Realistic searches on stretched exponential networks

We consider navigation or search schemes on networks which have a degree distribution of the form P(k) ∝ exp(−k ^γ). In addition, the linking probability is taken to be dependent on social distances and is governed by a parameter λ. The searches are realistic in the sense that not all search chains can be completed. An estimate of μ = ρ/s _d, where ρ is the success rate and s _d the dynamic path length, shows that for a network of N nodes, μ ∝ N ^−δ in general. Dynamic small world effect, i.e., δ ≃ 0 is shown to exist in a restricted region of the λ−γ plane.      

Weak chaos and metastability in a symplectic system of many long-range-coupled standard maps

We introduce, and numerically study, a system of N symplectically and globally coupled standard maps localized in a d=1 lattice array. The global coupling is modulated through a factor r^-α, being r the distance between maps. Thus, interactions are long-range (nonintegrable) when 0≤α≤1, and short-range (integrable) when α>1. We verify that the largest Lyapunov exponent λ_M scales as λ_M ∝ N^-κ(α), where κ(α) is positive when interactions are long-range, yielding weak chaos in the thermodynamic limit N↦∞ (hence λ_M→0). In the short-range case, κ(α) appears to vanish, and the behaviour corresponds to strong chaos. We show that, for certain values of the control parameters of the system, long-lasting metastable states can be present. Their duration t_c scales as t_c ∝N^β(α), where β(α) appears to be numerically in agreement with the following behavior: β>0 for 0 ≤α< 1, and zero for α≥1. These results are consistent with features typically found in nonextensive statistical mechanics. Moreover, they exhibit strong similarity between the present discrete-time system, and the α-XY Hamiltonian ferromagnetic model.     

Irreversible deposition of extended objects with diffusional relaxation on discrete substrates

Random sequential adsorption with diffusional relaxation of extended objects both on a one-dimensional and planar triangular lattice is studied numerically by means of Monte Carlo simulations. We focus our attention on the behavior of the coverage θ(t) as a function of time. Our results indicate that the lattice dimensionality plays an important role in the present model. For deposition of k-mers on 1D lattice with diffusional relaxation, we found that the growth of the coverage θ(t) above the jamming limit to the closest packing limit θ_CPL is described by the pattern θ_CPL-θ(t) ∝E_β[-(t/τ)^β], where E_β denotes the Mittag-Leffler function of order β∈(0,1). In the case of deposition of extended lattice shapes in 2D, we found that after the initial “jamming", a stretched exponential growth of the coverage θ(t) towards the closest packing limit θ_CPL occurs, i.e., θ_CPL - θ(t) ∝exp [-(t/τ)^β]. For both cases we observe that: (i) dependence of the relaxation time τ on the diffusion probability P_dif is consistent with a simple power-law, i.e., τ∝P_dif ^-δ; (ii) parameter β depends on the object size in 1D and on the particle shape in 2D.     

Magnetization of optically polarized gas atoms by an optical pulse train

The properties of the density matrix and the multipole moments arising in oriented and aligned atoms with zero nuclear spin through the interaction with strong resonant ultrashort pulses with wave vector k ₀ and circular or linear polarization have been found. Calculations have been made for the time-dependent light-induced magnetization μ(t′) of a gas of pre-oriented and prealigned atoms following the passage of a weak resonant elliptically polarized pulse with frequency ω and wave vector k collinear with k ₀. It is shown that for oriented atoms, μ(t′) is an even function of the detuning from resonance, ω-ω _ba, and can be split into two terms whose directions are a consequence of symmetry and are determined by the vectors k ₀ and k as well as by the direction of rotation of the electric fields corresponding to the pulses. For aligned atoms the vector μ(t′) is collinear with k, and the first term is an even function of ω-ω _ba. However, the second term is an odd function of ω-ω _ba and reverses direction when the sign of ω-ω _ba changes, as well as when the orientation of the axes of the polarization ellipse is changed. It is shown that if a series of weak linearly polarized pulses pass through the gas, the light-induced magnetization of the oriented and aligned gas atoms can be decomposed into three factors: the first determines the direction and is a consequence of the symmetry; the second (with the dimensions of magnetic moment) depends on the characteristics of the resonant transitions; and the third is a universal function of t′ and ω-ω _ba that does not depend on the underlying characteristics of the resonant transition. These vector factors and the universal functions are in principle different for oriented and aligned atoms. Zh. éksp. Teor. Fiz. 111, 63–92 (January 1997)     

