Dr. Mark Mineev-Weinstein (New Mexico Consortium, Los Alamos NM)

"Variational principle for pattern selections (no surface tension)"

Abstract: I present a stochastic growth theory for a variety of unstable processes. (*)

Applying the least action principle to a negative entropy (then it’s equivalent to maximum of entropy), we obtained after variation the deterministic evolution equation for the growth process in a form of Euler-Lagrange equations of motion. We also revealed the Hamiltonian structure, the Lagrangian, and the Hamilton-Jacobi PDE for this process.

Then we addressed pattern selection problems for the growth process by applying again the Maximum Entropy Principle, but already for the most probable (classical) deterministic scenario described by the obtained equations of motion mentioned above. As a result, we obtained experimentally observed asymptotic patterns, selected from a continuum of admissible steady state and/or self-similar solutions. All five patterns tested for selection belong to the two-dimensional Laplacian growth in a Hele-Shaw cell:

1. Saffman-Taylor finger in a rectangular Hele-Shaw channel (the selected relative width = 1/2).
2. Taylor-Saffman bubble moving in the same setting (the selected asymptotic velocity U=2V, where V is a background velocity of a viscous fluid).
3. Same bubble, but in unbound Hele-Shaw cell (the selected shape is a perfect circle from a continuum of all ellipses with the same area, and the selected velocity is U=2V, as above).
4. Self-similarly growing finger in a wedge (a formula for the selected ratio of the finger angle to a wedge angle is too long for the abstract, see instead the Figure attached).
5. Universal fjord opening angle (the selected value is 11.7 degrees in a wedge and 8-9 degrees in a general setting).

All five patterns selections by the Maximal Entropy Principle described above are in excellent agreement with experiments: 0% error for items 1-3 and .7% error for items 4-5.

We may add that, contrary to the common belief, surface tension is not needed for selection.

(*) Work done Jointly with O. Alekseev)