Strictly two-dimensional self-avoiding walks: Thermodynamic properties revisited

The density crossover scaling of various thermodynamic properties of solutions and melts of self-avoiding and highly flexible polymer chains without chain intersections confined to strictly two dimensions is investigated by means of molecular dynamics and Monte Carlo simulations of a standard coarse-grained bead-spring model. In the semidilute regime we confirm over an order of magnitude of the monomer density ?? the expected power law scaling for the interaction energy between different chains e _int ?? ?? ^21/8, the total pressure P ?? ?? ³ and the dimensionless compressibility gT = lim _q??0 S(q) ?? 1/?? ². Various elastic contributions associated to the affine and non-affine response to an infinitesimal strain are analyzed as functions of density and sampling time. We show how the size ??(??) of the semidilute blob may be determined experimentally from the total monomer structure factor S(q) characterizing the compressibility of the solution at a given wave vector q . We comment briefly on finite persistence length effects.      

Censored Glauber Dynamics for the Mean Field Ising Model

We study Glauber dynamics for the Ising model on the complete graph on n vertices, known as the Curie-Weiss Model. It is well known that at high temperature (β<1) the mixing time is Θ(nlog?n), whereas at low temperature (β>1) it is exp?(Θ(n)). Recently, Levin, Luczak and Peres considered a censored version of this dynamics, which is restricted to non-negative magnetization. They proved that for fixed β>1, the mixing-time of this model is Θ(nlog?n), analogous to the high-temperature regime of the original dynamics. Furthermore, they showed cutoff for the original dynamics for fixed β<1. The question whether the censored dynamics also exhibits cutoff remained unsettled.In a companion paper, we extended the results of Levin et al. into a complete characterization of the mixing-time for the Curie-Weiss model. Namely, we found a scaling window of order \(1/\sqrt{n}\) around the critical temperature β _c =1, beyond which there is cutoff at high temperature. However, determining the behavior of the censored dynamics outside this critical window seemed significantly more challenging.In this work we answer the above question in the affirmative, and establish the cutoff point and its window for the censored dynamics beyond the critical window, thus completing its analogy to the original dynamics at high temperature. Namely, if β=1+δ for some δ>0 with δ ² n→∞, then the mixing-time has order (n/δ)log?(δ ² n). The cutoff constant is (1/2+[2(ζ² β/δ?1)]^?1), where ζ is the unique positive root of g(x)=tanh?(β x)?x, and the cutoff window has order n/δ.     

New approach to analyzing and evaluating cross sections for partial photoneutron reactions

The presence of substantial systematic discrepancies between the results of different experiments devoted to determining cross sections for partial photoneutron reactions??first of all, (??, n), (??, 2n), and (??, 3n) reactions??is a strong motivation for studying the reliability and authenticity of these data and for developing methods for taking into account and removing the discrepancies in question. In order to solve the first problem, we introduce objective absolute criteria involving transitional photoneutron-multiplicity functions F ₁, F ₂, F ₃, ??; by definition, their values cannot exceed 1.0, 0.5, 0.33, ??, respectively. With the aim of solving the second problem, we propose a new experimental-theoretical approach. In this approach, reaction cross sections are evaluated by simultaneously employing experimental data on the cross section for the total photoneutron yield, ?? ^expt(??, xn) = ?? ^expt(??, n) + 2?? ^expt(??, 2n) + 3?? ^expt(??, 3n) + ??, which are free from drawbacks plaguing experimental methods for sorting neutrons in multiplicity, and the results obtained by calculating the functions F _theor ¹ , F _theor ² , F _theor ³ , ?? on the basis of the modern model of photonuclear reactions. The reliability and authenticity of data on the cross sections for (??, n), (??, 2n), and (??, 3n) partial reactions???? ^eval(??, in) = F _i ^theor ?? ^expt(??, xn)??were evaluated for the ⁹⁰Zr, ¹¹⁵In, ^{112,114,116,117,118,119,120,122,124}Sn, ¹⁵⁹Tb, and ¹⁹⁷Au nuclei.     

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.     

Relativistic effects in one-dimensional hydrogen atom

The dependence of stationary levels of a Dirac particle in the regularized Coulomb potential V _??(z) = ?q/(|z| + ??) on the cutoff parameter ?? is studied. It is shown that, in 1 + 1 D, the energy spectrum of a Dirac particle in such a potential reveals some specific features which nonanalytically depend on the coupling constant q and are essentially relativistic in nature. These properties turn out to be most important for ?? ? 1, explicitly demonstrating the existence of a physically reasonable energy spectrum for an arbitrarily small ?? > 0 and, at the same time, the absence of regular limit ?? ?? 0 (hence, the absence of a well-defined spectral problem for the Dirac equation without regularization for arbitrary q in 1 + 1 D).     

