraint for using Stirling's approximation with respect to the factorial function that appeared in his procedure\cite{boltzmann}. Meanwhile, the derivation of the precise entropy formula for the Boltzmann's system without using the infinity assumption seems unknown and hard to be accomplished by following his way. Here we show the entropy of the system composed of finite $N$ molecules of ideal gas is Tsallis entropy with the entropic index $q=\frac{D(N-1)-4}{D(N-1)-2}$ in $D$-dimensional space. This result partially reveals the detail behind Boltzmann entropy concealed by using the infinity assumption. In addition, Tsallis entropy is now established on a physical system which is as precise and fundamental as the foundation of Boltzmann entropy. By using the analogy of the $N$-body system, it is possible to obtain the entropic index of a combined system formed from subsystems having different entropic indexes. The number $N$ can be used for the physical measure of deviation from Boltzmann entropy.