arXiv subjects
Publications and source records attributed to Vasily E. Tarasov.
We use the fractional integrals to describe fractal media. We consider the fractal media as special ("fractional") continuous media. We discuss the possible experimental testing of the continuous medium model for fractal media that is suggested in Phys. Lett. A. 336 (2005) 167-174. This test is connected with measure of period of the Maxwell pendulum with fractal medium cylinder.
We use the fractional integrals in order to describe dynamical processes in the fractal media. We consider the "fractional" continuous medium model for the fractal media and derive the fractional generalization of the equations of balance of mass density, momentum density, and internal energy. The fractional generalization of Navier-Stokes and Euler equations are considered. We derive the equilibrium equation for fractal media. The sound waves in the continuous medium model for fractional media are considered.
We consider the fractional generalizations of equation that defines the medium mass. We prove that the fractional integrals can be used to describe the media with noninteger mass dimensions. Using fractional integrals, we derive the fractional generalization of the Chapman-Kolmogorov equation (Smolukhovski equation). In this paper fractional Fokker-Planck equation for fractal media is derived from the fractional Chapman-Kolmogorov equation. Using the Fourier transform, we get the Fokker-Planck-Zaslavsky equations that have fractional coordinate derivatives. The Fokker-Planck equation for the fractal media is an equation with fractional derivatives in the dual space.
We consider a fractional generalization of Hamiltonian and gradient systems. We use differential forms and exterior derivatives of fractional orders. We derive fractional generalization of Helmholtz conditions for phase space. Examples of fractional gradient and Hamiltonian systems are considered. The stationary states for these systems are derived.
We consider one-dimensional chain of coupled linear and nonlinear oscillators with long-range power-wise interaction. The corresponding term in dynamical equations is proportional to $1/|n-m|^{α+1}$. It is shown that the equation of motion in the infrared limit can be transformed into the medium equation with the Riesz fractional derivative of order $α$, when $0<α<2$. We consider few models of coupled oscillators and show how their synchronization can appear as a result of bifurcation, and how the corresponding solutions depend on $α$. The presence of fractional derivative leads also to the occurrence of localized structures. Particular solutions for fractional time-dependent complex Ginzburg-Landau (or nonlinear Schrodinger) equation are derived. These solutions are interpreted as synchronized states and localized structures of the oscillatory medium.
We derive the fractional generalization of the Ginzburg-Landau equation from the variational Euler-Lagrange equation for fractal media. To describe fractal media we use the fractional integrals considered as approximations of integrals on fractals. Some simple solutions of the Ginzburg-Landau equation for fractal media are considered and different forms of the fractional Ginzburg-Landau equation or nonlinear Schrodinger equation with fractional derivatives are presented. The Agrawal variational principle and its generalization have been applied.
The notion of fractional dynamics is related to equations of motion with one or a few terms with derivatives of a fractional order. This type of equation appears in the description of chaotic dynamics, wave propagation in fractal media, and field theory. For the fractional linear oscillator the physical meaning of the derivative of order $α<2$ is dissipation. In systems with many spacially coupled elements (oscillators) the fractional derivative, along the space coordinate, corresponds to a long range interaction. We discuss a method of constructing a solution using an expansion in $ε=n-α$ with small $ε$ and positive integer $n$. The method is applied to the fractional linear and nonlinear oscillators and to fractional Ginzburg-Landau or parabolic equations.
We consider the description of the fractal media that uses the fractional integrals. We derive the fractional generalizations of the equation that defines the medium mass. We prove that the fractional integrals can be used to describe the media with noninteger mass dimensions. The fractional equation of continuity is considered.
We consider the fractional generalizations of the phase volume, volume element and Poisson brackets. These generalizations lead us to the fractional analog of the phase space. We consider systems on this fractional phase space and fractional analogs of the Hamilton equations. The fractional generalization of the average value is suggested. The fractional analogs of the Bogoliubov hierarchy equations are derived from the fractional Liouville equation. We define the fractional reduced distribution functions. The fractional analog of the Vlasov equation and the Debye radius are considered.
