I have asked this question on [MathSE](https://math.stackexchange.com/questions/2341753/gaussian-distribution-maximum-entropy-and-the-heat-equation), but I got no replies, so I thought of trying here. 




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Consider the Gaussian distribution on $\mathbb{R}$ with mean $m$ and variance $t=\sigma^2$. This has the explicit density


$$ f(x,m,t) = \frac{1}{\sqrt{2\pi t}}e^{-\frac{(x-m)^2}{2t}} $$


The Gaussian distribution has many characterizations, and two of them are the following:


1. The Gaussian distribution maximizes entropy amongst all the distributions on $\mathbb{R}$ with mean $m$ and variance $t$.
2. The density $f(x,m,t)$ provides a solution of the heat equation. More precisely, consider the differential equation
$$ \begin{cases} \frac{\partial u}{\partial t} &= \frac{\partial^2 u }{\partial x^2} \quad \text{ for } x\in \mathbb{R},t>0 \\
u(x,0) &= \delta\_m(x) 
 \end{cases} $$ 
where $\delta\_m$ is the Dirac concentrated at $m$. Then the function $u(x,t)=f(x,m,2t)$ is a solution of this differential equation.


I would like to know whether there is a reason (intuitive or more formal) why the solution of the two problems (maximum entropy distribution and heat equation) is the same. Of course, an explanation is that we can solve explicitly both problems and the solution happens to be the same, but I wonder whether there is a more conceptual reason for this.



Gaussian distribution, maximum entropy and the heat equation
Both the Gaussian maximum entropy distribution and the Gaussian solution of the diffusion equation (heat equation) follow from the central limit theorem, that the limiting distribution of the sum of i.i.d. random variables with given average and variance is a Gaussian. The connection between the central limit theorem and the diffusion equation, which describes a random walk with zero mean and unit variance, is obvious; the connection between the central limit theorem and the maximum entropy distribution is less obvious. For pointers to the literature on the latter connection, see these [MO question and answers.](https://mathoverflow.net/questions/182752/central-limit-theorem-via-maximal-entropy)


