A mode coupling theory of higher order time correlation functions

Kawasaki's mode coupling theory [Ann. Phys. 61 (1970) 1] is used to compute time correlation functions of the form 〈A_k0(t₀) … A_kn(t_n)〉, where A_k(t) represents some slowly varying quantity. The Gaussian and Bare Vertex approximations are made, thus yielding extremely simple expressions for these higher order correlation functions. These do not contain any bare transport coefficients and suggest relatively simple tests by which the theory could be checked. Examples relating to light scattering in nonequilibrium systems and the hydrodynamics of simple fluids are presented.     

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.     

Some estimates for magnetic fluctuations in a turbulent medium

A new integral relationship between the fluctuations b(r, t) of a magnetic field and its mean B ₀(r, t) is derived for the steady-state magnetic field in a turbulent medium. This formula provides the estimate 〈b?curlb〉=?B ₀?curlB ₀. Simultaneously, the coefficient of amplification of the mean magnetic field α effect) is obtained: α=(η+β)B ₀? curlB ₀/B ₀ ² . The formula for α allows for a decrease in this coefficient owing to the back action of the magnetic field on the turbulent velocity field. It is shown that the Zel’dovich’s estimate 〈 b ²〉?β/η B ₀ ² for two-dimensional turbulence holds for magnetic fields at the instant the fluctuations 〈a ²〉 of the vector potential, rather than 〈b ²〉, reach a maximum. Here, η and β are the ohmic (molecular) and turbulent diffusion coefficients, respectively. This estimate is refined with allowance made for the fact that the condition for diffusion approximation itself relates the β, b, and B ₀ quantities to each other.     

Classical Motion in Force Fields with Short Range Correlations

We study the long time motion of fast particles moving through time-dependent random force fields with correlations that decay rapidly in space, but not necessarily in time. The time dependence of the averaged kinetic energy 〈p ²(t)〉/2 and mean-squared displacement 〈q ²(t)〉 is shown to exhibit a large degree of universality; it depends only on whether the force is, or is not, a gradient vector field. When it is, 〈p ²(t)〉～t ^2/5 independently of the details of the potential and of the space dimension. The stochastically accelerated particle motion is then superballistic in one dimension, with 〈q ²(t)〉～t ^12/5, and ballistic in higher dimensions, with 〈q ²(t)〉～t ². These predictions are supported by numerical results in one and two dimensions. For force fields not obtained from a potential field, the power laws are different: 〈p ²(t)〉～t ^2/3 and 〈q ²(t)〉～t ^8/3 in all dimensions d≥1.     

Sum rules for the bound-free transition form factors in hydrogenlike atoms

The product of two bound-free transition form factors of the type 〈k¦exp(i q·r)¦nlm〉 is analytically summed-up over all the degenerate stateslm. Using the genuine Coulomb wave 〈r¦k〉, together with the hydrogenlike wavefunction 〈r¦nlm〉, exact analytical expressions are derived for the sum rule. The results are conveniently expressed in terms of the finite linear combination of easily generated Appell and Lauricella hypergeometric polynomials of two and three variables. This highly desirable sum rule is most frequently required in broad applications within the distorted wave theory of particle-atom scattering.     

Magnetic Resonance Imaging of Diffusion in the Presence of Background Gradients and Imaging of Background Gradients

The measurement of the self-diffusion coefficient D by an NMR technique that uses an applied gradient G_A can be corrupted by systems that have a background magnetic field gradient G₀ and also by imaging gradients G₁, when used in an imaging mode. In a nonimaging mode, the corrupting cross term G_A · G₀can be eliminated in the diffusion measurement by use of an alternating-pulse-field-gradient (APFG) sequence that allows an accurate and uncorrupted measurement of D. A Carr-Purcell echo train enables the measurement of the expectation value, 〈DG²₀〉; assuming D and G₀ to be uncorrelated will allow 〈G²₀〉 to be determined. An image of D or of 〈DG²₀〉 may be obtained without the corrupting G_A · G₀ and G_A · G_I terms by appending a standard imaging sequence to an APFG sequence or a Carr-Purcell sequence, respectively; assuming D and G₀ to be uncorrelated will allow (〈G²₀〉 to be determined within each pixel. Measurements of D and 〈G²₀〉 and their images are made in apple flesh in which minute air bubbles are shown to produce the large 〈G²₀〉. Their values in an 81 g Golden Delicious apple at a measuring frequency of 100 MHz were D = 1.42 × 10⁻⁵ cm²/s and [formula] = 8.9 G/cm.     

Entropy generation in curved spaces as a diagnostic for particle creation

This paper defines a time-dependent entropy S(t) for a quantum field in a background cosmological spacetime, changes in which are connected directly with changes in the average particle number 〈n_p(t)〉 in each mode. Here the existence of an arrow of time, ds/dt > 0 and d〈n_p〉/dt>0, would not reflect the fact that the Universe is expanding, but, instead, the fact that the Universe started from a special state characterized either (a) very nearly by eigenstates of number, such as the vacuum, or (b) more generally, by very nearly random phases.     