Spectrum of Kelvin-wave turbulence in superfluids

We derive a type of kinetic equation for Kelvin waves on quantized vortex filaments with random large-scale curvature, that describes step-by-step (local) energy cascade over scales caused by 4-wave interactions. Resulting new energy spectrum E _LN(k) ∝ k ^−5/3 must replace in future theory (e.g., in finding the quantum turbulence decay rate) the previously used spectrum E _KS(k) ∝ k ^−7/5, which was recently shown to be inconsistent due to nonlocality of the 6-wave energy cascade.     

Dynamic and spectral mixing in nanosystems

In the framework of a simple spin-boson Hamiltonian we study an interplay between dynamic and spectral roots to stochastic-like behavior. The Hamiltonian describes an initial vibrational state coupled to discrete dense spectrum reservoir. The reservoir states are formed by three sequences with rationally independent periodicities 1; 1 ± δ typical for vibrational states in many nanosize systems (e.g., large molecules containing CH₂ fragment chains, or carbon nanotubes). We show that quantum evolution of the system is determined by a dimensionless parameter δΓ, where Γ is characteristic number of the reservoir states relevant for the initial vibrational level dynamics. When δΓ > 1 spectral chaos destroys recurrence cycles and the system state evolution is stochastic-like. In the opposite limit δΓ < 1 dynamics is regular up to the critical recurrence cycle k _c and for larger k > k _c dynamic mixing leads to quasi-stochastic time evolution. Our semi-quantitative analytic results are confirmed by numerical solution of the equation of motion. We anticipate that both kinds of stochastic-like behavior (namely, due to spectral mixing and recurrence cycle dynamic mixing) can be observed by femtosecond spectroscopy methods in nanosystems in the spectral window 10¹¹–10¹³ s⁻¹     

Reacting polymers with highly correlated initial conditions

We propose and theoretically study an experiment designed to measure short-time polymer reaction kinetics in melts or dilute solutions. The photolysis of groups centrally located along chain backbones, one group per chain, creates pairs of spatially highly correlated macroradicals. We calculate time-dependent rate coefficients κ(t) governing their first-order recombination kinetics, which are novel on account of the far-from-equilibrium initial conditions. In dilute solutions (good solvents) reaction kinetics are intrinsically weak, despite the highly reactive radical groups involved. This leads to a generalised mean-field kinetics in which the rate of radical density decay - ∼S(t), where S(t) ∼t ^{- (1 + g/3)} is the equilibrium return probability for 2 reactive groups, given initial contact. Here g≈ 0.27 is the correlation hole exponent for self-avoiding chain ends. For times beyond the longest coil relaxation time τ, - ∼S(t) remains true, but center of gravity coil diffusion takes over with rms displacement of reactive groups x(t) ∼t ^1/2 and S(t) ∼ 1/x ³(t). At the shortest times ( t 10^-6s), recombination is inhibited due to spin selection rules and we find ∼tS(t). In melts, kinetics are intrinsically diffusion-controlled, leading to entirely different rate laws. During the regime limited by spin selection rules, the density of radicals decays linearly, n(0) - n(t) ∼t. At longer times the standard result - ∼d ³(t)/d (for randomly distributed ends) is replaced by ∼d2x ³(t)/d ² for these correlated initial conditions. The long-time behavior, t > τ, has the same scaling form in time as for dilute solutions. Received 18 May 2000     

Distribution of a Particle’s Position in the ASEP with the Alternating Initial Condition

In this paper we give the distribution of the position of a particle in the asymmetric simple exclusion process (ASEP) with the alternating initial condition. That is, we find ℙ(X _m (t)≤x) where X _m (t) is the position of the particle at time t which was at m=2k−1, k∈ℤ at t=0. As in the ASEP with step initial condition, there arises a new combinatorial identity for the alternating initial condition, and this identity relates the integrand of the integral formula for ℙ(X _m (t)≤x) to a determinantal form together with an extra product.     

