created a stub for *holographic entanglement entropy* in reaction to this MO question.

I wanted to be able to point to *[[expectation value]]* without the link being broken. So I added a sentence there, but nothing more for the moment.

started an entry *Bohmian mechanics* (prompted by discussion in another thread)

I have split-off *Feynman diagram* from *perturbation theory*, gave it a brief Idea-section and added a pointer to an insightful reference.

I have created *random variable* with some minimum context.

In addition I have added pointers to Kolmogorov’s original book and to some modern lecture notes to *probability theory* and some related entries.

I have briefly cross-linked *probability space* with *possible worlds*, indicating a similarity of concepts and an overlap of implementations.

started a bare minimum at *Bloch region*

created a brief entry *rational thermodynamics*.

I haven’t actually seen yet the actual detail of this axiomatics (but see the citations given at the above link). What I currently care about is this historical fact, which I added to the Idea section:

What is called *rational thermodynamics* is a proposal (Truesdell 72) to base the physics of irreversible thermodynamics on a system of axioms and derive the theory from these formally.

The success of the axioms of rational thermodynamics as a theory of physical phenomena has been subject of debate. But the idea as such that continuum physics can be and should be given a clear axiomatic foundation seems to have inspired William Lawvere (see there for more), once an undergraduate student of Clifford Truesdell, to base continuum mechanics on constructions in topos theory, such as synthetic differential geometry and cohesion.

]]>There is a category in which an object is a single-variable probability distribution $P(a)$ (say, finitely supported) and a morphism from $P(a)$ to $P(b)$ is a two-variable distribution $P(a,b)$ which recovers the source and target distributions as marginals: $P(a)=\sum_b P(a,b)$ and $P(b)=\sum_a P(a,b)$.

These morphisms can be composed as

$P(a,c) = \sum_b \frac{P(a,b)P(b,c)}{P(b)}$This makes $a$ and $c$ conditionally independent given $b$ and therefore corresponds to the usual composition of conditional distributions $P(c|a)=\sum_b P(c|b)P(b|a)$ as it occurs e.g. in Markov processes.

In this way, we obtain a category of single-variable and two-variable distributions.

Now a natural question is: is it possible to introduce a higher category which contains distributions over any number of variables? In fact, the above data of composable morphisms actually yields a three-variable distribution

$P(a,b,c) = \frac{P(a,b)P(b,c)}{P(b)}$More generally, probability distributions can be “composed” as follows: let $X$ be a simplicial complex on vertices $\{a_1,\ldots,a_n\}$ with maximal faces $\{C_1,\ldots,C_k\}$. We say that $X$ has the running intersection property (RIP) if the ordering of the $C_j$ can be chosen such that for every $j$ there is an $i\lt j$ with

$S_j := C_j \cap (C_1\cup\ldots\cup C_{j-1}) \subseteq C_i .$Now suppose we have given distributions $P(C_i)$ over the variables in $C_i$ with compatible marginal distributions. If $X$ has the RIP, then a joint distribution of all variables $\{a_1,\ldots,a_n\}$ can be constructed as

$P(a_1,\ldots,a_n) = \frac{\prod_{i=1}^k P(C_i)}{\prod_{i=1}^k P(S_i)}$where $P(S_i)$ represents the marginal distribution. This we interpret as a filler of $X$ to a simplex which represents the composition of the given $P(C_i)$ to a joint distribution.

There are other examples of simplicial complexes, e.g. the three edges of a triangle, for which joint distributions cannot always be constructed.

**Question:** is there a higher categorical structure similar to quasi-categories, but with fillers for simplicial sets with the RIP rather than fillers for inner horns? Does every weak Kan complex have fillers for simplicial sets with the RIP?

created *volume*, just for completeness

stub for *[[moment]]*, just for completeness