Ballistic diffusion induced by non-Gaussian noise

In this letter,we have analyzed the diffusive behavior of a Brownian particle subject to both internal Gaussian thermal and external non-Gaussian noise sources.We discuss two time correlation functions C(t) of the non-Gaussian stochastic process,and find that they depend on the parameter q,indicating the departure of the non-Gaussian noise from Gaussian behavior:for q ≤ 1,C(t) is fitted very well by the first-order exponentially decaying curve and approaches zero in the longtime limit,whereas for q 1,C(t) can be approximated by a second-order exponentially decaying function and converges to a non-zero constant.Due to the properties of C(t),the particle exhibits a normal diffusion for q ≤ 1,while for q 1 the non-Gaussian noise induces a ballistic diffusion,i.e.,the long-time mean square displacement of the free particle reads [x(t)-x(t)]2∝t2.     

The correlation function of the Ising model in the MFA and their sum rule

It is shown that in the correctly performed molecular field approximation the correlation function 〈S(q) S(-q)〉 fulfills the sum rule N^-1Σ_q〈S(q) S(-q)〉 = 1. This can be proved for ferro- and antiferromagnets and the disordered phases of o-hydrogen.     

A generalized Faxén theorem for two-dimensional Brownian motion

The induced force method developed by Mazur and Bedeaux is applied here to two-dimensional Brownian motion. We obtain a generalized Faxén theorem which reduces to the Stokes-Basset drag force on a nonuniformly moving cylinder or disk in the special case where the fluid fluctuations are neglected. The resulting modified Langevin equation is solved numerically for the velocity autocorrelation Ø(t) and the expected long time result Ø(t)～1/t is obtained. It is perhaps surprising that the short time behavior of Ø(t) deviates considerably from that predicted on the basis of a modified Langevin equation incorporating the classic Oseen-Lamb drag force on a cylinder.     

Continuous sample paths in quantum field theory

Nelson's free Markoff field on ? ^l+1 is a natural generalization of the Ornstein-Uhlenbeck process on ?¹, mapping a class of distributions φ(x,t) on ? ^l ×?¹ to mean zero Gaussian random variables φ with covariance given by the inner product \(\left( {\left( {m^2 - \Delta - \frac{{\partial ^2 }}{{\partial t^2 }}} \right)^{ - 1} \cdot , \cdot } \right)_2 \) . The random variables φ can be considered functions φ〈q〉=∝ φ(x,t)q(x,t)d x dt on a space of functionsq(x,t). In the O.U. case,l=0, the classical Wiener theorem asserts that the underlying measure space can be taken as the space of continuous pathst →q(t). We find analogues of this, in the casesl>0, which assert that the underlying measure space of the random variables φ which have support in a bounded region of ? ^l+1 can be taken as a space of continuous pathst →q(·,t) taking values in certain Soboleff spaces.     

Consistent MFA correlation functions of Ising models for all temperatures

A correct calculation of the Ising model correlation function C(q) = 〈(S(q) ? 〈S(q)〉) (S(-q) ? 〈S(-q)〉)〉 in the MFA results in

$\text{C(q)=}\text{〈S}{}^2\text{〉?〈〉}{}^2\text{1?(〈S}{}^2\text{〉?〈S〉}{}^2\text{βJ(q}\text{1}\text{N}\text{∑}\text{q}\text{1}\text{1?〈S〉}{}^2\text{βJ(q)}{}^{\mathrm{?1}}\text{C(q) fulfills the exact sum rule N}{}^{-1}\text{Σ}{}_{\mathrm{q}}\text{C(q) = 〈S}{}^2\text{〉 ? 〈S〉}{}^2$

. Previous literature supposed a violation of this sum rule to be a characteristic disadvantage of this approximation.     

