Melnikov's criterion for nondifferentiable weak-noise potentials

Journal Article · Sat Feb 01 04:00:00 EST 1986 · J. Stat. Phys.; (United States)

DOI:https://doi.org/10.1007/BF01127729· OSTI ID:6957454

Jauslin, H R

The stationary probability density of Fokker-Planck models with weak noise has an asymptotic form containing a pseudopotential theta. If theta is smooth, it satisfies a Hamilton-Jacobi equation at zero energy and can be interpreted as the action of an associated Hamiltonian system. Under this assumption, theta has the properties of a Lyapunov function, and can be used, for example, as a thermodynamic potential in nonequilibrium steady states. The author considers systems having several attractors and shows, by applying Melnikov's method to the associated Hamiltonian, that in general theta is not differentiable. A small perturbation of a model with differentiable theta leads to a nondifferentiable theta. The method is illustrated on a model used in the treatment of the unstable mode in a laser.

• Research Organization: Univ. de Geneve, Geneve
• OSTI ID: 6957454
• Journal Information: J. Stat. Phys.; (United States), Journal Name: J. Stat. Phys.; (United States) Vol. 42:3; ISSN JSTPB
• Country of Publication: United States
• Language: English

Related Subjects

657002* -- Theoretical & Mathematical Physics-- Classical & Quantum Mechanics
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BOUNDARY CONDITIONS
DIFFERENTIAL EQUATIONS
ELECTROMAGNETIC RADIATION
EQUATIONS
EQUATIONS OF MOTION
FLUCTUATIONS
FOKKER-PLANCK EQUATION
FUNCTIONS
HAMILTONIANS
JACOBIAN FUNCTION
LASER RADIATION
LYAPUNOV METHOD
MATHEMATICAL OPERATORS
MECHANICS
NOISE
OPTICAL MODES
OSCILLATION MODES
PARTIAL DIFFERENTIAL EQUATIONS
PROBABILITY
QUANTUM MECHANICS
QUANTUM OPERATORS
RADIATIONS
STATISTICAL MECHANICS
TRANSPORT THEORY
VARIATIONS