Nonlinear fractional dynamics on a lattice with long range interactions
| Autor: | N. Laskin, George M. Zaslavsky |
|---|---|
| Rok vydání: | 2006 |
| Předmět: |
Statistics and Probability
Nonlinear Sciences - Exactly Solvable and Integrable Systems Quantum dynamics FOS: Physical sciences Propagator Nonlinear Sciences - Chaotic Dynamics Condensed Matter Physics Random walk 01 natural sciences 010305 fluids & plasmas Fractional calculus symbols.namesake Fractional dynamics Nonlinear system Quantum mechanics 0103 physical sciences symbols Exactly Solvable and Integrable Systems (nlin.SI) Chaotic Dynamics (nlin.CD) 010306 general physics Quantum Schrödinger's cat Mathematics |
| Zdroj: | Physica A: Statistical Mechanics and its Applications. 368:38-54 |
| ISSN: | 0378-4371 |
| Popis: | A unified approach has been developed to study nonlinear dynamics of a 1D lattice of particles with long-range power-law interaction. A classical case is treated in the framework of the generalization of the well-known Frenkel-Kontorova chain model for the non-nearest interactions. Quantum dynamics is considered following Davydov's approach for molecular excitons. In the continuum limit the problem is reduced to dynamical equations with fractional derivatives resulting from the fractional power of the long-range interaction. Fractional generalizations of the sine-Gordon, nonlinear Schrodinger, and Hilbert-Schrodinger equations have been found. There exists a critical value of the power s of the long-range potential. Below the critical value (s3 we have the well-known nonlinear dynamical equations with space derivatives of integer order. Long-range interaction impact on the quantum lattice propagator has been studied. We have shown that the quantum exciton propagator exhibits transition from the well-known Gaussian-like behavior to a power-law decay due to the long-range interaction. A link between 1D quantum lattice dynamics in the imaginary time domain and a random walk model has been discussed. 28 pages, submitted to Physica A |
| Databáze: | OpenAIRE |
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