we will show that in the diffusive scaling limit the particle density converges to the solution of the heat equation. The proof will be achieved by taking the limit in the rescaled entropy dissipation inequality and deducing the corresponding inequality for the heat equation.

A gradient flow formulation
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In this section we introduce a gradient flow formulation of non-homogeneous linear kinetic equations. Both for ease of presentation and for future use, we however first review the gradient flow formulation of the heat equation, that is here considered in somewhat different setting that includes the current as a dynamical variable.

Throughout the whole paper, the space domain is the $d$-dimensional torus ${{\mathbb T}}^d:= {{\mathbb R}}^d/{{\mathbb Z}}^d$ and we denote by ${\mathop{}\!\mathrm{d}}x$ the Haar measure on ${{\mathbb T}}^d$. The set of Borel probability measures on ${{\mathbb T}}^d$ is denoted by ${{\mathcal P}}({{\mathbb T}}^d)$ that we consider endowed with the (metrizable) topology induced by the weak convergence. Recall finally that the *entropy* is the convex lower semicontinuous functional $H\colon {{\mathcal P}}({{\mathbb T}}^d) \to [0,+\infty]$ defined by $H(\mu) =
\int\!{\mathop{}\!\mathrm{d}}x\, \rho \log \rho$ if ${\mathop{}\!\mathrm{d}}\mu=\rho\,{\mathop{}\!\mathrm{d}}x$ and $H(\mu)=+\infty$ otherwise.

Heat equation {#heat-equation .unnumbered}
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We start by an informal discussion. Consider the heat equation 