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If I have a field, $\phi(x)$, representing the concentration of particles in a one dimensional space, what is the Lagrangian for this field? My initial guess was

$$\mathcal{L} = \int dx[\frac{1}{2}\dot{\phi}^2 -\frac{1}{2}\phi'^2 - \mu\phi]$$

where $U(\phi)=\mu\phi$ was interpreted from the thermodynamic equation $U=TS-PV+\mu N$.

However, after looking at this post, I see that the kinetic energy term might not be so straightforward for diffusion systems.

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