Second moment of multiple-quantum NMR and a time-dependent growth of the number of multispin correlations in solids

The time evolution of multispin correlations (the growth of the number of correlated spins as a function of time) can be observed directly using the multiple-quantum nuclear magnetic resonance spectroscopy of solids. A quantity related to this number, namely, the second moment 〈n ²(t)〉 of the intensity distribution of coherences of different orders in the multiple-quantum spectrum can be calculated using the theory proposed in this work. An approach to the calculation of the four-spin time correlation function through which this moment is expressed is developed. The main sequences of contributions in the expansion of this function into a time power series are summed using the approximation of a large number of neighbors both for systems with a secular dipole-dipole interaction and for systems with a nonsecular effective interaction. An exponential dependence of 〈n ²(t)〉 is obtained. The value of 〈n ²(t)〉 is additionally calculated using an expansion in terms of orthogonal operators for three model examples corresponding to different limiting realizations of spin systems. It is shown that the results of the microscopic theory at least qualitatively agree with both the results obtained for model examples and experimental results obtained recently for adamantane.     

Hierarchy of mean multiplicities in pn and π+n interactions at 195 GeV/c

The dependence of the average charged multiplicity, 〈n_X〉, of the system X on |t| and M²_X is studied for the reactions p(π⁺)n→p_fast(p⁺_fast) + X and p(π⁺)n→p_slow + X at 195 GeV/c. For a fixed M²_X, 〈n_X〉 changes significantly among these reactions, yielding a systematic hierarchy of mean multiplicities. The results indicate that a proton emits on the average less charged particles than a pion and more than a neutron.     

Exact Results for Conditional Means of a Passive Scalar in Certain Statistically Homogeneous Flows

For a passive scalar T(r, t) randomly advected by a statistically homogeneous flow, the probability density function (pdf) of its fluctuation can in general be expressed in terms of two conditional means: 〈?² T|T〉 and 〈|?T|²|T〉. We find that in some special cases, either one of the two conditional means can be obtained explicitly from the equation of motion. In the cases when there is no external source and that the normalized fluctuation reaches a steady state or when a steady state results from a negative damping, 〈?² T|T〉=?(〈|?T|²〉/〈T ²〉)T. The linearity of the conditional mean in these cases follows directly from the fact that the advection equation of a passive scalar is linear. On the other hand, when the scalar is supported by a homogeneous white-in-time external source, 〈|?T|²|T〉=〈|?T|²〉.     

Slow dynamics in glycerol: collective de gennes narrowing and independent angstrom motion

The slow dynamics of microscopic density correlations in supercooled glycerol was studied by time-domain interferometry using ⁵⁷Fe-nuclear resonant scattering gamma rays of synchrotron radiation. The dependence of the relaxation time at 250 K on the momentum transfer q is maximum near the first peak of the static structure factor S(q) at q ～ 15 nm ^?1. The q-dependent behavior of the relaxation time known as de Gennes narrowing was confirmed in glycerol. Conversely, de Gennes narrowing around the second and third peaks of S(q) at q ～ 26 nm ^?1 and 54 nm ^?1 was not detected. The q dependence of the relaxation time was found to follow a power-law equation with power-law index of 1.9(2) in the q region well above the first peak of S(q) up to ～ 60 nm ^?1, which corresponds to angstrom scale, within experimental error. This suggests that in the angstrom-scale dynamics of supercooled glycerol, independent motions dominate over collective motion.     

Exactly Complete Solutions of the Pseudoharmonic Potential in N-Dimensions

We present analytically the exact solutions of the Schrödinger equation in the N-dimensional spaces for the pseudoharmonic oscillator potential by means of the ansatz method. The energy eigenvalues of the bound states are easily calculated from this eigenfunction ansatz. The normalized wavefunctions are also obtained. A realization of the ladder operators for the wavefunctions is studied and we deduced that these operators satisfy the commutation relations of the generators of the dynamical group SU(1,1). Some expectation values for 〈r ^?2〉, 〈r ²〉, 〈T〉, 〈V〉, 〈H〉, 〈p ²〉 and the virial theorem for the pseudoharmonic oscillator potential in an arbitrary number of dimensions are obtained by means of the Hellmann–Feynman theorems. Each solution obtained is dimensions and parameters dependent.     

