# Diffusion equation Lagrangian: what is the conjugate field?

Morse and Feshbach state without elaboration that the diffusion equation for temperature or concentration $$\psi$$ and its "conjugate" $$\psi^*$$ (quotation marks theirs) has Lagrangian density:

$$L=-\nabla\psi\cdot\nabla\psi^* -\frac{1}{2}a^2(\psi^*\frac{\partial\psi}{\partial t}-\psi\frac{\partial\psi^*}{\partial t}).$$

I don't understand what the conjugate field, $$\psi^*$$, 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, $$\psi^*=\psi$$, and only $$-\nabla\psi\cdot\nabla\psi$$ remains, which would be the Lagrangian for the Laplace equation (steady state diffusion).

• @CosmasZachos, do you have a reference (textbook?) where they work through this? – Travis Lee Feb 15 at 16:01
• – Cosmas Zachos Feb 15 at 16:33
• Which page in M&F? – Qmechanic Feb 15 at 17:50
• Wick-rotated version of question: physics.stackexchange.com/q/15242/2451 – Qmechanic Feb 15 at 17:53
• The reference is P. 313 of M&F. – MarkWayne Aug 1 at 21:25

The conjugate field ψ∗ is but the complex conjugate of ψ, so an extra degree of freedom to expedite derivation of the diffusion equation, $$\nabla^2 \psi = a^2 \partial_t \psi ,$$ analogous to the Lagrangian of the free Schroedinger equation, real in that case--only.
Alternatively, integrating by parts in the action and discarding the surface terms nets a Lagrangian density $$L= \psi^* (\nabla^2 -a^2 \partial_t )\psi,$$ so ψ∗ may be thought of as an extraneous Lagrange multiplier gimmick to brutally enforce the diffusion equation as is, and concentrate on its real solutions.
Note $$\int dx \psi$$ is a constant in time, as physically required for your diffusing quantity.
A central solution of this equation underlying its propagator is $$\psi({\mathbf x},t)= \frac{a^3}{8 (\pi t )^{3/2}} ~ e^{-a^2 {\mathbf x}^2 /4t}$$ which starts out at t =0 as a Dirac $$\delta ({\mathbf x})=\psi({\mathbf x},0)$$. As a result, any initial concentration profile f can be written as a linear superposition of such δs, $$\tilde \psi({\mathbf x},0)= \int d^3 y ~ f({\mathbf y} ) \delta ({\mathbf x} -{\mathbf y}) ,$$ and propagated through each component thereof, by the above solution, $$\tilde \psi({\mathbf x},t)= \frac{a^3}{8 (\pi t )^{3/2}} ~\int d^3 y ~ \tilde \psi({\mathbf y},0 ) ~ e^{-a^2 ({\mathbf x}-{\mathbf y})^2 /4t} .$$