Solvable RSOS models based on the dilute BWM algebra

In this paper we present representations of the recently introduced dilute Birman-Wenzl-Murakami algebra. These representations, labelled by the level-l B_n⁽¹⁾, C_n⁽¹⁾ and D_n⁽¹⁾ affine Lie algebras, are baxterized to yield solutions to the Yang-Baxter equation. The thus obtained critical solvable models are RSOS counterparts of the, respectively, D_n+1⁽²⁾, A_2n⁽²⁾ and B_n⁽¹⁾R-matrices of Bazhanov and Jimbo. For the D_n+1⁽²⁾ and B_n⁽¹⁾ algebras the RSOS models are new. An elliptic extension which solves the Yang-Baxter equation is given for all three series of dilute RSOS models.     

Einfaches Modell für monoatomare klassische Flüssigkeiten

A simple approximation for the conditional time-dependent pair distribution functionP(r′t ¦ rt) is proposed. With this approximation the van Hove-correlation-functionG _d (rt) is computed and compared with Ludwig's¹ computer results for liquid A1. Then we determine the velocity autocorrelationfunction ψ(t) under special assumptions for the particle motion. That function is compared with Rahman's² computer results for liquid Ar.     

Radiative transfer in the lyman α spectral line with self-consistent electron velocity distribution

Radiative transfer in the Ly α spectral line in a stationary, plane-parallel plasma of constant temperature and electron density is studied using model H-atoms with only two bound levels and a continuum. For this purpose, the equation of radiative transfer is solved simultaneously with the steady-state equations of the atomic levels and the kinetic equation of the electrons. The numerical results indicate that, in hydrogen plasmas with temperatures T ? 12,000°K and electron densities n_e ? 10¹⁶cm^?3, the high-energy tail of the electron velocity distribution deviates from a Maxwell distribution, even in cases of rather large optical thicknesses and that therefore the deviations from local thermodynamic equilibrium are increased compared with estimates based on the assumption of a Maxwellian electron velocity distribution. This qualitative conclusion should hold in spite of some deficiencies of the model which are discussed.     

Algebraic construction of higher-rank dilute A models

Starting from representations of the Birman-Wenzl-Murakami algebra underlying the critical B_n⁽¹⁾, C_n⁽¹⁾ and D_n⁽¹⁾ RSOS models of Jimbo, Miwa and Okado, we derive four series of solvable, critical RSOS models associated with the twisted affine Lie algebra A_n⁽²⁾. Two of these are the critical limit of the A_2n−1⁽²⁾ and A_2n⁽²⁾ models obtained previously by Kuniba. The other two series, again one of the A_2n−1⁽²⁾ and one of the A_2n⁽²⁾ type, are new, and the latter generalizes the dilute A models to arbitrary rank n. For the two new series we present an elliptic extension which satisfies the Yang-Baxter equation, and show that for certain values of the parameters the higher-rank dilute A models break the ℤ₂ symmetry of the underlying adjacency graphG, whereGis the level-l C_n⁽¹⁾ weight lattice.     

The ground state of two-dimensional electrons in a nonuniform magnetic field

An exact solution to the Schrödinger equation for the ground state of two-dimensional Pauli electrons in a nonuniform transverse magnetic field H is presented for two cases. In the first case, the field H depends on a single variable, H = H(y), while in the second case, the field is axially symmetric, H = H(ρ), ρ²=x ²+y ². The electron density distributions n = n(y) and n = n(ρ) that correspond to a completely filled lower level are found. For quasiuniform fields of fixed sign, the functions n(y) and n(ρ) are locally related to the magnetic field: n(y) = H(y)/?₀ and n(ρ) = H(ρ)/?₀, where ?₀ = hc/|e| is a magnetic flux quantum. Magnetic fields are considered that are periodic, singular, and bounded in the plane xy. Finite electron objects in a nonuniform magnetic field are analyzed.     

The exact solution of an elimination problem in kinetic theory: I. The one-dimensional hard sphere gas

A class of initial value problems for a one-dimensional hard sphere gas is considered where a specified particle has a given distribution f⁽¹⁾(z₁; 0) and the rest are in equilibrium at t=0. An exact expansion is obtained for a certain n-particle reduced distribution function f⁽ⁿ⁾(z₁;…;z_n; t) in terms of the 1-particle reduced distribution function f⁽¹⁾(z₁; t) for the specified particle by starting with separate expressions for these functions in terms of f⁽¹⁾(z₁; 0). Expansions for the corresponding cluster functions are first obtained and then graph theoretic methods applied to obtain a solution.     

