The theory of the magnetoresistance of the InSb quantum wire is presented.The quantum wire is cylindrical in shape with a symmetric pair of delta-barriers inside.A tunable magnetic field parallel with the axis of the quantum wire is considered.The dependence of the Fermi energy on the magnetic field is calculated.The Landauer formula is used in the calculation of the resistance of a quantum wire.When parameters of the quantum wire (its radius, amplitude of delta-barriers, and distance between them, donor concentration) are appropriately chosen, the theory predicts the dependence of resistance on the magnetic field manifesting well-defined minima.The minima are attributed to the resonant tunnelling of the conduction electrons through the double delta-barrier.

The stationary one-speed transport equation for neutral particles in the slab geometry is considered. The medium between two planes, z = 0 and z = L, is taken as absorbing and isotropically scattering. The extinction function sigma(r) is defined as a Gaussian random function with a constant mean value sigma = , a constant variance eta(2)(sigma) = <[sigma(r) - sigma](2)>, and a given autocorrelation function W-sigma (r(2) - r(1))=([sigma(r(2)) - sigma] [sigma(r(1)) - sigma]). The albedo omega (0 < omega < 1) is taken as a constant. Considering a perpendicular influx of particles from the left and no influx from the right, we focus attention on the solution I(zeta, mu) of the transport equation obtained within the framework of the Pomraning-Eddington approximation. Our boundary conditions read I(0, 1) = I-L and I(Z, - 1) = 0. (zeta = zeta(z) is the length of the projection of the optical path on the z-axis, and theta is the angle between the general flight direction and the z-axis, mu = cos theta.) Since the randomness of sigma(r) is Gaussian, the optical thickness Z = zeta(L) is also Gaussian. For finite values of L, we show that the transmission and reflection coefficients are con-elated random quantities. We calculate their first-order and second-order statistical moments.

The Waxman-Peck theory of population genetics is discussed in regard of soil bacteria. Each bacterium is understood as a carrier of a phenotypic parameter p. The central objective is the calculation of the probability density with respect to p, Phi(p,t;p(0)), of the carriers living at time t>0, provided that initially at t(0)=0, all bacteria carried the phenotypic parameter p(0)=0. The theory involves two small parameters: the mutation probability mu and a parameter gamma involved in a function w(p) defining the fitness of the bacteria to survive the generation time tau and give birth to an offspring. The mutation from a state p to a state q is defined by a Gaussian with a dispersion sigma(2)(m). The author focuses our attention on a function phi(p,t) which determines uniquely the function Phi(p,t;p(0)) and satisfies a linear equation (Waxman's equation). The Green function of this equation is mathematically identical with the one-particle Bloch density matrix, where mu characterizes the order of magnitude of the potential energy. (In the x representation, the potential energy is proportional to the inverted Gaussian with the dispersion sigma(2)(m)). The author solves Waxman's equation in the standard style of a perturbation theory and discusses how the solution depends on the choice of the fitness function w(p). In a sense, the function c(p)=1-w(p)/w(0) is analogous to the dispersion function E(p) of fictitious quasiparticles. In contrast to Waxman's approximation, where c(p) was taken as a quadratic function, c(p) approximately gammap(2), the author exemplifies the problem with another function, c(p)=gamma[1-exp(-ap(2))], where gamma is small but a may be large. The author shows that the use of this function in the theory of the population genetics is the same as the use of a nonparabolic dispersion law E=E(p) in the density-matrix theory. With a general function c(p), the distribution function Phi(p,t;0) is composed of a delta-function component, N(t)delta(p), and a blurred component. When discussing the limiting transition for t--> infinity, the author shows that his function c(p) implies that N(t)-->N( infinity ) not equal 0 in contrast with the asymptotics N(t)-->0 resulting from the use of Waxman's function c(p) approximately p(2).