An exactly solvable linear model giving indication of stochastic resonance

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. This work has been supported by the Slovak Grant Agency VEGA under contract No. 1/4319/97.     

AC Josephson effect in the long voltage-biased SINIS junction

Theory of non-stationary coherent effects is developed for superconductor-normal-superconductor (SNS) structures with relatively strong normal scattering on S/N interfaces (interface resistance is large compared to intrinsic resistance of N metal). Analytical expressions are found for the time-dependent anomalous Green’s functions induced in the N region under the fixed-voltage-bias. The amplitude of the current oscillations is determined in non-equilibrium conditions. Non-stationary correction to the distribution function is calculated in high-temperature limit and found to be slowly decreasing with the temperature, leading to the dominance of the second-harmonic term in the Josephson current, I _s (t) ∝ sin(4eVt) at high temperatures and low voltage. The article is published in the original.     

Rare t-quark decays t → clj+lk? and t → c?jvk induced by doublets of scalar leptoquarks within the minimal four-color-symmetry model

The rare t-quark decays t → cl _j ⁺ l _k ⁻ and t → cṽ _j k _k induced by scalar-leptoquark doublets are considered within the minimal model involving four-color quark-lepton symmetry and the Higgs mechanism of quark and lepton mass splitting. The partial widths with respect to the decays being considered and the total widths Γ(t → cl ⁺′l ⁻) = Σ_j,k Γ(t → cl _j ⁺ l _k ⁻ ) and Γ(t → cl ⁺′l ⁻) = Σ_j,kΓ(t → cṽ _j v _k ) with respect to, respectively, the charged leptonic and neutrino modes are calculated. It is shown that, at scalar-leptoquark masses higher than the t-quark mass (m _S > m _t), the branching ratios for these modes are Br(t → cl ⁺′l ⁻) ≈ (3.5−0.4) × 10⁻⁵ and Br(t → cṽ′v) ≈ (7.1−0.8) × 10⁻⁵ at m _s = 180–250 GeV and an appropriate value of the leptoquark-mixing angle (sin β ≈ 0.2) and can increase for m _S < m _t to Br(t → cl ⁺′l ⁻) ≈ 0.03−0.002 and Br(t → cl ⁺′l ⁻) ≈ 0.46−0.05 for the charged mode at m _S = 150–170 GeV for sin β ≈ 1 and sin β ≈ 0.2, respectively. In the cases being considered, t-quark decays to pairs of charged leptons can be accessible to detection at LHC. In the last case, these decays could manifest themselves (for example, in dilepton events) at the Tevatron as well. Original Russian Text ? P.Yu. Popov, A.D. Smirnov, 2006, published in Yadernaya Fizika, 2006, Vol. 69, No. 6, pp. 1006–1016.     

Effect of an electric field on the surface tension of a liquid at low temperatures

An external electric field changes the dispersion law of waves on the surface of a liquid. Besides the usual capillary term (∝k ³, k is the wave number) and gravitational term (∝k), a term quadratic in the wave vector appears in the expression for the square of the frequency in a homogeneous field. These excitations are associated with the variation of the coefficient of surface tension of the liquid at low temperatures. In the case of a large field tangent to the surface, the correction is proportional to T ^8/3, unlike the T ^7/3 correction in the absence of a field. Zh. éksp. Teor. Fiz. 111, 1369–1372 (April 1997)     

Asymptotic behavior of the β function in the ϕ<Superscript>4</Superscript> theory: A scheme without complex parameters

The previously-obtained analytical asymptotic expressions for the Gell-Mann-Low function β(g) and anomalous dimensions in the ϕ⁴ theory in the limit g → ∞ are based on the parametric representation of the form g = f(t), β(g) = f ₁(t) (where t ∝ g ₀^−1/2 is the running parameter related to the bare charge g ₀), which is simplified in the complex t plane near a zero of one of the functional integrals. In this work, it has been shown that the parametric representation has a singularity at t → 0; for this reason, similar results can be obtained for real g ₀ values. The problem of the correct transition to the strong-coupling regime is simultaneously solved; in particular, the constancy of the bare or renormalized mass is not a correct condition of this transition. A partial proof has been given for the theorem of the renormalizability in the strong-coupling region.     