Isometric immersions of pseudo-Riemannian space forms

In this paper we study local isometric immersions f:M_sⁿ(K)→N_s+q²ⁿ⁻¹(c) of a time-like n-submanifold M_sⁿ(K) with constant sectional curvature K and index s into a pseudo-Riemannian space form N_s+q²ⁿ⁻¹(c) with constant sectional curvature c and index s+q, where q≥0, 1≤s≤n−1 and K≠c. We first prove the existence of Chebyshev coordinates of a time-like submanifold M_sⁿ(K) in certain conditions. Afterwards, we generalize the classical Bäcklund theorem for space-like (or time-like) submanifolds in N_n−1²ⁿ⁻¹(c) and N₁²ⁿ⁻¹(c). Finally as an application, in the Chebyshev coordinates, we use the Bäcklund theorem to give a Bäcklund transformation and a permutability formula between the generalized sine-Laplace equation and the generalized sinh-Laplace equation.     

Effects of the velocity dependence of the collision frequency on the Dicke line narrowing

The effect of the velocity (v) dependence of the transport collision frequency ν_trv on the Dicke line narrowing is analyzed in terms of the strong-collision model generalized to velocity-dependent collision frequencies (the so-called kangaroo model). This effect has been found to depend on the mass ratio of the resonance (M) and buffer (M _b) particles, β = M _b/M: it is at a minimum for β ? 1 and reaches a maximum for β ? 3. A power-law particle interaction potential, U(r) ∝ r ^?n , is used as an example to show that, compared to ν_trv (v) = const (n = 4), the line narrows if ν_trv (v) decreases with increasing v (n < 4) and broadens if ν _trv (v) increases with v(n > 4). At β ? 3, the line width can increase [compared to ν_trv (v) = const] by 5 and 12% for the potentials with n = 6 and n ? 10, respectively; for the potentials with n = 1 (Coulomb potential) and n = 3, it can decrease by more than half and 6%, respectively. The line profile I(Ω) has been found to be weakly sensitive to ν_trv (v) at some detuning Ω_c of the radiation frequency Ω. Dicke line narrowing is used as an example to analyze the collisional transport of nonequilibrium in the resonance-particle velocity distribution in a laser field. The transport effect is numerically shown to be weak. This allows simpler approximate one-dimensional quantum kinetic equations to be used instead of the three-dimensional ones to solve spectroscopic problems in which it is important to take into account the velocity dependence of the collision frequency when the phase memory is preserved during collisions.     

The one-site distribution of Gibbs states on Bethe lattice are probability vectors of period⩽2 for a nonlinear transformation

We prove that the one-site distribution of Gibbs states (for any finite spin setS) on the Bethe lattice is given by the points satisfying the equation π=T ²π, whereT=h·A·?, with?(x)=x ^(q?1/q,h(x)=(x∥x∥ _q ) ^q ,A=(a(r, s)∶r, s∈S), and $$a(r,s) = \exp (K[r,s] + (1/q)[N,r + s])$$ We also show that forA a symmetric, irreducible operator the nonlinear evolution on probability vectorsx(n+1)=Ax(n) ^p ∥Ax(n) ^p ∥₁ withp>0 has limit pointsξ of period?2. We show thatA positive definite implies limit points are fixed points that satisfy the equationAξ ^p=λξ. The main tool is the construction of a Liapunov functional by means of convex analysis techniques.     

The Inviscid Burgers Equation with Initial Value of Poissonian Type

We study the discontinuities (shocks) of the solution to the Burgers equation in the limit of vanishing viscosity (the inviscid limit) when the initial value is the opposite of the standard Poisson process p. We show that this solution is only defined for t ε (0, 1). Let T ₀ = 0 and T _n , n≧1, be the successive jumps of p. We prove that for all M > 0 the inviscid limit is characterized on the region x ε (-∞, M], t ε (0, 1) by the increasing process $N(t) = \sup \{ n \in \mathbb{N} {\text{| }}M + nt > T_n \} $ and the random set I(x) = {n ε {0,..., N(t)}‖T _n -nt≦x<T _n+1 - nt}. The positions of shocks are given in a precise manner. We give the distribution of N(t) and also the distribution of its first jump. We also prove similar results when the initial value is u _μ(y, 0) = -μp(y/μ²) + μ^-1 max(y, 0), μ ε (0, 1).     

Flicker noise of hot electrons in silicon at T = 78 K

From flicker-noise and current-voltage measurements performed on an n⁺nn⁺ silicon planar device at T = 78 K we calculated Hooge's parameter α as a function of the electric field strength, E₀. We found that α(E₀) = α(0)/[1 + (E₀/E_c)²]. E_c is a critical field where the drift velocity equals the sound velocity, indicating the connection of the observed effect with acoustical phonon scattering.     

