oduction functional by $$I(\theta) = \int \theta {\left\vert{\nabla}\log \theta +  \eta\right\vert}^2 d\eta. \label{def:I_linear}$$ In the nonlinear case $m > 1$, the corresponding quantities are, $$H(\theta) = \frac{1}{m-1}\int \theta^m d\eta + \frac{1}{2}\int {\left\vert\eta\right\vert}^2 \theta d\eta, \label{def:H}$$ and the entropy production functional, $$I(\theta) = \int u{\left\vert\frac{m}{m-1}{\nabla}u^{m-1} + \eta\right\vert}^2 d\eta. \label{def:I}$$ In the nonlinear case, these entropies were originally introduced for studying in [@Newman84; @Ralston84]. Both and are displacement convex [@McCann97] and in fact, is a gradient flow for or in the Euclidean Wasserstein distance [@Otto01; @AmbrosioGigliSavare; @CarrilloMcCannVillani03; @CarrilloMcCannVillani06], and if $f(\tau,\eta)$ solves , then $$\frac{d}{d\tau}H(f(\tau)) = -I(f(\tau)).$$ For a given mass $M$, has a unique non-negative minimizer which is the ground state $\theta_M$. That is, if we define the relative entropy $$H(\theta|\theta_M) = H(\theta) - H(\theta_M),  \label{def:relativeH}$$ then $H(\theta|\theta_M) \geq 0$ with equality if and only if $\theta = \theta_M$ [@CarrilloToscani00; @DelPinoDolbeault02]. In order to estimate a convergence rate, it is therefore sensible to measure how quickly $H(\theta|\theta_M) \rightarrow 0$. Following the methods of [@CarrilloToscani00; @CarrilloEntDiss01; @CarrilloMcCannVillani03; @CarrilloMcCannVillani06], this is made possible by the following two theorems. The first relates the entropy production functional to the relative entropy . This represents a generalization of the Gross logarithmic inequality [@Gross75] (see also [@DelPinoDolbeault02]).

\[thm:rel\_entropy\] Let $f\in L_+^1({\mathbb R}^d)$ with ${\|f\|}_1 = M$ and let $\theta_M$ be the ground state Barenblatt solution with mass $M$. Then, $$H(f|\theta_M)  \leq \frac{1}{2} I(f). \label{ineq:relative_entropy}$$

For the Fokker-Plank equation , Theorem \[thm:rel\_entropy\] implies $H(\theta(\tau)|\theta_M) \lesssim e^{-2\tau}$. The (generalized)