Time evolution of a fractal distribution: Particle concentrations in free-surface turbulence

Steady-state turbulence is generated in a tank of water and the trajectories of particles forming a compressible system on the surface are tracked in time. The initial uniformly distributed floating particles coagulate and form a fractal structure, a rare manifestation of a strange attractor observable in real space. The surface pattern reaches a steady state in approximately 1 s. Measurements are made of the fractal dimensions D_q(t) (q=1 to 6) of the floating particles starting with the uniform distribution D_q(0)=2 for Taylor Microscale Reynolds number Re_λ?160. Focus is on the time evolution of the correlation dimension D₂(t) as the steady state is approached. This steady state is reached in several large eddy turnover times and does so at an exponential rate.     

Quantum transport in a time-dependent random potential with a finite correlation time

Using an earlier density matrix formalism in momentum space we study the motion of a particle in a time-dependent random potential with a finite correlation time τ, for 0 < t ? τ. Within this domain we consider two subdomains bounded by kinetic time scales (t _c ² = 2m? ^-1 c ², c ² = σ ², ξ ², σξ, with 2σ the width of an initial wavepacket and the correlation length of the gaussian potential fluctuations), where we obtain power law scaling laws for the effect of the random potential in the mean squared displacement 〈x ²〉 and in the mean kinetic energy 〈E _kin〉. At short times, ? min (t _σ ², 1/2t _ξ ²), 〈x ²〉 and 〈E _kin〉 scale classically as t ⁴ and t ², respectively. At intermediate times, t _σξ ? t ? 2t _σ ² and 1/2t _ξ ² ? t ? t _σξ, these quantities scale quantum mechanically as t ^3/2 and as √t, respectively. These results lie in the perspective of recent studies of the existence of (fractional) power law behavior of 〈x ²〉 and 〈E _kin〉 at intermediate times. We also briefly discuss the scaling laws for 〈x ²〉 and 〈E _kin〉 at short times in the case of spatially uncorrelated potential.     

An analysis of the new pion form factor data

A form factor F(t) for the pion is constructed which is compatible with analyticity and the data in the space-like and time-like region. For the mean square pion radius 〈r²〉 = (0.46_?0.08^+0.06) fm² is obtained. Typical errors of the extrapolated F(t) are given (e.g. F(t =?8 GeV²/c²) = 0.07_?0.10^+0.04). Assuming F(t) ≈ β/(?t)^α at the end of the space-like data region we obtain β = 0.31, α = 0.81 for t in GeV²/c² together with the error contours of (α, β). No conclusive answer on the existence of zeroes of F(t) can be given.     

Intermittency in the atmospheric surface layer: Unresolved or slowly varying?

We present streamwise velocity structure functions 〈δv_L(τ)〉=〈|v(t+τ)−v(t)|^p〉 (with p=1:5) obtained in the near neutral atmospheric surface layer at the Utah SLTEST site at the highest terrestrial Reynolds number Re_τ=O(10⁶). We show that the occurrence of very large scale coherent oscillations in the streamwise velocity throughout the wall region, interpreted as genuine structural features of the canonical turbulent boundary layer, affects the scaling exponents of the p>3 order structure functions. This results in a slight alteration of the intermittent behavior of the velocity field. It was found that for positive (fast) large scale oscillation of the low-pass filtered velocity signal, deviations from the Kolmogorov K41 prediction (absence of multiscaling) are more marked, as compared to negative (slow) excursion. The results are discussed in terms of convergence of statistics from atmospheric boundary layer measurements.     

