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From the energy functional, we can derive the Euler-Lagrange equation and its corresponding gradient flow equation. My question is, what is the physical unit for ``time'' in the gradient flow equation?

For example, the Oseen-Frank energy for liquid crystal is given by $$\int_{\Omega} \frac{1}{2} K |\nabla u|^2,$$ where $u$ is the angle of the director with the $x$-axis, and $K$ is an elastic constant with unit Newton. The corresponding gradient flow equation is $$\frac{\partial u}{\partial t} = K \triangle u.$$ But then what is the unit of time $t$? It seems that it does not have the unit of time. What am I missing here?

Thanks!

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    $\begingroup$ By the gradient flow equation, $[K] = L^2 T^{-1}$, where length and time are indicated. Meanwhile the Oseen-Frank "energy" (call it $\mathcal{E}$) has unit $[K] L^{-2} = T^{-1}$, not the physical energy unit $E = M L^2 T^{-2}$. This sort of elision of units is fairly typical of mathematical practice. Note that the unit of action is $[\hbar] = E T$, so $[\hbar \mathcal{E}] = E$. $\endgroup$
    – Steve Huntsman
    Commented Aug 17, 2011 at 15:47

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If you wish to interpret the gradient flow equation as an equation of physics (rather than mathematics), then you need to introduce a friction coefficient into your problem, which tells you how rapidly energy is dissipated in order to reach an equilibrium condition. The energy functional itself only tells you what the equilibrium state is, not how fast you will reach it if you start away from equilibrium. So just multiply the left-hand-side of your gradient flow equation by a friction coefficient $\gamma$ (units $Nsm^{-2}$) and integrate. You will then flow towards the equilibrium state having $\Delta u=0$, which minimizes the energy regardless of the value you chose for $\gamma$.

If you wish to learn more, see for example

http://wwwx.cs.unc.edu/~mn/classes/comp775/doc/calculusOfVariations.pdf

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  • $\begingroup$ But according to en.wikipedia.org/wiki/Friction#Coefficient_of_friction, the friction coefficient is a dimensionless scalar. Obviously your $\gamma$ and their "coefficient of friction" are different. Is there another name for physics quantity with units $N s m^{-2}$? $\endgroup$
    – Chong Luo
    Commented Aug 18, 2011 at 20:12
  • $\begingroup$ the friction coefficient $γ$ can also be called a viscosity (or, more precisely, dynamic viscosity); the unit $Nsm^{-2}$ is called a "Poise" in the hydrodynamical context (after the French physicist Poiseuille). en.wikipedia.org/wiki/Viscosity#Dynamic_viscosity $\endgroup$
    – Carlo Beenakker
    Commented Aug 19, 2011 at 0:38
  • $\begingroup$ You're welcome. Question: you just "unaccepted" my answer; was that intentional? $\endgroup$
    – Carlo Beenakker
    Commented Aug 30, 2011 at 13:46
  • $\begingroup$ Sorry, I clicked by accident. $\endgroup$
    – Chong Luo
    Commented Sep 7, 2011 at 16:19

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