A generalization of the Kronig-Penney problem is put forward with the potential energy V(x) = gamma Sigma(j) delta(x - ja), gamma > 0. A periodic multi-layer ... ABABABA ... is considered: layers A of thickness a are intercalated between layers B of much smaller thickness. In this superlattice, A and B symbolize, respectively, narrow-gap semiconductor layers and barrier layers. The conduction band of the semiconductor A is defined by the dispersion function E(k) which was derived in the Kane two-band theory. Owing to the non-zero value of the parameter gamma, the electron energies inside the interval corresponding to the conduction band of the semiconductor A are organized in minibands separated by forbidden gaps. With E(k) taken in the Kane form, the dispersion law epsilon = E(k) is non-parabolic if E-g (the width of the forbidden gap of the semiconductor A) is finite. This non-parabolicity affects the positions and widths of the minibands. If E-g tends to infinity, the original Kronig-Penney problem is recovered. If E-g decreases, the density of the minibands increases.

The one-dimensional version of the radiative transfer problem (i.e. the so-called rod model) is analysed with a Gaussian random extinction function σ(x). Then the optical length X = ∫0Ldxσ(x) is a Gaussian random variable. The transmission and reflection coefficients, T(X) and R(X), are taken as infinite series. When these series (and also when the series representing T2(X), T2(X), R(X)T(X), etc.) are averaged, term by term, according to the Gaussian statistics, the series become divergent after averaging. As it was shown in a former paper by the authors (in Acta Physica Slovaca (2003)), a rectification can be managed when a `modified' Gaussian probability density function is used, equal to zero for X > 0 and proportional to the standard Gaussian probability density for X > 0. In the present paper, the authors put forward an alternative, showing that if the m.s.r. of X is sufficiently small in comparison with \(\bar X\), the standard Gaussian averaging is well functional provided that the summation in the series representing the variable Tm-j(X)R j (X) (m = 1,2,..., j = 1,...,m) is truncated at a well-chosen finite term. The authors exemplify their analysis by some numerical calculations.

The 1D Kane oscillator is analysed. It is defined by the Schrodinger-Wannier equation in which the potential-energy term represents the zero-centred quadratic well and in which the 'kinetic-energy' term is chosen as an operator corresponding to the conduction-electron dispersion function of the Kane two-band theory. The author considers the approximate form of the Kane function for the conduction band well-known in the theory of narrow-gap semiconductors. Employing the momentum representation, he calculates the eigen-energies of the electrons in frame of the WKB approach. The eigen-energies are roots of a transcendental equation involving the complete elliptic integrals of the first and second kind. The author presents a detailed discussion of the dependence of the eigen-energies of the electrons on the Kane nonparabolicity of the conduction band.

The paper concerns the stationary solution of the one-speed transport equations of the rod model known in the theory of the neutron transport. The solution is presented in view of an arbitrary absorption cross section sigma(x) and of the scattering cross section equal to csigma(x) where c is taken as a constant. The author treats the 'albedo problem' with a given value of the influx of particles upon the rod from the left and with no influx from the right. He recalls, at first, the exact 'albedo solution', accentuating one important statement: the transmitted fraction T and the reflected fraction R of the fluxes of the particles are functions of one single variable, namely of the 'optical length' Lambda=integral(0)(L) dx sigma(x) of the rod. (L is the geometrical length of the rod.) If sigma(x) is defined as a random function, then Lambda and, consequently, T, R become random variables. The statistical moments <T-n> and <R-n> are calculated for n=1, 2. As T and R are nonlinear functionals of sigma(x), the random variables T and R need not be Gaussian even if sigma(x) is defined as a Gaussian random function. Conditions are discussed under which T and R may be approximated as two correlated Gaussian random variables.

Supercurrent in the weakly conducting c-axis direction of layered superconductors is studied. It is shown that in the presence of screw dislocations with Burgers vectors parallel to the c-axis, the critical current of small samples is enhanced with respect to the perfect crystal, each dislocation being able to carry a supercurrent I0.

