Exact results for two-dimensional coarsening

We consider the statistics of the areas enclosed by domain boundaries (‘hulls’) during the curvature-driven coarsening dynamics of a two-dimensional nonconserved scalar field from a disordered initial state. We show that the number of hulls per unit area, n _h (A, t)dA, with enclosed area in the range (A,A + dA), is described, for large time t, by the scaling form n _h (A, t) = 2c _h /(A + λ _h t)², demonstrating the validity of dynamical scaling in this system. Here $ c_h = {1 \mathord{\left/ {\vphantom {1 8}} \right. \kern-0em} 8}\pi \sqrt 3 $ is a universal constant associated with the enclosed area distribution of percolation hulls at the percolation threshold, and λ _h is a material parameter. The distribution of domain areas, n _d (A, t), is apparently very similar to that of hull areas up to very large values of A/λ _h t. Identical forms are obtained for coarsening from a critical initial state, but with c _h replaced by c _h /2. The similarity of the two distributions (of areas enclosed by hulls, and of domain areas) is accounted for by the smallness of c _h . By applying a ‘mean-field’ type of approximation we obtain the form n _d (A, t) ? 2c _d [λ _d (t+t ₀)] ^τ?2/[A+λ _d (t+t ₀)] ^τ , where t ₀ is a microscopic timescale and τ = 187/91 ? 2.055, for a disordered initial state, and a similar result for a critical initial state but with c _d → c _d /2 and τ → τ _c = 379/187 ? 2.027. We also find that c _d = c _h + O(c _h ² ) and λ _d = λ _h (1 + O(c _h )). These predictions are checked by extensive numerical simulations and found to be in good agreement with the data.     

Continuous time Black-Scholes equation with transaction costs in subdiffusive fractional Brownian motion regime

In this paper, we study the problem of continuous time option pricing with transaction costs by using the homogeneous subdiffusive fractional Brownian motion (HFBM) Z(t)=X(S_α(t)), 0<α<1, here dX(τ)=μX(τ)(dτ)^2H+σX(τ)dB_H(τ), as a model of asset prices, which captures the subdiffusive characteristic of financial markets. We find the corresponding subdiffusive Black-Scholes equation and the Black-Scholes formula for the fair prices of European option, the turnover and transaction costs of replicating strategies. We also give the total transaction costs.     

The Itô-Clifford integral III. The Markov property of solutions to stochastic differential equations

It is shown that the solution to the Itô-Clifford stochastic differential equationdX _t =F(X _t ,t)dΨ _t +dΨ _t G(X _t ,t)+H(X _t ,t)dt, whereF, G, H are suitable Lipschitz functions andΨ _t is the fermion martingale, satisfies a Markov property.     

Stochastic processes originating in deterministic microscopic dynamics

We investigate the probability distribution of the scaled trajectory of a test particle moving in an equilibrium fluid according to the laws of classical mechanics, i.e., ifQ(t) is the displacement of the test particle we letQ _A(t) =Q(At)/√A and consider the distribution of the trajectory Q_A(t) in the limit A→∞. The randomness of the motion is due entirely to the randomness of the initial state of the fluid, test particle, or both, and the process is generally non-Markovian. Nevertheless, it can be proven in some cases and we expect it to be true in many more that Q_A (t) looks like Brownian motion in the limit A→∞. Some results for simple model systems are presented.     

Magnetic relaxation near a fishtail peak of isothermal magnetization in B1 NbCy encapsulated in multiwall carbon nanocages

Isothermal magnetization near a fishtail peak in nanocrystalline B1 NbC_y encapsulated in multiwall carbon nanocages is studied within the time window of 100 < t < 4000 s. The current density J exhibits a linear logarithmic time decay. The effective activation energy U_eff increases linearly with temperature T and is independent of applied magnetic field H. The results of J(t) and U_eff (T, H) are consistent with the Anderson–Kim flux–creep model for thermally activated motion of uncorrelated vortices or vortex bundles over a net potential barrier U_eff. U_eff at a fishtail peak field H_fp evolves quickly above a fishtail peak temperature T_fp, but slowly below that temperature. The result suggests that a decrease of flux viscosity coefficient above T_fp at H_fp is the origin of the fishtail peak in nanocrystalline B1 NbC_y encapsulated in multiwall carbon nanocages.     

