# Lagrangian for a concentration field

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.