Linear quantum enskog equation II. Inhomogeneous quantum fluids

This is the second part of a work concerned with the quantum-statistical generalization of classical Enskog theory, whereby the first part is extended to spatially inhomogeneous fluids. In particular, working with Liouville operators and using cluster expansions and projection operators, we derive the inhomogeneous linear quantum Enskog equation and express the dynamic structure factor and the nonlocal mobility tensor in terms of the corresponding quantum Enskog collision operator. Thereby static correlations due to excluded volume effects and quantum-statistical correlations due to the fermionic (bosonic) character of the pairwise strongly interacting particles are treated exactly. When static correlations are neglected, this Enskog equation reduces to the inhomogeneous linear quantum Boltzmann equation (containing an exchange-modifiedt-matrix). In the classical limit, the well-known linear revised Enskog theory is recovered for hard spheres.     

Quantum-statistical approach to gross properties of peripheral collisions between heavy nuclei

Non-equilibrium quantum-statistical mechanics is applied to peripheral collisions between heavy nuclei (A?40) where a large number of degrees of freedom are involved during the process. By eliminating the relative motion from the explicit consideration, the transitions between different channels are determined by a Liouville equation with timedependent coupling matrix elements. The introduction of subsets of channels (coarse graining) leads to the definition of macroscopic variables which correspond to observable quantities. The equation of motion for the macroscopic variables become irreversible by assuming the values of the coupling matrix elements to be randomly distributed. The validity and possible applications of the resulting master equations are discussed.     

Complete sets of solutions of linear Lorentz covariant field equations with an infinite number of field components

We study field equations of the Gelfand-Yaglom type where transforms as a unitary representation of the inhomogeneous Lorentz group. We construct a complete set of solutions of this equation. This set includes solutions with spacelike momentum. Our method makes use of the decomposition of unitary representations of the homogeneous Lorentz group into unitary representations of the little groupsS U (2) andS U (1, 1). The covariant operators ^µ are written as differential operators on homogeneous spaces. For some classes of equations we calculate the mass spectrum explicitly.     

Dynamical Differential Equations Compatible with Rational <Emphasis Type="Italic">qKZ</Emphasis> Equations

For the Lie algebra _N we introduce a system of differential operators called the dynamical operators. We prove that the dynamical differential operators commute with the _N rational quantized Knizhnik–Zamolodchikov difference operators. We describe the transformations of the dynamical operators under the natural action of the _N Weyl group.Mathematics Subject Classifications (2000). 17B37, 17B80, 81R10.     

Regularized and renormalized Bethe-Salpeter equations: Some aspects of irreducibility and asymptotic completeness in renormalizable theories

Results on the links between 2-particle irreducibility and asymptotic completeness are presented in the framework of a renormalized Bethe-Salpeter formalism, introduced recently by J. Bros from an axiomatic viewpoint, for the most simple class of renormalizable theories. These results, which involve therenormalized 2-particle irreducible kernelG (i.e. from the perturbative viewpoint the sum of renormalized Feynman amplitudes of 2-particle irreducible graphs in the channel considered), complement the general quasi-equivalence previously established by Bros forregularized (non-renormalized) Bethe-Salpeter kernels. On the one hand, a formal derivation of (2-particle) asymptotic completeness from the irreducibility ofG is given. On the other hand, the links between regularized and renormalized kernels are investigated. This analysis provides in particular a converse derivation (up to some assumptions) of the 2-particle irreducibility ofG from asymptotic completeness. As a byproduct, it also provides a more explicit justification of previous heuristic derivations by K. Symanzik of integral equations betweenF and various differences of values ofG, and a simple alternative derivation of the recently proposed renormalized Bethe-Salpeter equation.     

Recursion operators and bi-Hamiltonian structures in multidimensions. II

We analyze further the algebraic properties of bi-Hamiltonian systems in two spatial and one temporal dimensions. By utilizing the Lie algebra of certain basic (starting) symmetry operators we show that these equations possess infinitely many time dependent symmetries and constants of motion. The master symmetries for these equations are simply derived within our formalism. Furthermore, certain new functionsT ₁₂ are introduced, which algorithmically imply recursion operators ₁₂. Finally the theory presented here and in a previous paper is both motivated and verified by regarding multidimensional equations as certain singular limits of equations in one spatial dimension.     

