Of course, dissipating gas is an example of entropy, as is the disappearance of details over time. So it shouldn't surprise us that the effects of Ricci flow have a good deal in common with entropy in physics.
For example, look at the heat diffusion equation <math>u_t = \Delta u \ </math>. This equation is precisely an expression for the change in the metric of a manifold over time in Ricci flow if we take <math>u = \ln g_{jj} \ </math>, where j is any number less than the dimension of the manifold and no summation is implied on that index, and g is a metric symmetric on the main diagonal (any relevant manifold can have a coordinate system which create such a metric).
Noticing this (and other similarities between the effect of Ricci flow on a manifolds metric and the effect of time on heat dispersion, Perelman formulated the concept of "Perelman entropy."