Morse and Feshbach state without elaboration that the diffusion equation for temperature or concentration ψ and its "conjugate" ψ∗ (quotation marks theirs) has Lagrangian density:
L=−∇ψ⋅∇ψ∗−12a2(ψ∗∂ψ∂t−ψ∂ψ∗∂t).
I don't understand what the conjugate field, ψ∗, is. Since the classical (non-Shrödinger) field should be real, I suspect the conjugation symbol * refers to something other than complex conjugation. With a real field, ψ∗=ψ, and only −∇ψ⋅∇ψ remains, which would be the Lagrangian for the Laplace equation (steady state diffusion).
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