Dynamic behaviours of 2∞ attractor and q-phase transitions at bifurcations in logistic map

Dynamic behaviours of the 2^∞ attractor at the accumulation of period doubling in the logistic map are studied by the sum of the local expansion rates S_n(x₁) of nearby orbits. The variance 〈[S_n(x)]²〉 and algebraic exponent ß_n(x₁) = S_n(x₁)/ln(n) exhibits self-similar structures. The critical bifurcations such as intermittency, band merging and crisis-sudden widening of the chaotic attractor are studied in terms of a q-weighted average Λ(q), (− ∞ < q < ∞) of the coarse-grained local expansion rates Λ of nearby orbitals.     

The Mixing Time Evolution of Glauber Dynamics for the Mean-Field Ising Model

We consider Glauber dynamics for the Ising model on the complete graph on n vertices, known as the Curie-Weiss model. It is well-known that the mixing-time in the high temperature regime (β < 1) has order n log n, whereas the mixing-time in the case β > 1 is exponential in n. Recently, Levin, Luczak and Peres proved that for any fixed β < 1 there is cutoff at time with a window of order n, whereas the mixing-time at the critical temperature β = 1 is Θ(n ^3/2). It is natural to ask how the mixing-time transitions from Θ(n log n) to Θ(n ^3/2) and finally to exp (Θ(n)). That is, how does the mixing-time behave when β = β(n) is allowed to tend to 1 as n → ∞. In this work, we obtain a complete characterization of the mixing-time of the dynamics as a function of the temperature, as it approaches its critical point β _c  = 1. In particular, we find a scaling window of order around the critical temperature. In the high temperature regime, β = 1 − δ for some 0 < δ < 1 so that δ ² n → ∞ with n, the mixing-time has order (n/δ) log(δ ² n), and exhibits cutoff with constant and window size n/δ. In the critical window, β = 1± δ, where δ ² n is O(1), there is no cutoff, and the mixing-time has order n ^3/2. At low temperature, β = 1 + δ for δ > 0 with δ ² n → ∞ and δ = o(1), there is no cutoff, and the mixing time has order . Research of J. Ding and Y. Peres was supported in part by NSF grant DMS-0605166.     

A Markovian Growth Dynamics on Rooted Binary Trees Evolving According to the Gompertz Curve

Inspired by biological dynamics, we consider a growth Markov process taking values on the space of rooted binary trees, similar to the Aldous-Shields (Probab. Theory Relat. Fields 79(4):509?C542, 1988) model. Fix n??1 and ??>0. We start at time 0 with the tree composed of a root only. At any time, each node with no descendants, independently from the other nodes, produces two successors at rate ??(n?k)/n, where k is the distance from the node to the root. Denote by Z _n (t) the number of nodes with no descendants at time t and let T _n =?? ^?1 nln(n/ln4)+(ln2)/(2??). We prove that 2^?n Z _n (T _n +n??), ?????, converges to the Gompertz curve exp(?(ln2)?e ^??|? ). We also prove a central limit theorem for the martingale associated to Z _n (t).     

Non-normal tracer diffusion from stirring by swimming microorganisms

The experimentally observed non-Gaussian form of passive tracer distributions in media stirred by active swimmers (Leptos et al., Phys. Rev. Lett. 103, 198103 (2009)) are analyzed in terms of continuous time random walks. The walks are characterized by a trapping time distribution ??(??) with long time behaviour ??(??) ?? ?? ^?1??? and a step size distribution p(??x) ?? (??x)^?2??? . The experimentally observed behaviour that ??x ²?? ?? t is obtained for a one-parameter family of exponents with ?? = 2??. However, the distribution function for this case is non-Gaussian and shows exponential tails. The shape of the distributions agrees rather well with the experimental observations from Leptos et al. and allows for the determination of the exponents.      

