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Today, I report a paper about Ricci flow, I saw entropy. As I know, entropy is a physical term.And I know it is used to describe how far the system from heat death.But I don't know the equation of entropy and the precise define.

For example, if there is a heat equation and suitable initial condition,how to define the entropy ? $$ \left\{ \begin{aligned} &u_t =\Delta u ~~~~\text{ in $~\Omega\times(0,T]$}\\ &u(x,0) =u_0(x) \end{aligned} \right. $$

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  • $\begingroup$ en.wikipedia.org/wiki/Differential_entropy $\endgroup$ – AHusain Dec 23 '15 at 10:44
  • $\begingroup$ @AHusain Sorry,I thinks it's useless for my question. I don't care about the entropy of information theory.But thanks too. $\endgroup$ – lanse7pty Dec 23 '15 at 11:18
  • $\begingroup$ They are the same thing. It is the probability distribution on the physical states. $\endgroup$ – AHusain Dec 24 '15 at 1:08
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Much too late, but there is a very nice script by Evans on the subject: https://math.berkeley.edu/~evans/entropy.and.PDE.pdf

The heat equation and, more generally parabolic equations, is treated on pp. 90 ff. Very roughly, the entropy is defined as $\log(u)$, where $u$ is the solution to the heat equation.

Hope that helps!

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  • $\begingroup$ should be a comment $\endgroup$ – воитель Oct 11 '18 at 23:37

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