Fractional statistical mechanics

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 the fractional Bogoliubov equation. The linear fractional kinetic equation for distribution of the charged particles is considered.     

Fractals and quantum mechanics

A new application of a fractal concept to quantum physics has been developed. The fractional path integrals over the paths of the Levy flights are defined. It is shown that if fractality of the Brownian trajectories leads to standard quantum mechanics, then the fractality of the Levy paths leads to fractional quantum mechanics. The fractional quantum mechanics has been developed via the new fractional path integrals approach. A fractional generalization of the Schrodinger equation has been discovered. The new relationship between the energy and the momentum of the nonrelativistic fractional quantum-mechanical particle has been established, and the Levy wave packet has been introduced into quantum mechanics. The equation for the fractional plane wave function has been found. We have derived a free particle quantum-mechanical kernel using Fox's H-function. A fractional generalization of the Heisenberg uncertainty relation has been found. As physical applications of the fractional quantum mechanics we have studied a free particle in a square infinite potential well, the fractional "Bohr atom" and have developed a new fractional approach to the QCD problem of quarkonium. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum mechanics. (c) 2000 American Institute of Physics.     

Fractional Zero Curvature Equation and Generalized Hamiltonian Structure of Soliton Equation Hierarchy

We presented the fractional zero curvature equation and generalized Hamiltonian structure by using of the differential forms of fractional orders. Example of the fractional AKNS soliton equation hierarchy and its Hamiltonian system are obtained.     

Renormalization group theory of anomalous transport in systems with Hamiltonian chaos

We present a general scheme to describe particle kinetics in the case of incomplete Hamiltonian chaos when a set of islands of stability forms a complicated fractal space-time dynamics and when there is orbit stickiness to the islands' boundary. This kinetics is alternative to the "normal" Fokker-Planck-Kolmogorov equation. A new kinetic equation describes random wandering in the fractal space-time. Critical exponents of the anomalous kinetics are expressed through dynamical characteristics of a Hamiltonian using the renormalization group approach. Renormalization transformation has been applied simultaneously for space and time and fractional calculus has been exploited.     

Generalized classical mechanics

Generalized classical mechanics has been introduced and developed as a classical counterpart of the fractional quantum mechanics. The Lagrangian of generalized classical mechanics has been introduced, and equation of motion has been obtained. Lagrange, Hamilton and Hamilton-Jacobi frameworks have been implemented. Oscillator model has been launched and solved in 1D case. A new equation for the period of oscillations of generalized classical oscillator has been found. The interplay between the energy dependency of the period of classical oscillations and the non-equidistant distribution of the energy levels for fractional quantum oscillator has been discussed. We discuss as well, the relationships between new equations of generalized classical mechanics and the well-known fundamental equations of classical mechanics.     

Hamiltonian formalism of fractional systems

In this paper we consider a generalized classical mechanics with fractional derivatives. The generalization is based on the time-clock randomization of momenta and coordinates taken from the conventional phase space. The fractional equations of motion are derived using the Hamiltonian formalism. The approach is illustrated with a simple-fractional oscillator in a free motion and under an external force. Besides the behavior of the coupled fractional oscillators is analyzed. The natural extension of this approach to continuous systems is stated. The interpretation of the mechanics is discussed.     

A new generalized fractional Dirac soliton hierarchy and its fractional Hamiltonian structure

Based on the differential forms and exterior derivatives of fractional orders,Wu first presented the generalized Tu formula to construct the generalized Hamiltonian structure of the fractional soliton equation.We apply the generalized Tu formula to calculate the fractional Dirac soliton equation hierarchy and its Hamiltonian structure.The method can be generalized to the other fractional soliton hierarchy.     

Optico-Mechanical Analogy: an Axiomatic Approach

An axiomatic characterization of a ‘two-level Hamiltonian structure’ is proposed, which expresses the optico-mechanical analogy by representing optics and mechanics as (disjoint) classes of models satisfying the axioms. There is the ‘Hamilton–Jacobi level’, which involves a differential manifold on which the characteristic function satisfying the Hamilton–Jacobi equation is defined; and the ‘symplectic level’, involving the Hamiltonian, defined on the cotangent bundle of the manifold. The two levels, with the (analogous) structures on them, concern both optics and mechanics.     