Non-linear scission/recombination kinetics of living polymerization

Living polymers are formed by reversible association of primary units (unimers). Generally the chain statistical weight involves a factor σ < 1 suppressing short chains in comparison with free unimers. Living polymerization is a sharp thermodynamic transition for σ ≪ 1 which is typically the case. We show that this sharpness has an important effect on the kinetics of living polymerization (one-dimensional association). The kinetic model involves i) the unimer activation step (a transition to an assembly-competent state); ii) the scission/recombination processes providing growth of polymer chains and relaxation of their length distribution. Analyzing the polymerization with no chains but unimers at t = 0 , with initial concentration of unimers M ≳ M ^* (M^* is the critical polymerization concentration), we determine the time evolution of the chain length distribution and find that: 1) for M ^* ≪ M ≪ M ^*/σ the kinetics is characterized by 5 distinct time stages demarcated by 4 characteristic times t₁, t₂, t₃ and t^*; 2) there are transient regimes (t ₁ ≲ t ≲ t ₃) when the molecular-weight distribution is strongly non-exponential; 3) the chain scissions are negligible at times shorter than t₂. The chain growth is auto-accelerated for t ₁ ≲ t ≲ t ₂ : the cut-off chain length (= polymerization degree 〈n〉_w N ₁ ∝ t ² in this regime. 4) For t ₂ < t < t ₃ the length distribution is characterized by essentially 2 non-linear modes; the shorter cut-off length N₁ is decreasing with time in this regime, while the length scale N₂ of the second mode is increasing. (5) The terminal relaxation time of the polymer length distribution, t^*, shows a sharp maximum in the vicinity of M^*; the effective exponent is as high as ∼ σ^-1/3 just above M^*.     

Structure of turbulent flows of incompressible fluids and the parametrization of turbulence

The structure of stationary isotropic, homogeneous turbulence in an incompressible fluid with Re ≫ 1 set into motion by a force with amplitude f ₀ and spatial and temporal time scales of r ₀ and τ ₀, respectively, is examined. It is found that, depending on the magnitude of the force that sets the fluid into motion, three fundamentally different turbulent stationary states of the fluid can develop and the dimensionless parameters responsible for transitions from one state to another, γ=f ₀ τ ₀ ² /r ₀ and Γ=γ ^4/3 Re, are determined. It is shown that for γ≪1 and Γ≪1 a Kolmogorov spectrum with E(k)∝1/k ^5/3 develops in the inertial range. During the transition to turbulent flows driven by large amplitude forces f ₀, i.e., during the transition to a regime with γ≪1 and Γ ≫ 1, a segment of the spectrum with E(k)∝1/k ² develops near the viscous range and “detaches” the Kolmogorov spectrum from the viscous range. Further increases in the amplitude f ₀ of the force, i.e., approaching the parameter range with γ≫1 and Γ≫1, causes the entire inertial range to be “occupied” by a spectrum E(k)∝1/k ², and outside the inertial range, large scale structures with a characteristic size extending to γ ^2/5 r ₀ begin to be generated. In the regime with Γ≪1, the power dissipated per unit mass of fluid is independent of the viscosity, but on going to turbulent regimes with Γ≫1, the viscous losses begin to depend on the viscosity of the fluid. The “turn-off” of viscous dissipation for Γ≫1 shows that a drag crisis can occur simply as the source power is increased, without any further conditions. With this method for the excitation of turbulence, the Loitsyanskii integral diverges for arbitrary values of γ and Γ. A physical mechanism is proposed to explain the readjustment of the spectrum of the turbulent fluctuations at different γ and Γ. These results have all been obtained neglecting intermittency. Zh. éksp. Teor. Fiz. 116, 1630–1647 (November 1999)     