Free energies of the spherical model of a spin glass

Free energies g(m, m_s) and f(m, q) of the spherical spin glass (SG) model due to Kosterlitz et al. are calculated explicitly as functions of the uniform magnetization m, and SG order parameter m_s and the Edwards-Anderson order parameter q. It is shown that g(0, m_s) and f(0, q) below the transition temperature T_g are constant in the ranges 0 ≦ m_s ≦ m_s⁰ and 0 ≦ q ≦ q₀ respectively, where $\text{q}{}_0\text{= (1 -?}\text{T}\text{T}{}_{\mathrm{g}}\text{) = m}{}^2{}_{\mathrm{s}}{}^0$. The proper equilibrium values of m_s( = m_s⁰) and q( d=q₀) are then fixed from the inspection of their behaviors under infinitesimal uniform field proproportional to N^-a($\text{a ≧ 0}$).     

Theoretical prediction of cosmological constant Λ in Veneziano ghost theory of QCD

Based on the Veneziano ghost theory of QCD, we predict the cosmological constant ??, which is related to energy density of cosmological vacuum by $ \Lambda = \frac{{8\pi G}} {3}\rho _\Lambda $ . In the Veneziano ghost theory, the vacuum energy density ?? _?? is expressed by absolute value of the product of quark vacuum condensate and quark current mass: $ \rho _\Lambda = \frac{{2N_f H}} {{m_{\eta '} }}c|m_q < 0|:\bar qq:|0 > | $ . We calculate the quark local vacuum condensates ??0|: $ \bar q $ q: |0?? by solving Dyson-Schwinger Equations for a fully dressed confining quark propagator S _f (p) with an effective gluon propagator G _???? ^ab (q). The quark current mass m _q is predicted by use of chiral perturbation theory. Our theoretical result of ??, with the resulting ??0|: 471-4 q: |0?? = ?(235 MeV)³ and light quark current mass m _q ? 3.29?C6.15 MeV, is in a good agreement with the observable of the ?? ?? 10^?52 m^?2 used widely in a great amount of literatures.     

Investigation of near-threshold eta-meson production in the reaction π ? p→ ηn

Differential and total cross sections for eta-meson production in the reaction ?? ^? p ?? ??n were measured within the experimental program eta-meson physics implemented in the pion channel of the synchrocyclotron of the Petersburg Nuclear Physics Institute (PNPI, Gatchina). These measurements were performed at incident-pion momenta (700, 710, 720, and 730 MeV/c) in the vicinity of the threshold for the process under study by using the neutral-meson spectrometer designed and created at the Meson Physics Laboratory of PNPI. It is shown that, in the immediate vicinity of the threshold (685 MeV/c), the process of eta-meson production proceeds predominantly via S ₁₁(1535)-resonance formation followed by the decay S ₁₁(1535) ?? ??n (the respective branching fraction is Br ?? 60%), but that, as the momentum of incident pions increases, the role of the D wave becomes ever more important. A detailed analysis of this effect indicates that it is due to the increasing contribution of the D ₁₃(1520) resonance. Although the branching fraction of the decay of this resonance through the ??n channel is assumed to be very small (BR ?? 0.24%), the effect is enhanced owing to the interference between the D wave and the dominant resonance S ₁₁(1535).     

Spatial evolutionary game with bond dilution

In this paper, we study numerically the prisoner’s dilemma game (PDG) and snowdrift game (SG) on a two-dimensional square lattice with both quenched and annealed bond dilution. For quenched bond dilution, the system undergoes a dynamical transition at the critical occupation probability q^∗, which is higher than the bond percolation transition point for a square lattice. In the critical region, the defined order parameter has a scaling form as P_e∼(q−q^∗)^β for q<q^∗ with the critical exponents β=1.42 for PDG and β=1.52 for SG, which differ from those with quenched site dilution. For annealed bond dilution, the system exhibits a distinct cooperative behavior. We find that the cooperation is much enhanced in the range of small payoff parameters on a lattice with slightly annealed bond dilution.     

Linear and nonlinear liftings of states of quantum systems

In this paper, we study the representability of an arbitrary quantum state ?? ?? ??(H) as the reduction of a vector state r ?? ??(H) of the extended system. We extend the operation of lifting from the set of states ?? _n (H) to the set of generalized states ??(H). A method of constructing the Hilbert space H and the affine linear lifting ??(H) ?? ??(H) is studied. The construction of individual expansion H _?? of the space H for which the state ?? is a reduction of a vector state H _?? is of special interest.     

