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

1$\begingroup$ 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. $\endgroup$– Deane YangMay 18 '13 at 14:43

1$\begingroup$ it seems to be little more than a formal correspondence, judging from page 11 of arxiv.org/abs/math.DG/0211159 $\endgroup$– Carlo BeenakkerMay 18 '13 at 15:25

4$\begingroup$ Although it might seem like nothing more than a formal correspondence, the power of using entropytype 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. $\endgroup$– Deane YangMay 18 '13 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 $\tau=Tt$ and $\Delta f=rR.$ ($f$ 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.
Added December 12, 2013: For all of the following, see Perelman, Ni, Topping, etal. Let $\mathcal{N} =\int fe^{f}d\mu$ be the classical entropy. Then, under $\frac{\partial }{\partial t}g=2(\operatorname{Ric}+\nabla^{2}f)$ and $\frac{\partial f}{\partial t}=\Delta fR$, we have $\frac{d\mathcal{N}}{dt}=\mathcal{F} (g,f)\doteqdot\int(R+\left\vert \nabla f\right\vert ^{2})e^{f}d\mu$ (Perelman's energy). If the solutions are on $[0,T)$, then $\mathcal{F} (t)\leq\frac{n}{2\left( Tt\right) }\int e^{f}d\mu$, which implies that $\frac{d}{dt}(\mathcal{N}(\frac{n}{2}\int e^{f}d\mu)\log(Tt))\geq0$. Let $\mathcal{W}(g,f,\tau)=(4\pi\tau)^{n/2}\left( \tau\mathcal{F}+\mathcal{N} \right) n\int(4\pi\tau)^{n/2}e^{f}d\mu$ (Perelman's entropy). Under $\frac{\partial}{\partial t}g=2\operatorname{Ric}$ and $\frac{\partial f}{\partial t}=\Delta f+\nabla f^{2}R+\frac{n}{2\tau}$, we have have $\frac{d\mathcal{F}}{dt}=2\int\operatorname{Ric}+\nabla^{2}f^{2}e^{f} d\mu\frac{n}{2\tau}\mathcal{F}$ and $\frac{d\mathcal{N}}{dt}=\mathcal{F} +\frac{n}{2\tau}\int e^{f}d\mu\frac{n}{2\tau}\mathcal{N}$. So, by coupling with $\frac{d\tau}{dt}=1$, we obtain (Perelman's entropy formula) \begin{align*} (4\pi\tau)^{n/2}\frac{d\mathcal{W}}{dt} & =\frac{n}{2\tau}\left( \tau\mathcal{F}+\mathcal{N}\right) \mathcal{F}+\tau\frac{d\mathcal{F}} {dt}+\frac{d\mathcal{N}}{dt}\\ & =2\tau\int\operatorname{Ric}+\nabla^{2}f\frac{1}{2\tau}g^{2}e^{f}d\mu. \end{align*}
Perelman himself wrote about his entropy formula for the Ricci flow that "The interplay of statistical physics and (pseudo)riemannian geometry occurs in the subject of Black Hole Thermodynamics, developed by Hawking et al. Unfortunately, this subject is beyond my understanding at the moment."
Subsequently, this connection has been explored in some detail in two papers by Samuel and Chowdury: "Geometric flows and black hole entropy" and "Energy, entropy and the Ricci flow". For a discussion of both these papers, see chapter 6 of "On the Emergence Theme of Physics" by Robert Carroll.
Perelman has given a gradient formulation for the Ricci flow, introducing an "entropy function" which increases monotonically along the flow. We pursue a thermodynamic analogy and apply Ricci flow ideas to general relativity. We investigate whether Perelman's entropy is related to (BekensteinHawking) geometric entropy as familiar from black hole thermodynamics. From a study of the fixed points of the flow we conclude that Perelman entropy is not connected to geometric entropy. However, we notice that there is a very similar flow which does appear to be connected to geometric entropy. The new flow may find applications in black hole physics suggesting for instance, new approaches to the Penrose inequality.

$\begingroup$ These articles are quite interesting, but I missed the point where there were supposed to be 'new approaches to the Penrose inequality'. It seems like they suggested they would get to it at the end, then did not suggest anything concrete. $\endgroup$ Jul 16 '19 at 12:28
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 LottVillani and Sturm nonnegative Ricci curvature can be characterized by the property that the Boltzmannentropy is convex along optimal transportation. This is very intuitive, imagine e.g. a pile of sand being transported from the south to the northpole 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 Boltzmannentropy (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.