The sigma model approach to the closed bosonic string on the affine-metric manifold is considered. The two-loop metric counterterms for the nonlinear two-dimensional sigma model with affine-metric target manifold are calculated. The correlation of the metric and affine connection is considered as the result of the ultraviolet finiteness (or beta-function vanishing) condition for the nonlinear sigma model. The examples of the nonflat nonRiemannian manifolds resulting in the trivial metric beta-function are suggested.
Using Liouville space and superoperator formalism we consider pure stationary states of open and dissipative quantum systems. We discuss stationary states of open quantum systems, which coincide with stationary states of closed quantum systems. Open quantum systems with pure stationary states of linear oscillator are suggested. We consider stationary states for the Lindblad equation. We discuss bifurcations of pure stationary states for open quantum systems which are quantum analogs of classical dynamical bifurcations.
In this paper fractional generalization of Liouville equation is considered. We derive fractional analog of normalization condition for distribution function. Fractional generalization of the Liouvile equation for dissipative and Hamiltonian systems was derived from the fractional normalization condition. This condition is considered considered as a normalization condition for systems in fractional phase space. The interpretation of the fractional space is discussed.
In this paper we discuss a model of quantum computer in which a state is an operator of density matrix and gates are general quantum operations, not necessarily unitary. A mixed state (operator of density matrix) of n two-level quantum systems is considered as an element of 4^n-dimensional operator Hilbert space (Liouville space). It allows to use a quantum computer model with four-valued logic. The gates of this model are general superoperators which act on n-ququat state. Ququat is a quantum state in a four-dimensional (operator) Hilbert space. Unitary two-valued logic gates and quantum operations for an n-qubit open system are considered as four-valued logic gates acting on n-ququat. We discuss properties of quantum four-valued logic gates. In the paper we study universality for quantum four-valued logic gates.
Using Liouville space and superoperator formalism we consider pure stationary states of open and dissipative quantum systems. We discuss stationary states of open quantum systems, which coincide with stationary states of closed quantum systems. Open quantum systems with pure stationary states of linear oscillator are suggested. We consider stationary states for the Lindblad equation. We discuss bifurcations of pure stationary states for open quantum systems which are quantum analogs of classical dynamical bifurcations.
A generalization of canonical quantization which maps a dynamical operator to a dynamical superoperator is suggested. Weyl quantization of dynamical operator, which cannot be represented as Poisson bracket with some function, is considered. The usual Weyl quantization of observables is a specific case of suggested quantization. This approach allows to define consistent quantization procedure for non-Hamiltonian and dissipative systems. Examples of the harmonic oscillator with friction (generalized Lorenz-Rossler-Leipnik-Newton equation), the Fokker-Planck-type system and Lorenz-type system are considered.
In this paper we derive the canonical distribution as a stationary solution of the Liouville equation for the classical dissipative system. Dissipative classical systems can have stationary states look like canonical Gibbs distributions. The condition for non-potential forces which leads to this stationary solution is very simple: the power of the non-potential forces must be directly proportional to the velocity of the Gibbs phase (phase entropy density) change. The example of the canonical distribution for a linear oscillator with friction is considered.
Usually models for quantum computations deal with unitary gates on pure states. In this paper we generalize the usual model. We consider a model of quantum computations in which the state is an operator of density matrix and the gates are quantum operations, not necessarily unitary. A mixed state (operator of density matrix) of n two-level quantum systems is considered as an element of $4^{n}$-dimensional operator Hilbert space. Unitary quantum gates and nonunitary quantum operations for n-qubit system are considered as generalized quantum gates acting on mixed state. In this paper we study universality for quantum computations by quantum operations on mixed states.
In this paper we generalize the usual model of quantum computer to a model in which the state is an operator of density matrix and the gates are general superoperators (quantum operations), not necessarily unitary. A mixed state (operator of density matrix) of n two-level quantum system (open or closed n-qubit system) is considered as an element of $4^{n}$-dimensional operator Hilbert space (Liouville space). It allows to use quantum computer (circuit) model with 4-valued logic. The gates of this model are general superoperators which act on n-ququats state. Ququat is quantum state in a 4-dimensional (operator) Hilbert space. Unitary two-valued logic gates and quantum operations for n-qubit open system are considered as four-valued logic gates acting on n-ququats. We discuss properties of quantum 4-valued logic gates. In the paper we study universality for quantum four-valued logic gates.