Schrödinger方程的辛结构与量子力学的辛算法

冯康开创的哈密顿力学的辛算法取得了惊人的成功.这是因为哈密顿力学的数学框架是辛几何,一个合理的离散方法自然应使离散哈密顿力学保持辛结构.本文指出,经过适当的变换,Schrödinger方程也具有辛结构,从而把哈密顿力学的辛算法,推广用到量子力学.作为例子计算了中子在旋转磁场中的演化.计算结果表明,辛算法明显优于通常算法,特别是对演化时间长的情况.     

Fractional sequential mechanics — models with symmetric fractional derivative

The symmetric fractional derivative is introduced and its properties are studied. The Euler-Lagrange equations for models depending on sequential derivatives of type are derived using minimal action principle. The Hamiltonian for such systems is introduced following methods of classical generalized mechanics and the Hamilton’s equations are obtained. It is explicitly shown that models of fractional sequential mechanics are non-conservative. The limiting procedure recovers classical generalized mechanics of systems depending on higher order derivatives. The method is applied to fractional deformation of harmonic oscillator and to the case of classical frictional force proportional to velocity. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001.     

On Wigner functions and a damped star product in dissipative phase-space quantum mechanics

Dito and Turrubiates recently introduced an interesting model of the dissipative quantum mechanics of a damped harmonic oscillator in phase space. Its key ingredient is a non-Hermitian deformation of the Moyal star product with the damping constant as deformation parameter. We compare the Dito-Turrubiates scheme with phase-space quantum mechanics (or deformation quantization) based on other star products, and extend it to incorporate Wigner functions. The deformed (or damped) star product is related to a complex Hamiltonian, and so necessitates a modified equation of motion involving complex conjugation. We find that with this change the Wigner function satisfies the classical equation of motion. This seems appropriate since non-dissipative systems with quadratic Hamiltonians share this property.     

Fractional generalization of Liouville equations

In this paper fractional generalization of Liouville equation is considered. We derive fractional analog of normalization condition for distribution function. Fractional generalization of the Liouville equation for dissipative and Hamiltonian systems was derived from the fractional normalization condition. This condition is considered as a normalization condition for systems in fractional phase space. The interpretation of the fractional space is discussed.     

Painlevé IV coherent states

A simple way to find solutions of the Painlevé IV equation is by identifying Hamiltonian systems with third-order differential ladder operators. Some of these systems can be obtained by applying supersymmetric quantum mechanics (SUSY QM) to the harmonic oscillator. In this work, we will construct families of coherent states for such subset of SUSY partner Hamiltonians which are connected with the Painlevé IV equation. First, these coherent states are built up as eigenstates of the annihilation operator, then as displaced versions of the extremal states, both involving the related third-order ladder operators, and finally as extremal states which are also displaced but now using the so called linearized ladder operators. To each SUSY partner Hamiltonian corresponds two families of coherent states: one inside the infinite subspace associated with the isospectral part of the spectrum and another one in the finite subspace generated by the states created through the SUSY technique.     

Coupled Higgs field equation and Hamiltonian amplitude equation: Lie classical approach and (G′/G)-expansion method

In this paper, coupled Higgs field equation and Hamiltonian amplitude equation are studied using the Lie classical method. Symmetry reductions and exact solutions are reported for Higgs equation and Hamiltonian amplitude equation. We also establish the travelling wave solutions involving parameters of the coupled Higgs equation and Hamiltonian amplitude equation using (G??/G)-expansion method, where G?=?G(??) satisfies a second-order linear ordinary differential equation (ODE). The travelling wave solutions expressed by hyperbolic, trigonometric and the rational functions are obtained.     

Basic Theory of Fractional Conformal Invariance of Mei Symmetry and its Applications to Physics

In this paper, we present a basic theory of fractional dynamics, i.e., the fractional conformal invariance of Mei symmetry, and find a new kind of conserved quantity led by fractional conformal invariance. For a dynamical system that can be transformed into fractional generalized Hamiltonian representation, we introduce a more general kind of single-parameter fractional infinitesimal transformation of Lie group, the definition and determining equation of fractional conformal invariance are given. And then, we reveal the fractional conformal invariance of Mei symmetry, and the necessary and sufficient condition whether the fractional conformal invariance would be the fractional Mei symmetry is found. In particular, we present the basic theory of fractional conformal invariance of Mei symmetry and it is found that, using the new approach, we can find a new kind of conserved quantity; as a special case, we find that an autonomous fractional generalized Hamiltonian system possesses more conserved quantities. Also, as the new method’s applications, we, respectively, find the conserved quantities of a fractional general relativistic Buchduhl model and a fractional Duffing oscillator led by fractional conformal invariance of Mei symmetry.     

