A statistical analogy

In this section we show that the functional ${\cal W},$ introduced in section 3, is in a sense analogous to minus entropy.

5.1 Recall that the partition function for the canonical ensemble at temperature $\beta^{-1}$ is given by $Z=\int{exp(-\beta E)d\omega(E)},$ where $\omega(E)$ is a "density of states" measure, which does not depend on $\beta.$ Then one computes the average energy $<E>=-\frac{\partial}{\partial\beta}\log Z,$ the entropy $S=\beta<E>+\log Z,$ and the fluctuation $\sigma=<(E-<E>)^2>=\frac{\partial^2}{(\partial\beta)^2}\log Z.$

Now fix a closed manifold $M$ with a probability measure $m$, and suppose that our system is described by a metric $g_{ij}(\tau),$ which depends on the temperature $\tau$ according to equation $(g_{ij})_\tau=2(R_{ij}+\nabla_i\nabla_j f),$ where $dm=udV, u=(4\pi\tau)^{-\frac{n}{2}}e^{-f},$ and the partition function is given by $\log Z=\int{(-f+\frac{n}{2})dm}.$ (We do not discuss here what assumptions on $g_{ij}$ guarantee that the corresponding "density of states" measure can be found) Then we compute

$$<E>=-\tau^2\int_M{(R+\vert\nabla f\vert^2-\frac{n}{2\tau})dm},$$

$$S=-\int_M{(\tau(R+\vert\nabla f\vert^2)+f-n)dm},$$

$$\sigma=2\tau^4\int_M{\vert R_{ij}+\nabla_i\nabla_j f-\frac{1}{2\tau}g_{ij}\vert^2dm}$$

Alternatively, we could prescribe the evolution equations by replacing the $t$-derivatives by minus $\tau$-derivatives in (3.3 ), and get the same formulas for $Z, <E>, S, \sigma,$ with $dm$ replaced by $udV.$

Clearly, $\sigma$ is nonnegative; it vanishes only on a gradient shrinking soliton. $<E>$ is nonnegative as well, whenever the flow exists for all sufficiently small $\tau>0$ (by proposition 1.2). Furthermore, if (a) $u$ tends to a $\delta$-function as $\tau\to 0,$ or (b) $u$ is a limit of a sequence of functions $u_i,$ such that each $u_i$ tends to a $\delta$-function as $\tau\to\tau_i>0,$ and $\tau_i\to 0,$ then $S$ is also nonnegative. In case (a) all the quantities $<E>, S, \sigma$ tend to zero as $\tau\to 0,$ while in case (b), which may be interesting if $g_{ij}(\tau)$ goes singular at $\tau=0,$ the entropy $S$ may tend to a positive limit.

If the flow is defined for all sufficiently large $\tau$ (that is, we have an ancient solution to the Ricci flow, in Hamilton's terminology), we may be interested in the behavior of the entropy $S$ as $\tau\to\infty.$ A natural question is whether we have a gradient shrinking soliton whenever $S$ stays bounded.

5.2 Remark. Heuristically, this statistical analogy is related to the description of the renormalization group flow, mentioned in the introduction: in the latter one obtains various quantities by averaging over higher energy states, whereas in the former those states are suppressed by the exponential factor.

5.3* An entropy formula for the Ricci flow in dimension two was found by Chow [C]; there seems to be no relation between his formula and ours.

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.