**Motivation:**

This is a toy model of how a closed system will always evolve towards the distribution of maximal entropy, where no further transfer of heat/energy is possible.

**Problem set up:**

Fix a positive integer $N$, and denote by $[N]$ the set $\{1, \dots, N\}$.

Let $\mathcal L := [N] \times [N]$ be a 2D lattice.

We model a flow of heat as follows - Initially at time $0$, all heat is concentrated at $(1, 1)$. At each time step $t$, for $t \in \mathbb Z_+$, an element of $\mathcal L$ is picked uniformly at random. At that stage, it and all its immediate neighbours to its east, west, north and south average their heat.

Formally, we have a sequence of iid uniformly distributed $\mathcal L$-valued random variables $\varepsilon_n$, for $n \in \mathbb Z$, and a sequence $X_n$ of functions $\Omega \times \mathcal L \to [0, 1]$, defined as follows:

$X_0 = \mathbf 1_{(1, 1)}$, almost surely.

Assume now $X_0, \dots, X_n$ have already been defined.

Write $X_n = \sum_{z \in \mathcal L} \lambda_z \mathbf 1_{z}$ for (random) $\lambda_z \in [0, 1]$ with $\sum_{x \in \mathcal L} \lambda_z = 1$, and $\mathcal N(\varepsilon_n)$ for the set consisting of $\varepsilon_n$ and its two to four, depending on the location of $\varepsilon_n$, neighbours.

Then define $X_{n+1} := \left[\underset{x \in \mathcal N(\varepsilon_n)}{\sum} \underset{y \in \mathcal N(\varepsilon_n)}{\sum} \frac{\lambda_y}{|\mathcal N(\varepsilon_n)|} \mathbf 1_x \right]+ \underset{z \in \mathcal L \setminus \mathcal N(\varepsilon_n)}{\sum} \lambda_z \mathbf 1_z$.

where $|S|$ denotes the cardinality of a finite set $S$.

Question:Let $Y$ be the “uniform distribution of heat” given by $Y := \underset{z \in \mathcal L}{\sum} \frac{1}{|\mathcal L|} \mathbf 1_z$. Is it true that almost surely, $X_n \to Y$ uniformly?

Thus the system evolves almost surely toward a distribution where no further transfer of heat is possible.