On the Generalized Phase Space Approach to Duffin-Kemmer-Petiau Particles

We present a general derivation of the Duffin-Kemmer-Petiau (D.K.P) equation on the relativistic phase space proposed by Bohm and Hiley. We consider geometric algebras and the idea of algebraic spinors due to Riesz and Cartan. The generators ^(p) of the D.K.P algebras are constructed in the standard fashion used to construct Clifford algebras out of bilinear forms. Free D.K.P particles and D.K.P particles in a prescribed external electromagnetic field are analized and general Liouville type equations for these cases are obtained. Choosing particular values for the label p we classify the different types of the D.K.P Liouville operators.     

Flow equations for the spin-boson problem

Using continuous unitary transformations recently introduced by Wegner [1], we obtain flow equations for the parameters of the spin-boson Hamiltonian. Interactions not contained in the original Hamiltonian are generated by this unitary transformation. Within an approximation that neglects additional interactions quadratic in the bath operators, we can close the flow equations. Applying this formalism to the case of Ohmic dissipation at zero temperature, we calculate the renormalized tunneling frequency. We find a transition from an untrapped to trapped state at the critical coupling constant α _c =1. We also obtain the static susceptibility via the equilibrium spin correlation function. Our results are both consistent with results known from the Kondo problem and those obtained from mode-coupling theories. Using this formalism at finite temperature, we find a transition from coherent to incoherent tunneling atT ₂ ^* ≈2T ₁ ^* , whereT ₁ ^* is the crossover temperature of the dynamics known from the NIBA.     

Transport theory of dissipative heavy-ion collisions

The non-perturbative quantum-statistical theory of dissipative heavy-ion collisions introduced earlier, is generalized by including explicitly the relative motion of the colliding nuclei. We start from the Liouville equation in the Wigner representation which allows for useful and illustrative interpretations of the resulting quantities and equations. Using the randomness of the coupling matrix elements and the semi-classical approximation for the relative motion we derive a general time-dependent transport equation for the macroscopic Wigner functions (phase-space distribution functions). The limits of weak and strong coupling are discussed.     

Computing multi-valued physical observables for the high frequency limit of symmetric hyperbolic systems

We develop a level set method for the computation of multi-valued physical observables (density, velocity, energy, etc.) for the high frequency limit of symmetric hyperbolic systems in any number of space dimensions. We take two approaches to derive the method.The first one starts with a weakly coupled system of an eikonal equation for phase S and a transport equation for density ρ:

The main idea is to evolve the density near the n-dimensional bi-characteristic manifold of the eikonal (Hamiltonian–Jacobi) equation, which is identified as the common zeros of n level set functions in phase space . These level set functions are generated from solving the Liouville equation with initial data chosen to embed the phase gradient. Simultaneously, we track a new quantity f = ρ(t,x,k)|det(_k)| by solving again the Liouville equation near the obtained zero level set = 0 but with initial density as initial data. The multi-valued density and higher moments are thus resolved by integrating f along the bi-characteristic manifold in the phase directions.The second one uses the high frequency limit of symmetric hyperbolic systems derived by the Wigner transform. This gives rise to Liouville equations in the phase space with measure-valued solution in its initial data. Due to the linearity of the Liouville equation we can decompose the density distribution into products of function, each of which solves the Liouville equation with L^∞ initial data on any bounded domain. It yields higher order moments such as energy and energy flux.The main advantages of these new approaches, in contrast to the standard kinetic equation approach using the Liouville equation with a Dirac measure initial data, include: (1) the Liouville equations are solved with L^∞ initial data, and a singular integral involving the Dirac-δ function is evaluated only in the post-processing step, thus avoiding oscillations and excessive numerical smearing; (2) a local level set method can be utilized to significantly reduce the computation in the phase space. These methods can be used to compute all physical observables for multi-dimensional problems.Our method applies to the wave fields corresponding to simple eigenvalues of the dispersion matrix. One such example is the wave equation, which will be studied numerically in this paper.     

High-temperature exchange third virial coefficient for hard spheres via an asymptotic method for path integrals

The exchange part of the third cluster integral can be divided into two parts:b ₃(exch-1), which arises from the exchange of two particles, andb ₃(exch-2), which arises from the cyclic exchange of all three particles. The first few terms ofb ₃(exch-1) are calculated by arguing thatb ₃(exch-1) =-[9a³/(4³)]b₂(exch)[1 + O(/a)], whereb ₂(exch) is the exchange second cluster integral, is the thermal de Broglie wavelength, anda is the hardsphere diameter. The first three terms ofb ₃(exch-2) are calculated by writing it in path integral form and expanding about the shortest path.     