d-symmetry superconductivity due to valence bond correlations

It is shown that d-symmetry superconductivity due to valence bond correlations is possible. Valence bond correlations are compatible with antiferromagnetic spin order. In order to explictly construct a homogeneous state with the valence bond structure in the two-dimensional Hubbard model for an arbitrary doping, we have used the variational method based on unitary local transformation. Attraction between holes in the d-channel is due to modulation of hopping by the site population in course of the valence bond formation, and corresponding parameters have been calculated variationally. An important factor for the gap width is the increase in the density of states on the Fermi level due to antiferromagnetic splitting of the band. The gap width and its ratio to the T _c are 2Δ≃0.1t and 2Δ/kT _c≃4.5−4 for U/t≃8. The correspondence between the theoretical phase diagram and experimental data is discussed. The dependence of T _c on the doping δ=|n−1| and the Fermi surface shape are highly sensitive to the weak interaction t′ leading to diagonal hoppings. In the case of t′>0 and p-doping, the peak on the curve of T _c(δ) occurs at the doping δ _opt, when the energy of the flattest part of the lower Hubbard subband crosses the Fermi level at k∼(π,0). In underdoped samples with δ<δ _opt, the anisotropic pseudogap in the normal state corresponds to the energy difference |E(π,0)−μ| between this part of the spectrum and the Fermi level. Zh. éksp. Teor. Fiz. 114, 985–1005 (September 1998)     

Systematics of perturbative semiclassical quantum defect expansions probed by RKR-QDTand a Fisher-information-based criterion

The systematics of perturbative semiclassical quantum defect expansions corresponding to a hydrogenic potential plus a perturbing term of the form -A/2r^κ, k\geqslant 2\kappa \geqslant 2, are studied as a function of expansion order N. Towards this task the expansions μ _Nare first used as input for constructing associated N-dependent atomic RKR-QDT potential curves. Subsequently the coordinate Fisher information for the energy levels supported by those curves as well as its rate ε with respect to N is semiclassically computed. Then, the plot of relative quantum defect error between successive orders, δμ _N+1,N, with respect to ε serves as convergence indicator for both approximate potentials and quantum defects. For a given κ and when the quantum defect expansion proves to be of limited accuracy the plot reveals an A- and N-dependent scatter of points and “saturation” (the relative error remains almost constant with respect to ε). More importantly, when ε is equal to or lower than the value of ε (N=1) for which πμ ₁\leqslant 1/2_{1}\leqslant 1/2 the relative error exhibits a κ-, A- and N-independent power-law dependence, δμ _N+1,N∝ ε ^m, clearly distinguishing the N=1 order (m=1/2) from all other N>1 orders (m=1). These power-laws may be employed for setting-up confidence level bounds on perturbative expansions.     

Distribution function for large velocities of a two-dimensional gas under shear flow

The high-velocity distribution of a two-dimensional dilute gas of Maxwell molecules under uniform shear flow is studied. First we analyze the shear-rate dependence of the eigenvalues governing the time evolution of the velocity moments derived from the Boltzmann equation. As in the three-dimensional case discussed by us previously, all the moments of degreek⩾4 diverge for shear rates larger than a critical valuea _c ^(k) , which behaves for largek asa _c ^(k) ∼k ⁻¹. This divergence is consistent with an algebraic tail of the formf(V) ∼V ^−4-σ(a), where σ is a decreasing function of the shear rate. This expectation is confirmed by a Monte Carlo simulation of the Boltzmann equation far from equilibrium.     

A Construction of Blow up Solutions for Co-rotational Wave Maps

The existence of co-rotational finite time blow up solutions to the wave map problem from ${\mathbb{R}^{2+1} \to N}The existence of co-rotational finite time blow up solutions to the wave map problem from \mathbbR²⁺¹ ? N{\mathbb{R}^{2+1} \to N} , where N is a surface of revolution with metric d ρ ² + g(ρ)² dθ², g an entire function, is proven. These are of the form u(t,r)=Q(l(t)t)+R(t,r){u(t,r)=Q(\lambda(t)t)+\mathcal{R}(t,r)} , where Q is a time independent solution of the co-rotational wave map equation −u _tt  + u _rr  + r ⁻¹ u _r  = r ⁻² g(u)g′(u), λ(t) = t ^−1-ν, ν > 1/2 is arbitrary, and R{\mathcal{R}} is a term whose local energy goes to zero as t → 0.