Relativistic quantum and classical kinetic equations in the tomographic-probability representation

Using the Radon integral transform of the relativistic kinetic equation for a spin-zero particle, we obtain the classical and quantum evolution equations for the tomographic probability density (tomogram) describing the states of the particle in both the classical and quantum pictures. The Green functions (propagators) of the evolution equations of a free particle are constructed. The examples of the evolution of Gaussian tomogram is considered.     

Probability Representation of Kinetic Equations for Open Quantum Systems

The tomographic probability distribution is used to describe the kinetic equations for open quantum systems. Damped oscillator is studied. The purity parameter evolution for different damping regimes is considered.     

General Bell-CHSH type and entropic inequalities based on quantum tomograms

Review of Bell-CHSH type and entropic inequalities in composite quantum correlated systems in the probability representation of states is presented. The upper bounds for some new Bell-CHSH type inequalities within the framework of classical probability theory and in quantum tomography are compared. Violation of Bell-CHSH type inequalities are shown explicitly using the method of averaging in tomographic picture of quantum states. Joint tomographic entropies of multiqubit systems are studied. Limitations on inequalities for tomographic entropies are obtained. A negative result of possible connection between the violation of entropic and Bell-CHSH type inequalities in multi-partite states is reported.     

Bell-type inequalities in classical probability theory

A review of the tomographic probability representation for qudit sates is presented. Properties of related stochastic matrices are considered. Tomograms of two qubits and three qubits are used to study the Bell-type inequalities. The Bell-type inequalities in the standard classical probability theory are discussed. Joint probability distributions of classical systems with several random variables and their properties in the case of factorized distribution functions are considered.     

Tomographic representation of kinetic equations in classical statistical mechanics

A tomographic representation of kinetic equations is constructed using the Radon transform. Liouville’s equation is considered for one and many particles. The reduced Liouville’s equation is obtained in the tomographic representation and the Bogolyubov chain is investigated in this representation. An example of the relativistic kinetic equation in the tomographic representation is considered.     

Probability Description and Entropy of Classical and Quantum Systems

Tomographic approach to describing both the states in classical statistical mechanics and the states in quantum mechanics using the fair probability distributions is reviewed. The entropy associated with the probability distribution (tomographic entropy) for classical and quantum systems is studied. The experimental possibility to check the inequalities like the position–momentum uncertainty relations and entropic uncertainty relations are considered.     

Nonequilibrium evolution thermodynamics of vacancies

An alternative approach, nonequilibrium evolution thermodynamics, is compared with the classical Landau approach. A statistical justification of the approach is done with the help of a probability distribution function on an example of a solid with vacancies. Two kinds of kinetic equations are derived in terms of the internal energy and the modified free energy.     

Calculations of the propagator within the tomographic-probability framework

We show that the ambiguity in choosing the tomographic propagator for an evolving linear quantum system is related to the homogeneity properties of the system symplectic tomogram. We study in detail an example of the driven harmonic oscillator. We prove that two formally different propagators of the quantum kinetic equation for the oscillator are identical on the domain of homogeneous tomographic probability distributions.     

Observables,Evolution Equation,and Stationary States Equation in the Joint Probability Representation of Quantum Mechanics

Symplectic and optical joint probability representations of quantum mechanics are considered, in which the functions describing the states are the probability distributions with all random arguments (except the argument of time). The general formalism of quantizers and dequantizers determining the star product quantization scheme in these representations is given. Taking the Gaussian functions as the distributions of the tomographic parameters the correspondence rules for most interesting physical operators are found and the expressions of the dual symbols of operators in the form of singular and regular generalized functions are derived. Evolution equations and stationary states equations for symplectic and optical joint probability distributions are obtained.     

Tomographic representation of minisuperspace quantum cosmology and noether symmetries

The probability representation, in which cosmological quantum states are described by a standard positive probability distribution, is constructed for minisuperspace models selected by Noether symmetries. In such a case, the tomographic probability distribution provides the classical evolution for the models and can be considered an approach to select “observable” universes. Some specific examples, derived from Extended Theories of Gravity, are worked out. We discuss also how to connect tomograms, symmetries and cosmological parameters.     

Herleitung kinetischer gleichungen mit dem verallgemeinerten Stratonovich-Verfahren

The kinetic equations for the 2-time conditional probability density are derived for Coulomb systems and coupled one-dimensional harmonic oscillators. The coupled oscillators are also treated exactly. The exact second central moment of the space coordinate is compared with that derived from the kinetic equation. This shows which approximations of the generalized Stratonovich method can be responsible for the possibly irreversible character of the derived kinetic equations. Using the approximation of long difference times the kinetic equations for Coulumb systems with and without homogeneous external magnetic field are transformed into the well-known Balescu-Lenard equations.     

Bell-type inequalities and upper bounds for multiqudit states

Multiqudit systems are studied in the tomographic-probability representation of quantum states. Results of calculations for the Bell-type numbers within the framework of classical probability theory and in quantum tomography are compared. Violations of the Bell-type inequalities are shown explicitly using the method of averaging in the tomographic picture of quantum states.     

Fluid dynamic limits of kinetic equations. I. Formal derivations

The connection between kinetic theory and the macroscopic equations of fluid dynamics is described. In particular, our results concerning the incompressible Navier-Stokes equations are based on a formal derivation in which limiting moments are carefully balanced rather than on a classical expansion such as those of Hilbert or Chapman-Enskog. The moment formalism shows that the limit leading to the incompressible Navier-Stokes equations, like that leading to the compressible Euler equations, is a natural one in kinetic theory and is contrasted with the systematics leading to the compressible Navier-Stokes equations. Some indications of the validity of these limits are given. More specifically, the connection between the DiPerna-Lions renormalized solution of the classical Boltzmann equation and the Leray solution of the Navier-Stokes equations is discussed.This paper is dedicated to Joel Lebowitz on his 60th-birthday.     

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.     

Linear relaxation processes governed by fractional symmetric kinetic equations

The fractional symmetric Fokker-Planck and Einstein-Smoluchowski kinetic equations that describe the evolution of systems influenced by stochastic forces distributed with stable probability laws are derived. These equations generalize the known kinetic equations of the Brownian motion theory and involve symmetric fractional derivatives with respect to velocity and space variables. With the help of these equations, the linear relaxation processes in the force-free case and for the linear oscillator is analytically studied. For a weakly damped oscillator, a kinetic equation for the distribution in slow variables is obtained. Linear relaxation processes are also studied numerically by solving the corresponding Langevin equations with the source given by a discrete-time approximation to white Levy noise. Numerical and analytical results agree quantitatively.     

The driven-oscillator evolution in the tomographic-probability representation

We consider the problem of the driven harmonic oscillator in the probability representation of quantum mechanics, where the oscillator states are described by fair nonnegative probability distributions of position measured in rotated and squeezed reference frames in the system??s phase space. For some specific oscillator states like coherent states and nth excited states, the tomographic-probability distributions (called the state tomograms) are found in an explicit form. The evolution equation for the tomograms is discussed for the classical and quantum driven oscillators, and the tomographic propagator for this equation is studied.     

Stationary perturbation theory in the probability representation of quantum mechanics

In the probability representation of quantum mechanics, the eigenvalue problems in Hilbert space appear as *-genvalue equations. We show the possibility of employing the nondegenerate stationary perturbation method in the probability representation of quantum mechanics. The perturbed eigentomograms and the eigenvalues of energy are shown to be computed ab initio in terms of tomographic symbols of the operators involved.