A normalization problem is discussed, regarding the use of the Green function G(r,v,L/r(0),v(0)) in the statistical theory of configurations of a 'partially flexible polymer' of length L thermalized with its environment. The polymer is defined with a (constant) stiffness parameter eta > 0. The Green function G(r, v, L/r(0), v(0)) is expressed as a path-integral with paths r(s), v(s) in a 6-dimensional space {r, v}, v(s) = dr(s)/ds. The basic question which the author analyzes in detail is the accommodation of the Green function G(r,v, L\r(0), v(0)) with the concept of the end-to-end probability density P-c(r, <(Omega)over right arrow>, L/r(0), <(Omega)over right arrow>(0)) defined under the constraints \v\ = \v(0)\ = 1. The vectors r, r(0) and the unit vectors v = <(Omega)over right arrow>, v(0) = <(Omega)over right arrow>(0) define, respectively, the positions and the tangential directions of the polymer ends. (r(0) = r(0), r(L) = r, v(0) = <(Omega)over right arrow>(0), v(L) = <(Omega)over right arrow>.) An explicit expression for P-c(r, <(Omega)over right arrow>, L/r(0), <(Omega)over right arrow>(0)) is derived as a function of the stiffness parameter eta and the temperature T = 1/(k(B)beta). An exact formula is derived for the mean-square end-to-end distance (R-2)(L) as a function of eta and beta. A thorough description is presented concerning the statistics of the tangential direction <(Omega)over right arrow> of the polymer end with s = L provided that the tangential direction <(Omega)over right arrow>(0) is given. When elucidating this statistics, a special attention is devoted to very stiff polymers for which the author defines the 'paraxial approximation'.

This paper, following directly from Part I, continues to demonstrate the use of the paraxial approximation in solving the time-dependent wave equation for the propagation of signals in slightly dispersive optical materials with some Gaussian randomness of the permittivity. With the same emphasis as in Part I, a path-integral formulation is employed. Attention is focused on a parabolic graded-index fibre with (1/n(k0)(r)) = (1 + alpha r(perpendicular to)(2)/2) n(k0), alpha > 0. Arising from the possibility of replacing the autocorrelation function of the permittivity by a quadratic function, some final concise expressions are derived. It is shown that if a signal begins propagating with a Gaussian power density (\u(r, 0)\(2)), the Gaussianity of the function (\u(r, t)\(2)) is maintained at all later times t. The longitudinal shape of the function (\u(r, t)\(2)) exhibits attenuation and broadening dependent on the dispersiveness of the optical medium under consideration. The perpendicular shape of the function (\u(r, t)\(2)) oscillates with the angular frequency Omega perpendicular to = [cv(x)(k(v))alpha n(k0)](1/2) where c and v(x)(k(0)) are, respectively, the velocity of light in vacuum and the axial group velocity of light corresponding to k(x) = k(0). Special attention is devoted to the limiting case when alpha --> 0, i.e. to propagation in a bulk dispersive optical medium.

Diffusion of dopants in crystalline structures with arrays of delta-doped layers is studied theoretically. Two types of the structures are treated. The first type is deterministic,defined by a periodic location of the delta-layers. the second type is stochastic, with a Poissonian location of the delta-layers. In the deterministic case, the r.m.s. deviation eta(t) of the concentration of the diffusants exhibits an exponential long-time behaviour: eta(det) similar to exp(-4 pi(2) dT/a(2)) where D > 0 is the diffusion coefficient and a > 0 is the spacing between the delta-layers. In the stochastic case. eta(t) decreases with a much slower rate: eta(stoch)(t) similar to t(-1/1). A general qualitative (semi-quantitative) discussion is given laying emphasis on stochasticity as a prerequisite of any realistic theory of the diffusional homogenization of sintered materials.

A simple kinetic theory of the spontaneous flattening of randomly corrugated solid surfaces is presented, based on Mullins' equation partial derivative zeta/partial derivative t = A del(2)zeta - B(del(2))(2)zeta for idealized surfaces given by the equation z = zeta(z,y,t), t > 0. We define zeta(a,y,t) as a statistically homogeneous random function centred at zero, [zeta(x, y,t)] = 0. The coefficients A > 0 and B > 0 are proportional to the surface tension and correspond to the evaporation-condensation and surface-diffusion mechanism, respectively. The autocorrelation function of zeta(z, y, 0) (at time to = 0) is modelled as a function of the Gaussian shape. Two situations are treated: the case where zeta is a statistically isotropic (in lateral directions) random function and the case where zeta is a random function in the x-direction, not dependent on the perpendicular coordinate y. Attention is focussed on the r.m.s. deviation eta(t) = {[zeta(x, y, t)](2)}(1/2). It is shown for both the situations that the long-time dependence of eta(t) is of the form t(-mu),mu > 0, in contrast with the exponential time-dependence which was formerly derived by Mullins for sinusoidally corrugated surfaces.

