Free energy in Allen-Cahn PDE I am a mathematician and I am taking a mathematical physics course. In the part of reaction-diffusion equations, there is something that I do not understand. I have been defined the Allen-Cahn equation as $$u_t=u_{xx}+u-u^3.$$ Then the professor wrote the equation as $$u_t=u_{xx}-W'(u),$$ where $$W(u)=1/4-u^2/2+u^4/4.$$ The free energy was defined as
$$ H[u]=\int_\Omega \left(\frac12 u_x^2+W(u)\right)\,\mathrm{d}x. $$
I do not understand what this free energy tries to model. I read on Wikipedia that the part $1/2\int_\Omega u_x^2\,\mathrm{d}x$ is called Dirichlet energy, but I cannot find any physical interpretation of this. Could you provide any simple explanation of $H[u]$ for a theoretical mathematician?
 A: $H$ is like a potential and the system will always "roll" to smaller values of $H$ (unlike in Hamiltonian mechanics but much rather as in "gradient flows"), i.e. $H$ is a Lyapunov function of the Allen-Cahn dynamics: it decreases in time unless a stationary condition is reached. In this case, the time-evolution will try to smoothen inhomogeneities (i.e. decrease $(\partial_x u)^2$) and there will be an attraction towards the values $u=\pm 1$ (since those are the minima of $W$). Depending on the initial conditions, a minimum of $H$ may or may not be reached (If the initial data are e.g. $u(t=0,x\geq0)=1=-u(t=0,x<0)$, then the resulting solution can not reach a minimum of $H$. 
A free energy is typically not simply a Lyapunov function. We understand it as a kind of Boltzmann entropy (say there is an underlying microscopic dynamics with a local detailed balance property which reduces to Allen-Cahn for macroscopic densities): the free energy counts the number of microstates that are in agreement with the observed macroscopic density profile.