On estimating nanoparticle size with the help of the Mössbauer effect

A method for estimating nanoparticle size by determining the distribution function p(H _n ) of hyperfine magnetic fields from data on the Mössbauer spectrum of ⁵⁷Fe nuclei is described. The idea of the method stems from the fact that, owing to the breaking of exchange bonds for surface atoms, their contribution to the total area bounded by the p(H _n ) curve can be singled out. The potentialities of the method are illustrated using the data obtained in experiments with nanoparticles of magnetite.     

No Existence and Smoothness of Solution of the Navier-Stokes Equation

The Navier-Stokes equation can be written in a form of Poisson equation. For laminar flow in a channel (plane Poiseuille flow), the Navier-Stokes equation has a non-zero source term (∇²u(x, y, z) = F_x (x, y, z, t) and a non-zero solution within the domain. For transitional flow, the velocity profile is distorted, and an inflection point or kink appears on the velocity profile, at a sufficiently high Reynolds number and large disturbance. In the vicinity of the inflection point or kink on the distorted velocity profile, we can always find a point where ∇²u(x, y, z) = 0. At this point, the Poisson equation is singular, due to the zero source term, and has no solution at this point due to singularity. It is concluded that there exists no smooth orphysically reasonable solutions of the Navier-Stokes equation for transitional flow and turbulence in the global domain due to singularity.     

Existence of a phase-transition in a one-dimensional Ising ferromagnet

Existence of a phase-transition is proved for an infinite linear chain of spins μ _j =±1, with an interaction energy $$H = - \sum J(i - j)\mu _i \mu _j ,$$ whereJ(n) is positive and monotone decreasing, and the sums ΣJ(n) and Σ (log logn) [n ³ J(n)]^?1 both converge. In particular, as conjectured byKac andThompson, a transition exists forJ(n)=n ^?α when 1 < α < 2. A possible extension of these results to Heisenberg ferromagnets is discussed.     

Role of metastable atoms and molecules of helium in excitation transfer to hydrogen atoms

The afterglow of a discharge in helium with a small admixture of hydrogen is studied spectroscopically (p=40 Torr, [e]≤10¹¹ cm^?3). The time-resolved measurements of intensities of the first four lines of the Balmer series are performed. The concentrations of metastable helium atoms and molecules are evaluated from the relative intensity of the absorption lines. The ratios of excitation transfer rates from atoms He(2 ³ S ₁) k ₁(n) and molecules of helium He₂(a 2sσ ³Σ _u ⁺ ) k ₂(n) to atomic hydrogen H*(n) are measured to be k ₁(n=3)/k ₂(n=3)=0.04±0.02 and k ₁(n=4)/k ₂(n=4)=0.01±0.02. The ratios of excitation rate constants k ₂(n) corresponding to different states H(n) are measured to be k ₁(n=4)/k ₂(n=3)=0.023±0.01; k ₁(n=5)/k ₂(n=3)≤0.013; and k ₁(n=6)/k ₂(n=3)≤0.007.     

Bloch electron dynamics

New equations of motion for a Bloch electron [momentum p=h k,energy ε _n(p),zone number n, charge -e]: $$m_j \frac{{dv_j }}{{dt}} = - e(E + v \times B)_j $$ are proposed, where v≡?ε_n(p)/?p is the velocity, and {m_j}are the principal masses m _j ^? 1=?²ε_n/?p _j ² along the normal and the two principal axes of curvatures at each point of the constant-energy surface represented by ε=ε_n(p).Their advantages over the prevalent equations of motion where the left-hand-side is replaced by hk _j are demonstrated by examining de Haas-van Alphen oscillations and orientation-dependent cyclotron resonance peaks.     

The velocity distribution and the density near a particle of an equilibrium two component plasma

The electronic and ionic velocity distributions and densities of an equilibrium two component plasma at distances from a test particle (electron or ion) smaller than the interparticle distance, are derived using a technique developed recently by the author. In the case of particles repelled by the test particle the results agree with the results obtained by integrating the Gibbs distribution. In the case of particles attracted by the test particle however, the velocity distribution has the form of a translated maxwellian with an empty sphere in the middle and the asymptotic form of the density as r → 0 is given by n_a(r) ～r^{$\text{-1}\text{2}$}. By the latter formula the classical problem of stability of matter is locally resolved without using short distance repulsive potentials.     

Two remarks on the painlevé test

Using the example of two previously discussed autonomous differential equations it is pointed out that the Painlevé test with an ansatz w = Σ_n=p^∞w_n(z − z₀)ⁿ, p < 0, integer, will not reveal movable branch points if all leading order contribution arise from terms with equal degree of nonlinearity. Just contrary to this case, for the reduced damped and driven φ⁴ equation with p = −1 two resonance conditions arise at order n = 3 which cannot be satisfied simultaneously.