On locally continuous cocycles

Cohomology groups of locally continuous cocycles of a topological group G with values in a topological G-module A_Ψ are considered and the role of those of degree 2 in the theory of topological extensions of G by A_Ψ is analyzed. The exactness of long sequence is proven. It is shown that, if G and A_Ψ are Polish and if all topological extensions of G by A_Ψ are fibered, the cohomology groups of Borel cocycles and those of locally continuous Borel cocycles are isomorphic for degree 0,1,2.     

The resonance decay function method in the determination of the pre-factor of the Néel relaxation time of single-domain nanoparticles

In its simple form, the relaxation time of the Néel relaxation process of the magnetic moment of single-domain particles is given by τ_N=τ_0Nexp(σ), σ being the ratio of anisotropy energy to thermal energy. The pre-factor, τ_0N, is normally given a value of 10⁻⁹ s, but values ranging from 10⁻⁸ to 10⁻¹² s have been reported in literature. Here, by means of the field and frequency dependence of the complex magnetic susceptibility, χ(ω,H)=χ′(ω,H)−iχ″(ω,H), of a magnetic fluid sample, in the MHz-GHz range, in conjunction with the determination of the sample decay function, b(t), the pre-factor τ_0N is determined. b(t) is readily obtained through the inverse Fourier transformation relationship, which exists between b(t) and χ″(ω).     

Time dependence of isothermal magnetization in single-phase B1 MoCy encapsulated in multiwall carbon nanocages

Isothermal magnetization M(t) in nanocrystalline single-phase B1 MoC_y encapsulated in multiwall carbon nanocages is studied within the time window of 100 < t < 5000 s. The current density J exhibits a linear logarithmic time decay. The effective activation energy U_eff increases linearly with increasing temperature T, and decreases linearly with increasing J. The behaviors of J(t), U_eff(T), and U_eff(J) can be described by the Anderson–Kim flux-creep model for thermally activated motion of uncorrelated vortices or vortex bundles over a net potential barrier U_eff. The slower relaxation of current density above the broad peak field in the isothermal magnetization curves suggests that the peak is a result of vortex dynamics.     

Scaling and memory effect in volatility return interval of the Chinese stock market

We investigate the probability distribution of the volatility return intervals τ for the Chinese stock market. We rescale both the probability distribution P_q(τ) and the volatility return intervals τ as to obtain a uniform scaling curve for different threshold value q. The scaling curve can be well fitted by the stretched exponential function , which suggests memory exists in τ. To demonstrate the memory effect, we investigate the conditional probability distribution P_q(τ|τ₀), the mean conditional interval 〈τ|τ₀〉 and the cumulative probability distribution of the cluster size of τ. The results show clear clustering effect. We further investigate the persistence probability distribution P_±(t) and find that P₋(t) decays by a power law with the exponent far different from the value 0.5 for the random walk, which further confirms long memory exists in τ. The scaling and long memory effect of τ for the Chinese stock market are similar to those obtained from the United States and the Japanese financial markets.     

A multi-channel scattering theory for some time dependent Hamiltonians,charge transfer problem

Scattering theory for time dependent HamiltonianH(t)=?(1/2) Δ+ΣV _j (x?q _j (t)) is discussed. The existence, asymptotic orthogonality and the asymptotic completeness of the multi-channel wave operators are obtained under the conditions that the potentials are short range: |V _j (x)|≦C _j (1+|x|)^?2?ε, roughly spoken; and the trajectoriesq _j (t) are straight lines at remote past and far future, and |q _j (t)?q _k (t)| → ∞ ast → ± ∞ (j≠k).     

