The Thermodynamic Extremal Principle

A unified variational framework for coupled dissipative systems

Wolfgang Flachberger

The Thermodynamic Extremal Principle

\[ \delta \left(\dot{\mathcal{F}} + \tfrac{1}{2} \mathcal{P}\right) = 0 \]

$\mathcal{F}[q]$ — free energy, a functional of the state variables $q$

$\mathcal{P}[\dot{q}]$ — dissipation, a positive-definite quadratic form in the rates $\dot{q}$

The Lagrangian

\[ \mathcal{L}[\dot{q}] = \dot{\mathcal{F}}[q] + \tfrac{1}{2}\,\mathcal{P}[\dot{q}] \]

Variation is spatial (over rates at a fixed instant), not temporal.

Fundamentally distinct from Hamilton's principle.

Naturally leads to minimum energy state for closed systems.

Derived from one framework

  • Stokes & Navier-Stokes flow
  • Linear creep law
  • Diffusion (Fick's law)
  • Cahn-Hilliard (phase separation)
  • Coupled thermo-mechanics

Example: Diffusion

Free energy and dissipation

\[ \mathcal{F}[\phi] = \int_V \tfrac{1}{2}\phi^2 \; dV \qquad\qquad \mathcal{P}[\underline{j}] = \int_V \frac{|\underline{j}|^2}{D} \; dV \]

The Lagrangian density

\[ \mathscr{L}\!\left(\tfrac{\partial \phi}{\partial t},\, \underline{j},\, \mu\right) = \phi\,\frac{\partial \phi}{\partial t} + \frac{|\underline{j}|^2}{2D} + \mu\left(\frac{\partial \phi}{\partial t} + \nabla \cdot \underline{j}\right) \]

Euler-Lagrange Equations

\[\begin{aligned} \frac{\partial \mathscr{L}}{\partial (\partial\phi/\partial t)} &= 0 & \quad\Rightarrow\quad & \mu = -\phi \\[8pt] \frac{\partial \mathscr{L}}{\partial \underline{j}} - \nabla\!\left(\frac{\partial \mathscr{L}}{\partial (\nabla \cdot \underline{j})}\right) &= 0 & \quad\Rightarrow\quad & \underline{j} = D\,\nabla\mu = -D\,\nabla\phi \\[8pt] \frac{\partial \mathscr{L}}{\partial \mu} &= 0 & \quad\Rightarrow\quad & \frac{\partial \phi}{\partial t} + \nabla \cdot \underline{j} = 0 \end{aligned}\]

Result

\[ \frac{\partial \phi}{\partial t} = D\,\nabla^2\phi \]

The classical diffusion equation, derived from the TEP.

The TEP can be used to automatically derive variational formulations.

Very helpful for coupled problems with many field variables — the principle captures everything in the most compact way.

Thank you.

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