  We study the connection between a system of many independent Brownian
particles on one hand and the deterministic diffusion equation on the other.
For a fixed time step $h>0$, a large-deviations rate functional $J_h$
characterizes the behaviour of the particle system at $t=h$ in terms of the
initial distribution at $t=0$. For the diffusion equation, a single step in the
time-discretized entropy-Wasserstein gradient flow is characterized by the
minimization of a functional $K_h$. We establish a new connection between these
systems by proving that $J_h$ and $K_h$ are equal up to second order in $h$ as
$h\to0$. This result gives a microscopic explanation of the origin of the
entropy-Wasserstein gradient flow formulation of the diffusion equation.
Simultaneously, the limit passage presented here gives a physically natural
description of the underlying particle system by describing it as an entropic
gradient flow.