Change in the shape of a symmetric light pulse passing through a quantum well

A theory for the response of a 2D two-level system to irradiation by a symmetric light pulse is developed. Under certain conditions, such an electron system approximates an ideal solitary quantum well in a zero field or a strong magnetic field H perpendicular to the plane of the well. One of the energy levels is the ground state of the system, while the other is a discrete excited state with energy ?ω₀, which may be an exciton level for H=0 or any level in a strong magnetic field. It is assumed that the effect of other energy levels and the interaction of light with the lattice can be ignored. General formulas are derived for the time dependence of the dimensionless “coefficients” of the reflection ?(t), absorption A(t), and transmission ?(t) for a symmetric light pulse. It is shown that the ?(t), A(t), and ?(t) time dependences have singular points of three types. At points t ₀ of the first type, A(t ₀)=T(t ₀)=0 and total reflection takes place. It is shown that for γ_r?γ, where γ_r and γ are the radiative and nonradiative reciprocal lifetimes, respectively, for the upper energy level of the two-level system, the amplitude and shape of the transmitted pulse can change significantly under the resonance ω_l=ω₀. In the case of a long pulse, when γ_l<γ_r, the pulse is reflected almost completely. (The quantity γ_l characterizes the duration of the exciting pulse.) In the case of an intermediate pulse duration γ_l?γ_r, the reflection, absorption, and transmission are comparable in value and the shape of the transmitted pulse differs considerably from the shape of the exciting pulse: the transmitted pulse has two peaks due to the existence of the point t ₀ of total reflection, at which the transmission is zero. If the carrier frequency ω_l of light differs from the resonance frequency ω₀, the oscillating ?(t), A(t), and ?(t) time dependences are observed at the frequency Δω=ωl?ω₀. Oscillations can be observed most conveniently for Δω?γ_l. The position of the singular points of total absorption, reflection, and transparency is studied for the case when ω_l differs from the resonance frequency.     

On Asymptotics for the Airy Process

The Airy process t→A(t), introduced by Prähofer and Spohn, is the limiting stationary process for a polynuclear growth model. Adler and van Moerbeke found a PDE in the variables s ₁,s ₂, and t for the probability Pr(A(0)≤s ₁, A(t)≤s ₂). Using this they were able, assuming the truth of a certain conjecture and appropriate uniformity, to obtain the first few terms of an asymptotic expansion for this probability as t→∞, with fixed s ₁ and s ₂. We shall show that the expansion can be obtained by using the Fredholm determinant representation for the probability. The main ingredients are formulas obtained by the author and C. A. Tracy in the derivation of the Painlevé II representation for the distribution function F ₂ plus a few others obtained in the same way.     

Dislocation dynamics in pulse-loaded NaCl crystals

The mean path l of freshly introduced dislocations in NaCl crystals under the effect of triangular loading pulses τ(t) is shown to depend only on the pulse amplitude τ_m and be insensitive to the rate of stress growth. The replacement of triangular pulses with trapezoidal ones with a constant-load plateau (τ=const) extension of up to 60 min only insignificantly changes the l(τ_m) dependence. The data obtained are interpreted on the basis of the concept of quasi-static relaxation in a nonequilibrium system of dislocations subject to a combined effect of time-dependent applied stresses τ(t), coordinate-dependent internal stresses τ_i(x), and “dry friction” τ_p due to the pinning of dislocations by point defects. In such a model, the l(τ_m) dependence should saturate at τ_m<2τ_p; this is in fact observed in the range of 0.2τ_m<τ_m<0.3τ_y (here, τ_y is the yield stress), which gives an estimate for the pinning stress τ_p≈0.1τ_y. Based on the model suggested, a series of experimentally confirmed predictions were obtained, e.g., a recipe of “switching-off” of anomalies. Thus, a preliminary treatment of the samples by a series of stress pulses or holding in a magnetic field, which transforms the system of fresh dislocations into a more equilibrium state, sharply decreases the density of mobile dislocations quasi-statically responding to a pulsed load. It is shown that the discussed anomalies of dislocation mobility should be observed only in sufficiently pure crystals, where , and should be absent in contaminated crystals, where τ_p ～ τ_y.     