Morphogenesis of sliced N-soliton solutions

Consider an instantaneous severing interaction which at t = t_s transforms is given N-soliton solution q⁰ into two new solutions, q^L and q^R, with discontinuous anitial conditions at t = t_s such that q^L(q^R) is equal to q⁰ to the left (right) of the severing point x_s and vanishes to the right (left) of x_s. The soliton content of q^L and q^R is studied in the context of the cubic Schrödinger equation and the sine-Gordon equation by solving the direct scattering problem. It is shown that the motion constants C_n, n = 1, 2 …, are related by , Furthermore, if q⁰ contains N solitons and no contribution from the continuous spectrum, then q^L(q^R) contains N^L(N^R) solitons with 0 ? N^L, N^R ? N and N^L + N^R = N. The slicing puts a soliton into q^L or q^R, respectively as the slicing point x_s is taken to the right or left of the soliton's “center” at t = t_s.     

Phase space analysis of interacting dark energy in f(T) cosmology

In this paper, we examine the interacting dark energy model in f(T) cosmology. We assume dark energy as a perfect fluid and choose a specific cosmologically viable form f(T) = ????T. We show that there is one attractor solution to the dynamical equation of f(T) Friedmann equations. Further we investigate the stability in phase space for a general f(T) model with two interacting fluids. By studying the local stability near the critical points, we show that the critical points lie on the sheet u* = (c ? 1)v* in the phase space, spanned by coordinates (u, v, ??, T). From this critical sheet, we conclude that the coupling between the dark energy and matter c ?? (?2, 0).     

Muon Capture on Deuteron and 3He

We present the results of a recent simultaneous study of the muon capture reactions ²H(?? ^?, ?? _?? )nn and ³He(?? ^?, ?? _?? )³H. The initial and final A = 2 and 3 nuclear wave functions are obtained from the Argonne v ₁₈ or chiral N3LO two-nucleon potential, in combination with, respectively, the Urbana IX or chiral N2LO three-nucleon potential in the case of A?=?3. The weak current consists of polar- and axial-vector components. The former are related to the isovector piece of the electromagnetic current via the conserved-vector-current hypothesis. These and the axial currents are derived either in a meson-exchange or in a chiral effective field theory (??EFT) framework. In the first case, there is one parameter which is fixed by reproducing the Gamow-Teller matrix element in tritium ??-decay (GT^EXP). In the second case, the low-energy constants, two in the polar and one in the axial-vector current, are fixed, respectively, by reproducing the A?=?3 magnetic moments and GT^EXP. The total rates are found to be 392.0 ± 2.3 s^?1 for A?=?2, and 1484 ± 13 s^?1 for A?=?3, where the spread accounts for the model dependence relative to the adopted interactions and currents (and cutoff sensitivity in the ??EFT currents).     

Classification of representations of the algebra U′q(so 3) through examples II

This paper completes series of articles devoted to classification of the representations of the nonstandard deformation U′_q(so ₃) providing examples of such representations in low dimensions. The classification differs substantially when the deformation parameter q is/is not root of unity (q ⁿ=1). When it is a root of unity, the situation differs for odd and even n. The examples presented here cover the first nontrivial case when n is even (namely, n=4), from which the general case follows easily.     

Quantum integrable systems related to lie algebras

Some quantum integrable finite-dimensional systems related to Lie algebras are considered. This review continues the previous review of the same authors [83] devoted to the classical aspects of these systems. The dynamics of some of these systems is closely related to free motion in symmetric spaces. Using this connection with the theory of symmetric spaces some results such as the forms of spectra, wave functions, S-matrices, quantum integrals of motion are derived. In specific cases the considered systems describe the one-dimensional n-body systems interacting pairwise via potentials g²v(q) of the following 5 types: v_I(q) = q^?2, v_II(q) = sinh^?2q, v_III(q) = sin^?2q, $\text{v}{}_{\mathrm{<mtext>IV</mtext>}}\text{(q) =}\text{P}\text{(q)}$, v_V(q) = q^?2 + ω²q². Here $\text{P}\text{(q)}$ is the Weierstrass function, so that the first three cases are merely subcases of the fourth. The system characterized by the Toda nearest-neighbour potential exp(q_jq_{j+ 1}) is moreover considered.This review presents from a general and universal point of view results obtained mainly over the past fifteen years. Besides, it contains some new results both of physical and mathematical interest.     

Phenomenological implications of light stop and higgsinos

We examine the phenomenological implications of light $\tilde t_R $ and higgsinos in the Minimal Supersymetric Standard Model, assuming tan² ??<m _t m _b and heavy $\tilde t_L $ and gauginos. In this simplified setting, we study the contributions to ??m _{B d},?? _K,BR(b??s??),R _b???(Z??bb) /??(Z??hadrons),BR(t??bW), and their interplay.