The linear stochastic equation dx β /dt+[1+f β (t)]x β (t)=A sin (Ωt) is discussed. The functionƒ β (t) is defined as a Poissonian noise dependent on a parameterβ>0,ƒ β (t)=β Σ j [δ(t − t j + ) −δ (t − t j − )]. The mean frequency of the delta-pulses is chosen asβ-dependent in the formλ(β)=2γ(β −2 + 1) exp(−β) whereγ is a constant from the interval (0, 0.974). With the stochastic functionƒ β (t) defined in this way, attention is paid on the oscillational term of the averaged function 〈x(t)〉, 〈x(t)〉osc=Āsin(Ωt − α). It is found that the dependenceĀ=Ā(β) exhibits one maximum and one minimum. The occurrence of these extrema seems to affirm the presence of stochastic resonance.

The one-dimensional propagator problem is solved for conduction electrons near a planar defect imitated by a delta-barrier or by a delta-well in a semiconductor. On condition that interband transitions may be neglected, the propagator is derived for a wide class of (possibly non-parabolic) functions fitting in with the dispersion function E(k) in the bottom part of the conduction band.

The transmission coefficient T(k0) is calculated for conduction electrons incident with a wave vector k0 upon a double barrier (double well) formed of two equal delta-barriers (of two equal delta-wells) embedded in a one-dimensional (1-D) semiconductor or in a 1-D metal. The stationary Schrödinger–Wannier equation E(−i∂/∂x)ψ+V(x)ψ=ℰψ is solved for V(x)=γ[δ(x+a/2)+δ(x−a/2)] (with real and time-independent parameters γ, a) and ℰ=E(k0)>0. (The interband transitions are neglected.) The operator E(−i∂/∂x) corresponds to a given (possibly nonquadratic) dispersion function E(k) of the conduction electrons [E(0)=0]. It is shown that T(k0) is an oscillating function reaching the maximum value [T(k0)→1] on an infinite set {K(j)} of values of k0. The shape of T(k0) depends on the shape of the dispersion function E(k) in a simple way: T(k0)=Tpar(mv(k0)/ℏ)) where Tpar(k0) means the transmission coefficient in the special case when the dispersion function is quadratic, Epar(k)=ℏ2k2/2m, and v(k)=(1/ℏ)∂E(k)/∂k is the group velocity due to E(k). [Here E(k) is taken as an increasing function.]

A class of one-dimensional diffusions is studied based on the equation q(τ) = -f0- f1(τ) where f0 > 0 is a (deterministic) constant and f1(τ) a stationary stochastic process defined by one-sided Poissonian shocks, f1(τ) = ΣQjδ(τ - τj), Qj⪰0, 0⪯τ⪯t. The amplitudes Qj are random, distributed with an arbitrary probability density S(Q). With q(t) = x, q(0) = x0, an explicit expression is derived for the probability density P(x,t¦x0). If q(τ) represents a variable characterising the quality of an object undergoing wear, the lifetime λ > 0 of the object is defined by the equation q(λ) = x. Here x < x0 is the lowest value of q still tolerable for the functioning of the object. Statistical properties of λ are studied (its mean, variance and probability density L(x, λ¦x0)). With two-sided Poissonian shocks distributed with a symmetric amplitude probability density, S(Q) = S(-Q), a new representation for the density matrix of quasiparticles with an arbitrary dispersion law E = E(k)⪰0 is indicated.

A Poisson-modified Wiener process is considered. Its conditional probability density is calculated exactly. Various forms of the evolution equation are derived for the case when the initial probability density is arbitrary. A generalization is also treated when this equation contains a term analogous to the potential energy term in the Schrodinger equation. The Green function of this equation is derived in the form of a functional integral which may be considered as a direct generalization of the Feynman-Kac integral. An application is suggested in the theory of quasiparticles with a non-parabolic dispersion law.