Friday, June 21, 2013

From combinatorics to entropy

From combinatorics to entropy:
Let $N = n_1 + ... + n_k$ and $p_i = \frac{n_i}{N}$.
$$\log ( \frac{N!}{n_1 ! ... n_k ! } ) \approx - N \sum_i p_i \log p_i$$
by Stirling's formula.

I wonder if this was the first time ever in human history that such an expression $$\sum_i p_i \log p_i$$ appeared! Entropy is often too abstract to me. The approximation above is a link between counting combinations and entropy, and it seems to provide the most concrete grasp~
This is the genius of Boltzmann, Maxwell and Gibbs which leads to the development of statistical mechanics.

Energy, entropy, free energy, enthalpy, Legendre transform, etc. are still difficult to understand to me.

Renyi/generalized entropy:
$$D_q = \frac{1}{q-1} \log \sum_i p_i^q$$
It is related to the generalized dimension $\dim_q(\mu)$ and $L^q$-spectrum $\tau(q) := \liminf_{r \to 0} \frac{1}{\log r} \log \sup \sum_i \mu(B_i)^q$:
$$\dim_q (\mu) = \lim_{r \to 0} \frac{D_q}{\log r} = \frac{\tau(q)}{q-1} .$$