Influence of inter cell resonant tunneling on the out-of-plane electronic transport behavior in layered high T<Subscript>c</Subscript> cuprates

The influence of inter unit cell resonant tunneling between the copper-oxygen planes on the c-axis electronic conductivity (σ_c) in normal state of optimal doped bilayer high T_c cuprates like Bi₂Sr₂CaCu₂O_8+x is investigated using extended Hubbard Hamiltonian including resonant tunneling term (T₁₂) between the planes in two adjoining cells. The expression for the out-of-plane (c-axis) conductivity is calculated within Kubo formalism and single particle Green's function by employing Green's function equations of motion technique within meanfield approximation. On the basis of numerical computation, it is pointed out that the renormalized c-axis conductivity increases exponentially with the increment in inter cell resonant tunneling. The effect of T₁₂ on renormalized c-axis conductivity is found to be prominent at low temperatures as compared to temperatures above room temperature (~300 °K). The Coulomb correlation suppresses the variation of renormalized c-axis conductivity with temperature, while renormalized c-axis conductivity increases on increasing carrier concentration. These theoretical results are viewed in terms of existing c-axis transport measurements.     

Practical Kedem-Katchalsky equations and their modification

The research problem presented in this work concerns modification of the Kedem-Katchalsky (K-K) equation for volume flow (J _v ) through system (h|M|l), consisting of a membrane M and boundary layers h and l. Such boundary layers appear in the vicinity of the membrane on both sides due to the lack of mixing of solutions. This paper also includes the derivation of the equation for volume flow (J _vr ) dissipated on concentration boundary layers h and l. The derivation of these equations concerns the case in which the substance transport through the membrane is generated by the osmotic pressure gradient . On the basis of the equations for the volume flows (J _v ) and (J _vr ), some calculations for a nephrophane membrane, used in medicine, and for aqueous glucose solutions have been carried out. In order to test the equations for (J _v ) and (J _vr ), we have also carried out calculations for the volume flow (J′ _v ) that is transferred through the membrane in the case of mixed solutions on both sides of the membrane. This volume flux has been calculated on the basis of the original (K-K) equation. The results are presented in Fig. 2.      

Quantum Kinematic Theory of the Poincare Group in Two-Dimensional Spacetime

Non-Abelian quantum kinematics is applied to thePoincare group P ₊ (1, 1),as an example of the quantization-through-the-symmetryapproach to quantum mechanics. Upon quantizing thegroup, generalized Heisenberg commutation relations are obtained, and aclosed Heisenberg–Weyl algebra follows. Then,according to the general theory, the three basicquantum-kinematic invariant operators are calculated;these afford the superselection rules for diagonalizing theincoherent rigged Hilbert space H(P ₊ ) of the regularrepresentation. This paper examines only one of thesediagonalization schemes, while introducing a irreducible spacetime representation carried by isotopicplane-wave eigenvectors of two compatible superselectionoperators (which define a Poincare-invariant linear2-momentum). Thereafter, the principle of microcausality produces massive 2-spinor isotopic states in 1+ 1 Minkowski space. The Dirac equation is thus deducedwithin the quantum kinematic formalism, and the familiarJordan–Pauli propagation kernel in 2-dimensional spacetime is also obtained as a Hurwitzinvariant integral over the group manifold. The maininterest of this approach lies in the adoptedgroup-quantization technique, which is a strictlydeductive method and uses exclusively the assumed Poincaresymmetry.     

A Markov Process Associated with a Boltzmann Equation Without Cutoff and for Non-Maxwell Molecules

Tanaka,⁽¹⁸⁾ showed a way to relate the measure solution {P _t } _t of a spatially homogeneous Boltzmann equation of Maxwellian molecules without angular cutoff to a Poisson-driven stochastic differential equation: {P _t } is the flow of time marginals of the solution of this stochastic equation. In the present paper, we extend this probabilistic interpretation to much more general spatially homogeneous Boltzmann equations. Then we derive from this interpretation a numerical method for the concerned Boltzmann equations, by using easily simulable interacting particle systems.     

The solution space of the unitary matrix model string equation and the Sato Grassmannian

The space of all solutions to the string equation of the symmetric unitary one-matrix model is determined. It is shown that the string equation is equivalent to simple conditions on pointsV ₁ andV ₂ in the big cell Gr⁽⁰⁾ of the Sato Grassmannian Gr. This is a consequence of a well-defined continuum limit in which the string equation has the simple form matrices of differential operators. These conditions onV ₁ andV ₂ yield a simple system of first order differential equations whose analysis determines the space of all solutions to the string equation. This geometric formulation leads directly to the Virasoro constraintsL _n (n0), whereL _n annihilate the two modified-KdV -functions whose product gives the partition function of the Unitary Matrix Model.