Principles of generalized spin-echo spectroscopy

The concept is proposed for determining the total dynamic scattering function of an object under study, representing a sum of odd and even parts measured by the generalized neutron spin-echo method in the form of the signals S _odd(q, t) ～ ΣS(ω, q)sin(ωt)dω and S_even(q, t) ～ ΣS(ω,q)cos(ωt)dω as functions of the momentum q transferred to the neutron and the time t corresponding to the frequency ω and the transferred energy ?ω. The principle of the generalized spin echo and the results of mathematical modeling are confirmed in experiments on inelastic scattering on magnetic fluids and polymer solutions. The developed method makes it possible to study the features of the dynamics of atomic and molecular systems, e.g., to analyze soft vibrational spectra of nanoparticle ensembles against the background of intense relaxation processes, which is inaccessible for classical spin-echo spectrometry.     

Simple tests of quark-nucleon reciprocity

The inclusive cross-section for the production of a single hadron in deep inelastic electroproduction is studied in a dual resonance model. The Bjorken scaling behaviour in the virtual photon fragmentation region for finite x (≡ 2p_L^c.m./√s) is (1/σ^{T,L))d3σT,L/E?}d³p ～ (1/q²) F(x,p_⊥²/q²) and thus the transverse momentum grows like q², whereas in the parton model (1/σ^T,L)d³σ^T,L/E^?1d³p ～ F(x,p_⊥²). A related effect is the absence of two-jet structure in e⁺e^? annihilation. We believe that dual model results may give a more reliable indication of the deep inelastic behaviour for composite hadrons than the parton model.     

Spin autocorrelation of Ni in K2Mn0.975Ni0.025F4 studied by nuclear spin-lattice relaxation

From the nuclear spin-lattice relaxation of the out-of-layer ¹⁹F nuclei in magnetic fields perpendicular to the c-axis the low-frequency component of the autocorrelation function 〈S^z(t)S^z(O)〉 of Ni in ordered K₂Mn_0.975Ni_0.025F₄ is found to be substantially reduced relative to the Mn host. The experimental rates vs temperature are in accord with those for relaxation involving two spin excitations calculated with local Green's functions.     

A study of self-organizing processes of nonlinear stochastic variables

A general theory is given for the time evolution of nonlinear stochastic variables a(t) = {a_i(t)} whose statistical distribution is changing due to the self-organization of “macroscopic” order. The dynamics of a(t) is conveniently expressed by self-consistent equations for the ensemble average x(t) = 〈a(t)〉, the supersystem, and for the deviations ξ(t) = a(t)?x(t), the subsystem; the systems are connected to each other by feedback loops in their dynamics. The time dependence of the variance and the correlation function ofξ(t) are studied in terms of relaxation toward local equilibrium underx(t) and dynamical coupling withx(t). A special example shows that the stochastic motions of subsystems are pulled together by the motion of the supersystem through feedback loops, and that this pull-together phenomenon occurs when symmetry-breaking instability exists in nonlinear systems.     

Long-time relaxation phenomena in Pb<Subscript>0.94</Subscript>Ba<Subscript>0.06</Subscript>Sc<Subscript>0.5</Subscript>Nb<Subscript>0.5</Subscript>O<Subscript>3</Subscript> (PBSN-6) single crystals

Long-time polarization relaxation in the temperature range where PBSN-6 single crystals reside in the relaxor state was studied. An analysis of the time dependence of the permittivity ε′(t) performed at measuring frequencies from 1 Hz to 1 kHz in weak electric fields E₀ showed that the relaxation (or freezing) times derived by extrapolating relations of the type ε′(t) ～ log(t/t₀) and ε′(t) ～ exp{?[ln(t/t₀)]_β} range from 10⁸ to 10¹¹ min and depend substantially on the bias voltage applied to the sample. A study of the pattern of the dielectric response in moderate and strong infralow-frequency fields revealed that, after a sample was maintained under a bias lower than the coercive force, it no longer exhibited the additional anomalies in the amplitude dependences of the effective loss tangent tan_eff(E₀) than were observed in a thermally recuperated sample.     

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).