Large-scale structure of isotropic homogeneous turbulence

It is shown that the longitudinal correlation functionf is asymptotically proportional tor ^?3 asr→∞ and the energy spectrum function is asymptotically proportional toκ ² asκ→0 if and only if 0<〈(f u d ³ x)·u〉<∞. Moreover, the latter finiteness condition is shown to be essentially equivalent to 〈(fy·ud ³ x)²〉<∞ for nonstochasticyεL ²(R₃). Confirmed by recent experimental measurements, the larger dependencef ∝r ^?3 is concomitant with anO(r ^?6)=O(f ²) fall-off of the viscous force term in the Kármán-Howarth equation.     

Some exact solutions of the local induction equation for the motion of a vortex in a Bose–Einstein condensate with a Gaussian density profile

The dynamics of a vortex filament in a Bose–Einstein condensate whose equilibrium density in the reference frame rotating at the angular velocity Ω is Gaussian with the quadratic form r·D?r has been considered. It has been shown that the equation of motion of the filament in the local-induction approximation permits a class of exact solutions in the form R(β, t) = βM(t) + N(t) of a straight vortex, where β is the longitudinal parameter and is the time. The vortex slips over the surface of an ellipsoid, which follows from the conservation laws N · D?N=C ₁ and M · D?N=C ₀=0. The equation of the evolution of the tangential vector M(t) appears to be closed and has integrals of motion M ·D?M=C ₂ and (|M| ? M· G?Ω) = C, with the matrix G? = 2(I?TrD? ? D?)^?1. Crossing of the respective isosurfaces specifies trajectories in the phase space.     

Computable upper bounds for the adiabatic approximation errors

For a given Hermitian Hamiltonian H(s)(s∈[0,1])with eigenvalues Ek(s)and the corresponding eigenstates|Ek(s)(1 k N),adiabatic evolution described by the dilated Hamiltonian HT(t):=H(t/T)(t∈[0,T])starting from any fixed eigenstate|En(0)is discussed in this paper.Under the gap-condition that|Ek(s)-En(s)|λ0 for all s∈[0,1]and all k n,computable upper bounds for the adiabatic approximation errors between the exact solution|ψT(t)and the adiabatic approximation solution|ψadi T(t)to the Schr¨odinger equation i|˙ψT(t)=HT(t)|ψT(t)with the initial condition|ψT(0)=|En(0)are given in terms of fidelity and distance,respectively.As an application,it is proved that when the total evolving time T goes to infinity,|ψT(t)-|ψadi T(t)converges uniformly to zero,which implies that|ψT(t)≈|ψadi T(t)for all t∈[0,T]provided that T is large enough.     

Resonant paramagnetic enhancement of the thermal and zero-point Nyquist noise

The interaction between a very thin macroscopic solenoid, and a single magnetic particle precessing in a external magnetic field B₀, is described by taking into account the thermal and the zero-point fluctuations of stochastic electrodynamics. The inductor belongs to a RLC circuit without batteries and the random motion of the magnetic dipole generates in the solenoid a fluctuating current I_dip(t), and a fluctuating voltage ε_dip(t), with spectral distribution quite different from the Nyquist noise. We show that the mean square value 〈I_dip²〉 presents an enormous variation when the frequency of precession approaches the frequency of the circuit, but it is still much smaller than the Nyquist current in the circuit. However, we also show that 〈I_dip²〉 can reach measurable values if the inductor is interacting with a macroscopic sample of magnetic particles (atoms or nuclei) which are close enough to its coils.     

Moments of the spectra for rotational transitions induced by collisions or by external perturbations

Considered first is the cross sectiond σ(J′←J)/dω for the rotational transitionJ′←J of a linear rotator in collisions. The set of cross sections for a givenJ and for allJ′ defines a discrete spectral distribution as a function of the transition energy ΔE=E(J′)?E(J). In both the classical and the quantum mechanical sudden approximations the moments \(S(\mu ;J) = \sum\limits_{J'} {(\Delta {\rm E}} )^\mu d\sigma (J' \leftarrow J)/d\omega \) of ordersμ=2n andμ=2n+1 (n=0,1,...) are polynomials of degreen in the rotational energyE(J). The special cases forn=0 indicate thatS(0;J) andS(1;J) are independent ofJ. This is a theorem proved elsewhere. Explicit expressions forS(μ; J) of low orders are presented. They are derived on the assumption of equal relative velocitiesv andv′ before and after the collision. Correction onS(μ;J) for the difference betweenv andv′ is made to the lowest order in terms ofS(μ+1;J) forv=v′. The quantum mechanical results for linear rotators are extended to spherical-top and symmetric-top rotators. These results apply not only to the collision cross section but also to a physical quantity expressible in the form ¦〈Γ′¦F¦Γ〉¦² of a transition probability withany F that is independent of the initial and final rotational states ¦Γ〉 and ¦Γ′〉.     

