Statistical analysis of complex systems with nonclassical invariant measures

Abstract

I investigate the problem of finding a statistical description of a complex many-body system whose invariant measure cannot be constructed stemming from classical thermodynamics ensembles. By taking solitons as a reference system and by employing a general formalism based on the Ablowitz-Kaup-Newell-Segur scheme, I demonstrate how to build an invariant measure and, within a one dimensional phase space, how to develop a suitable thermodynamics. A detailed example is provided with a universal model of wave propagation, with reference to a transparent potential sustaining gray solitons. The system shows a rich thermodynamic scenario, with a free energy landscape supporting phase transitions and controllable emergent properties. I finally discuss the origin such behavior, trying to identify common denominators in the area of complex dynamics.

9 pages, 4 figures

References in corpus (14)

• Dispersive, superfluid-like shock waves in nonlinear optics
• On Dispersive and Classical Shock Waves in Bose-Einstein Condensates and Gas Dynamics
• Shocks in nonlocal media
• Kinetic equation for a dense soliton gas
• Statistical mechanics of a nonlinear discrete system
• Anomalous properties of the acoustic excitations in glasses on the mesoscopic length-scale
• Dissipationless shock waves in repulsive Bose-Einstein condensates
• Formation of shock waves in a Bose-Einstein condensate
• The theory of optical dispersive shock waves in photorefractive media
• Solar Flare Intermittency and the Earth's Temperature Anomalies
• Multi-Scaling Comparative Analysis of Time Series and a Discussion on "Earthquake Conversations" in California
• Lévy scaling: the Diffusion Entropy Analysis applied to DNA sequences
• Dynamics of Kinks: Nucleation, Diffusion and Annihilation
• Free-energy transition in a gas of non-interacting nonlinear wave-particles