Towards a theory of growing surfaces: Mapping two-dimensional Laplacian growth onto Hamiltonian dynamics and statistics

Conference · Mon Nov 01 00:00:00 EST 1993

OSTI ID:10104153

Blumenfeld, R

I show that the evolution of a two dimensional surface in a Laplacian field can be described by Hamiltonian dynamics. First the growing region is mapped conformally to the interior of the unit circle, creating in the process a set of mathematical zeros and poles that evolve dynamically as the surface grows. Then the dynamics of these quasi-particles are transformed into a seperable action-angle Hamiltonian that describes an orbital motion on a torus. A specific case is discussed explicitly, which demonstrates the integrability of the surface-tension-free Laplacian growth process. This formulation holds as long as the singularities of the map are confined to within the unit circle. This approach further allows for surface tension to be introduced as an energetic term in the resulting Hamiltonian which effects repulsion between the quasi-particles and the surface. These results are used here to formulate a first-principles statistical theory of pattern formation in stochastic growth, where noise is a key player.

• Research Organization: Los Alamos National Lab., NM (United States)
• Sponsoring Organization: USDOE, Washington, DC (United States)
• DOE Contract Number: W-7405-ENG-36
• OSTI ID: 10104153
• Report Number(s): LA-UR-93-3591; CONF-9309185-4; ON: DE94002636; TRN: 94:000126
• Resource Relation: Conference: Fluctuations and order: the new synthesis workshop,Los Alamos, NM (United States),9-12 Sep 1993; Other Information: PBD: [1993]
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE
SURFACES
CONFORMAL MAPPING
EQUATIONS OF MOTION
LAPLACIAN
HAMILTONIANS
NONLINEAR PROBLEMS
DYNAMICS
DIFFERENTIAL CALCULUS
STATISTICS
661300
990200
OTHER ASPECTS OF PHYSICAL SCIENCE
MATHEMATICS AND COMPUTERS