The semihard processes γγ→ΨX, γγ→ΨΨ, γγ→ϱΨ

The cross sections for Ψ-production in the semihard collisions are calculated at s ⪢ |t|, m_Ψ² and at not too large a jet effective mass m_X²≲m_Ψ² + |t| for the process γγ→ΨX. We found these cross sections in the lowest nontrivial QCD order (two-gluon exchange in the t-channel): σ(γγ→ΨX) ≈ 10³⁵ cm², σ(γγ→ΨΨ)=2×10⁻³⁶ cm². The polarization of Ψ in these processes is transversal (contrary to the longitudinal one of light mesons ϱ⁰, ω, ϕ in similar processes).     

Percolation magnetoresistance in ferromagnetic superconductors

It is shown that the magnetoresistance (as a function of magnetic field H) in polycrystalline magnetic superconductors has the percolation character which is the consequence of the anisotropy of magnetic susceptibility. The magnetoresistance R_m(H), and the upper critical field H_c2(T) of ErRh₄B₄ are evaluated and compared with the experimental data.     

Coevolutionary extremal dynamics on gasket fractal

We considered a Bak-Sneppen model on a Sierpinski gasket fractal. We calculated the avalanche size distribution and the distribution of distances between subsequent minimal sites. To observe the temporal correlations of the avalanche, we estimated the return time distribution, the first-return time, and the all-return time distribution. The avalanche size distribution follows the power law, P(s)∼s−τ, with the exponent τ=1.004(7). The distribution of jumping sites also follows the power law, P(r)∼r−π, with the critical exponent π=4.12(4). We observe the periodic oscillation of the distribution of the jumping distances which originated from the jumps of the level when the minimal site crosses the stage of the fractal. The first-return time distribution shows the power law, Pf(t)∼t−τf, with the critical exponent τf=1.418(7). The all-return time distribution is also characterized by the power law, Pa(t)∼t−τa, with the exponent τa=0.522(4). The exponents of the return time satisfy the scaling relation τf+τa=2 for τf?2.     

The correlation function for a quantum oscillator in a low-temperature heat bath

The momentum autocorrelation function c(t) for a quantum oscillator coupled with harmonic forces to a heat bath of oscillators is calculated at low temperatures. It is found that c(t) contains two distinct terms: one, the zero-point contribution c₀(t), is temperature independent, and the other, c₁(t), does depend on temperature. We concentrate our attention on the low-temperature case. An expression for c₁(t) is obtained, which is valid for arbitrary strenghts of the coupling and for arbitrary times. It is shown that c₁(t) is governed by the low-frequency behaviour of F(λ) = A²(λ)ϱ(λ), where ϱ(λ) is the density of normal modes and A(λ) is the central-oscillator component of the λth normal mode; other details of the problem are irrelevant. It is found that c₁(t) decays in time as an inverse-power law, with a relaxation time t_q ≈ ħ/kT.     

On the time and cell dependence of the coarse-grained entropy. I

We consider a finite, thermally isolated, classical system which passes from an equilibrium state $\text{A}$ by the removal of an internal constraint to another equilibrium state $\text{B}$ after an empirical relaxation time. In the phase space of the system, cells are introduced according to the set of measuring instruments used and their experimental inaccuracies. It is shown that the coarse-grained entropy S_cg(t) tends to its new equilibrium value in general faster than the expectation values of the macroscopic variables to their new equilibrium values. We then investigate the dependence of S_cg(t) on the size of the phase cells. For fixed t, we find a lower bound on S_cg(t) by doing to the limit of infinite accuracy of the measuring instruments. In the limit t → ∞, this lower bound on S_cg(t) also converges to the equilibrium entropy of $\text{B}$. These properties strongly support the opinion that S_cg(t) is a proper microscopic expression for the entropy for equilibrium and nonequilibrium. Finally, explicit calculations of S_cg(t) for the model of a point particle enclosed in a one-dimensional box are presented which confirm the general results.