Gibbs measure

Remarks

Example: on the full shift $\set{1, \dots, k}^{\N}$ take $\varphi = 0$, and let $\mu$ be the uniform Bernoulli measure, under which the coordinates are independent and uniform. Then $\mu([x_0 \dots x_{n-1}]) = k^{-n}$ and $S_n \varphi = 0$, so the ratio is $1$ with $P = \log k$, and $C = 1$ works.

Non-example: for the same shift and $\varphi = 0$, the Dirac measure at the constant sequence $(1, 1, 1, \dots)$ is invariant. It gives measure $0$ to the cylinder $[2]$, so the lower bound fails.

The constant $P$ is then the topological pressure of $\varphi$. For a topologically mixing $\Sigma_A$ and a Hölder continuous $\varphi$, a Gibbs measure exists, is unique, and is the unique equilibrium state of $\varphi$ (Bowen).

In statistical mechanics, consider a finite system with states $s$ and energies $E(s)$. The Gibbs measure at inverse temperature $\beta$ gives $s$ the probability $e^{-\beta E(s)} / Z$, where $Z = \sum_s e^{-\beta E(s)}$ is the partition function. For infinite lattice systems, Gibbs measures are defined instead by the Dobrushin–Lanford–Ruelle condition on conditional probabilities, and there may be several of them.