Fractional Statistical Mechanics

Abstract

The Liouville and first Bogoliubov hierarchy equations with derivatives of noninteger order are derived. The fractional Liouville equation is obtained from the conservation of probability to find a system in a fractional volume element. This equation is used to obtain Bogoliubov hierarchy and fractional kinetic equations with fractional derivatives. Statistical mechanics of fractional generalization of the Hamiltonian systems is discussed. Liouville and Bogoliubov equations with fractional coordinate and momenta derivatives are considered as a basis to derive fractional kinetic equations. The Fokker-Planck-Zaslavsky equation that has fractional phase-space derivatives is obtained from fractional Bogoliubov equation. The linear fractional kinetic equation for distribution of the charged particles is considered.

18 pages, LaTeX, 1 figure

References in corpus (23)

• Fractional Quantum Mechanics
• Time fractional Schrodinger equation
• Continuous Medium Model for Fractal Media
• Fractional Hydrodynamic Equations for Fractal Media
• Fractional dynamics of systems with long-range interaction
• Fractional dynamics of coupled oscillators with long-range interaction
• Nonlinear Fractional Dynamics on a Lattice with Long Range Interactions
• Fractional Generalization of Gradient and Hamiltonian Systems
• Fractional Generalization of Liouville Equations
• Fractional Systems and Fractional Bogoliubov Hierarchy Equations
• Dynamics with Low-Level Fractionality
• Fractional Variations for Dynamical Systems: Hamilton and Lagrange Approaches
• Fractional Liouville and BBGKI Equations
• Coupled oscillators with power-law interaction and their fractional dynamics analogues
• Fractional Fokker-Planck Equation for Fractal Media
• Electromagnetic field of fractal distribution of charged particles
• Possible Experimental Test of Continuous Medium Model for Fractal Media
• Magnetohydrodynamics of Fractal Media
• Transport Equations from Liouville Equations for Fractional Systems
• Multipole Moments of Fractal Distribution of Charges
• Fractional Generalization of Gradient Systems
• Classical Canonical Distribution for Dissipative Systems
• Stationary Solutions of Liouville Equations for Non-Hamiltonian Systems

Cited by in corpus (9)

• Fractional Vector Calculus and Fractional Maxwell's Equations
• Conservation Laws and Hamilton's Equations for Systems with Long-Range Interaction and Memory
• Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics
• A Fractional Fokker-Planck Model for Anomalous Diffusion
• Fractional Liouville Equation on Lattice Phase-Space
• Power-law Spatial Dispersion from Fractional Liouville Equation
• Non-Linear Langevin and Fractional Fokker-Planck Equations for Anomalous Diffusion by Levy Stable Processes
• Generalized diffusion equation with fractional derivatives within Renyi statistics
• Fractional Einstein field equations in $2+1$ dimensional spacetime