Imprints of the Quantum World in Classical Mechanics

The imprints left by quantum mechanics in classical (Hamiltonian) mechanics are much more numerous than is usually believed. We show that the Schrödinger equation for a nonrelativistic spinless particle is a classical equation which is equivalent to Hamilton’s equations. Our discussion is quite general, and incorporates time-dependent systems. This gives us the opportunity of discussing the group of Hamiltonian canonical transformations which is a non-linear variant of the usual symplectic group.     

质点的相对论式的汉密尔敦运动式与相对论式的哈生堡方程式

在本文中,作者推得一组相对论式的汉密尔敦运动式;并根据此运动式,详细地讨论了一质点之运动;由此还可以很自然地看出,在量子力学中,狄拉克电子方程式似乎是一个必然的波动方程式。在狄拉克理论中的取平方根步骤在这里找到它在古典物理学中的对照。同时根据了上面的理论,作者还推得到一个相对论式的哈生堡方程式,最后则讨论了此方程式在狄拉克电子理论中的一些应用;同时并指出,在相对论的观点上,此方程式可以引导出一些比较对称的及有比较普遍形式的物理的量。     

Size-dependent geometrically nonlinear free vibration analysis of fractional viscoelastic nanobeams based on the nonlocal elasticity theory

In recent decades, mathematical modeling and engineering applications of fractional-order calculus have been extensively utilized to provide efficient simulation tools in the field of solid mechanics. In this paper, a nonlinear fractional nonlocal Euler–Bernoulli beam model is established using the concept of fractional derivative and nonlocal elasticity theory to investigate the size-dependent geometrically nonlinear free vibration of fractional viscoelastic nanobeams. The non-classical fractional integro-differential Euler–Bernoulli beam model contains the nonlocal parameter, viscoelasticity coefficient and order of the fractional derivative to interpret the size effect, viscoelastic material and fractional behavior in the nanoscale fractional viscoelastic structures, respectively. In the solution procedure, the Galerkin method is employed to reduce the fractional integro-partial differential governing equation to a fractional ordinary differential equation in the time domain. Afterwards, the predictor–corrector method is used to solve the nonlinear fractional time-dependent equation. Finally, the influences of nonlocal parameter, order of fractional derivative and viscoelasticity coefficient on the nonlinear time response of fractional viscoelastic nanobeams are discussed in detail. Moreover, comparisons are made between the time responses of linear and nonlinear models.     

Quantization of Brownian Motion

Following our work on the quantization of nonconservative systems using fractional calculus, the canonical quantization of a system with Brownian motion is carried out according to the Dirac method. A suitable Lagrangian corresponding to the Langevin equation is set up. Further, a Hamiltonian is constructed and is transformed to Schrödinger's equation which is solved.     

Fractional corresponding operator in quantum mechanics and applications: A uniform fractional Schrödinger equation in form and fractional quantization methods

In this paper we use Dirac function to construct a fractional operator called fractional corresponding operator, which is the general form of momentum corresponding operator. Then we give a judging theorem for this operator and with this judging theorem we prove that R–L, G–L, Caputo, Riesz fractional derivative operator and fractional derivative operator based on generalized functions, which are the most popular ones, coincide with the fractional corresponding operator. As a typical application, we use the fractional corresponding operator to construct a new fractional quantization scheme and then derive a uniform fractional Schrödinger equation in form. Additionally, we find that the five forms of fractional Schrödinger equation belong to the particular cases. As another main result of this paper, we use fractional corresponding operator to generalize fractional quantization scheme by using Lévy path integral and use it to derive the corresponding general form of fractional Schrödinger equation, which consequently proves that these two quantization schemes are equivalent. Meanwhile, relations between the theory in fractional quantum mechanics and that in classic quantum mechanics are also discussed. As a physical example, we consider a particle in an infinite potential well. We give its wave functions and energy spectrums in two ways and find that both results are the same.