# The relations between the Perelman's entropy functional and notions of entropy from statistical mechanics

I am looking for the relations and analogies between the Perelman's entropy functional,$\mathcal{W}(g,f,\tau)=\int_M [\tau(|\nabla f|^2+R)+f-n] (4\pi\tau)^{-\frac{n}{2}}e^{-f}dV$, and notions of entropy from statistical mechanics. Would you please explain it in details?

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The more standard notions of entropy, notably Boltzmann and Shannon are roughly of the form $\int u\log u\,d\mu$ and if in Perelman's definiton you set $u = e^{-f}$ you get one term like this. The gradient term looks to me more like Fisher information, which can be viewed as the derivative of entropy with respect to time under Brownian motion. I suppose that the scalar curvature arises because everything is on a curved instead of flat space. The constant term arises from normalization. But this is all just my speculation. –  Deane Yang May 18 at 14:43
it seems to be little more than a formal correspondence, judging from page 11 of arxiv.org/abs/math.DG/0211159 –  Carlo Beenakker May 18 at 15:25
Although it might seem like nothing more than a formal correspondence, the power of using entropy-type functionals for certain types of elliptic and parabolic PDE's indicates strongly to me that there is a deeper connection to the physical and information theoretic definitions of entropy than what we currently understand. –  Deane Yang May 18 at 15:39

For metrics on $S^{2}$ with positive curvature, Hamilton introduced the entropy $N\left( g\right) =-\int\ln(R\operatorname{Area})Rd\mu.$ If the initial metric has $R>0,$ he proved that this is nondecreasing under the Ricci flow on surfaces; note that $Rd\mu$ satisfies $(\frac{\partial}{\partial t}-\Delta )(Rd\mu)=0.$ Let $T$ be the singular time; then $$\frac{d}{dt}N\left( g\left( t\right) \right) =2\int\left\vert \operatorname{Ric}+\nabla^{2}f-\frac{1}{2\tau}g\right\vert ^{2}d\mu+4\int% \frac{\left\vert \operatorname{div}(\operatorname{Ric}+\nabla^{2}f-\frac {1}{2\tau}g)\right\vert ^{2}}{R}d\mu,$$ where $\tau=T-t$ and $\Delta f=r-R.$ ($f$ satisfies $\frac{\partial f}{\partial t}=\Delta f+rf$; since $n=2$, $\operatorname{Ric}=\frac{1}{2}Rg$)

Perelman's entropy has the main term: $\int fe^{-f}d\mu,$ which is the classical entropy with $u=e^{-f}$ as Deane Yang wrote. (Besides Section 5 of Perelman, further discussion of entropy appeared later in some of Lei Ni's papers as well as elsewhere.) Even though this term is lower order (in terms of derivatives), geometrically it is the most significant as can be seen by taking the test function to be the characteristic function of a ball (multiplied by a constant for it to satisfy the constraint); technically, one chooses a cutoff function. Thus Perelman proved finite time no local collapsing below any given scale only assuming a local upper bound for $R,$ since the local lower Ricci curvature bound (control of volume growth is needed to handle the cutoff function) can be removed by passing to the appropriate smaller scale.

Heuristically (ignoring the cutoff issue), since the constraint is $\int(4\pi\tau)^{-n/2}e^{-f}d\mu=1,$ if we take $\tau=r^{2}$ and $e^{-f}=c\chi_{B_{r}},$ then $c\approx\frac{r^{n}% }{\operatorname{Vol}B_r}.$ So, if the time and scale are bounded from above, by Perelman's monotonicity, we have $$-C\leq\mathcal{W}(g,f,r^{2})\lessapprox r^{2}% \max_{B_{r}}R+\ln\frac{\operatorname{Vol}B_r}{r^{n}},$$ yielding the volume ratio lower bound.

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There is a nice interpretation of Perelman's monotonicity formulas in terms of optimal transportation, see e.g. these lecture notes by Peter Topping

http://homepages.warwick.ac.uk/~maseq/grenoble_20100324.pdf

It seems helpful to look at the elliptic case first. As discovered by Lott-Villani and Sturm nonnegative Ricci curvature can be characterized by the property that the Boltzmann-entropy is convex along optimal transportation. This is very intuitive, imagine e.g. a pile of sand being transported from the south to the north-pole on the sphere.

The idea for the Ricci flow is similar (being a (super)Ricci flow can be viewed as parabolic version of having nonnegative Ricci curvature), but the details are a bit more complicated. The $W$-functional can be written as derivative of a suitable Boltzmann-entropy (see Section 5 in Perelman's first paper) and the monotonicity of $W$ can be interpreted as convexity of this entropy, see the above lecture notes for details.

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