Diamagnetism and the dispersion of the magnetic permeability

It is well known that the usual Kramers–Kronig relations for the relative permeability function μ(ω) are not compatible with diamagnetism (μ(0) < 1) and a positive imaginary part (Im μ(ω) > 0 for ω > 0). We demonstrate that a certain physical meaning can be attributed to μ for all frequencies, and that in the presence of spatial dispersion, μ does not necessarily tend to 1 for high frequencies ω and fixed wavenumber k. Taking the asymptotic behavior into account, diamagnetism can be compatible with Kramers–Kronig relations even if the imaginary part of the permeability is positive. We provide several examples of diamagnetic media and metamaterials for which μ(ω, k) ?  1 as ω →∞.     

Diffusion in very chaotic hamiltonian systems

We study nonintegrable hamiltonian dynamics: H(I,θ) = H₀(I) + kH₁(I,θ), for large k, that is, far from integrability. An integral representation is given for the conditional probability P(I,θ, t¦I₀, θ₀, t₀) that the system is at I, θ at t, given it was at I₀, θ₀ at t₀. By discretizing time into steps of size ?, we show how to evaluate physical observables for large k, fixed ?. An explicit calculation of a diffusion coefficient in a two degrees of freedom problem is reported. Passage to ? = 0, the original hamiltonian flow, is discussed.     

Theoretical integrated intensities for the 2ν2 and 2ν2-ν2 bands of HCN and DCN

Dipole moment functions, both perpendicular and parallel to the molecular axis, are calculated from the SCF and MRD-CI results of a previous study for the normal ν₂ bending vibrations of HCN and DCN. Vibrationally averaged dipole moments and the infrared transition matrix elements are then obtained from the dipole moment functions and vibrational wave functions. MRD-CI results, with known experimental values in parentheses, for HCN are 〈0|μ|0〉 = ?2.954(?2.985) D, 〈1|μ|1〉 = ?2.915(±2.942) D, 〈0|μ|1〉 = 0.148(0.147) D, 〈0|μ|2〉 = ?0.027 D, 〈1|μ|2〉 = 0.210 D. Calculated absolute intensities at 1 atm and 0°C for the (02⁰0) ← (000), (02⁰0) ← (010), and (02²0) ← (010) bands of HCN are 25 (40 ± 10 as estimated from spectra), 8.5, and 17.0 atm^?1 cm^?2, respectively. Results for DCN are also reported.     

New bounds for dilepton production from heavy neutral leptons

Bounds on 〈E_?ⁿ〉/〈E₊ⁿ〈, 〉E₊E_?〈/〉E₂²〈 and 〈E₊E_?〉/〈E₊〉〈E_?〉 are direved for the processes ν_μN → μ^?μ⁺(e⁺) + X and μN → μ^?μ⁺ + X if dileptons are mediated by a $\text{spin}\text{-}\text{1}\text{2}$ heavy neutral lepton L₀. The bounds are shown to be independent of the production mechanism and mass of L₀. Useful conditional bounds are obtained relating the bounded quantities, which give information about the structure of the weak current responsible for L₀ decay.     

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.     

Correlation functions of the transverse Ising chain at the critical field for large temporal and spatial separations

We study the behavior of 〈σ₀^x(t)σ_n^x(0)〉 and 〈σ₀^y(t)σ_n^y(0)〉 for the transverse Ising chain at the critical magnetic field at T = 0. Explicit results are obtained for the three distinct regions where t → ∞ and n → ∞with $\text{0 ?}\text{n}\text{t}\text{<1}$, $\text{1 <}\text{n}\text{t}$, or $\text{t = n + n}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{(}\text{z}\text{2}\text{)}$ where z is fixed of order one. In this latter region the general Painlevé V solution is shown to reduce to a Painlevé II function. We use our results to discuss the general problem of long-time behavior of Toda equations with slowly decaying initial values.     

Isomer shifts and changes of the nuclear charge radii in99Ru and101Ru

The recoilless nuclear gamma resonance of the 127 keV γ-rays of¹⁰¹Ru was observed in ruthenium metal, RuO₂ and [Ru(NH₃)₄(HSO₃)₂]. By comparison of the isomer shifts observed in these materials for the 127 keV absorption line with the corresponding shifts of the 90keV γ-rays of⁹⁹Ru one obtains δ〈r ²〉 [127 keV]/ δ〈r ²〉 [90 keV]=1.78±0.26 for the ratio of the changes of the mean square nuclear charge radii between the first excited and the ground states in these nuclei. An estimate of electron density differences based on free-ion relativistic self-consistent field calculations yields δ〈r ²〉[90keV]≈+1.4·10^?3 for⁹⁹Ru and δ〈r ²〉/〈r ²〉 [127 keV]≈+2.4·10^?3 for the¹⁰¹Ru case. These results are discussed in terms of the core excitation model.