background - thermal ensembles

Definition

An ensemble is data

• A set of microstates $M_{\mu }$
• A set of macroscopic observables ${\varphi }_i:M_{\mu }\to X_i$ , $i\in I$. $X_i$ is a set.

Constructions

• The set of macrostates is the product of all the target spaces for observables: $X:={\prod }_i{X}_i$
• The observables are collected into a map $\varphi :M_{\mu }\to {\prod }_i{X}_i:=X$
• Given a macrostate $x\in X$, the fiber at $x$, $Fi{b}_x(M_{\mu }\stackrel{\varphi }{\to }X):=E_x$, encodes the space of all microstates corresponding to the macrostate $x$. The cardinality of this space is a form of the entropy of the macrostate (we can then take log of this number to reproduce something proportional to Boltzman entropy).

Remark

The definition given above makes sense in any $\infty$-category. For instance, if we work in the category of anima, we can count the “cardinality” of the fiber $E_x$ by using Euler characteristics on the K(n)-cohomology, and performing an appropriate sum (Yanovski’s form of Baez-Dolan homotopy cardinality).