# How define the entropy of heat equation?

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.

• en.wikipedia.org/wiki/Differential_entropy – AHusain Dec 23 '15 at 10:44
• @AHusain Sorry,I thinks it's useless for my question. I don't care about the entropy of information theory.But thanks too. – lanse7pty Dec 23 '15 at 11:18
• They are the same thing. It is the probability distribution on the physical states. – AHusain Dec 24 '15 at 1:08

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.