On quantum analogue of dynamical stabilisation of inverted harmonic oscillator by time periodical uniform field

Quantum analogue of stabilised forced oscillations around an unstable equilibrium position is explored by solving the non-stationary Schrödinger equation (NSE) of the inverted harmonic oscillator (IHO) driven periodically by spatial uniform field of frequency \(\Omega \), amplitude \(F_{0}\) and phase \(\phi \), i.e. the system with the Hamiltonian of \(\hat{{H}}=(\hat{{p}}^{2}/2m)-(m\omega ^{2}x^{2}/2)-F_0 x\sin \) \(\left( {\Omega t+\phi } \right) \). The NSE has been solved both analytically and numerically by Maple 15 in dimensionless variables \(\xi = x\sqrt{m\omega /\hbar }\hbox {, }f_0 =F_0 /\omega \sqrt{\hbar m\omega }\) and \(\tau =\omega t\). The initial condition (IC) has been specified by the wave function (w.f.) of a generalised Gaussian type which suits well the corresponding quantum IC operator. The solution obtained demonstrates the non-monotonous behaviour of the coordinate spreading \(\sigma \left( \tau \right) \hbox { =}\sqrt{\big ( {\overline{\Delta \xi ^{2}\big ( \tau \big )} } \big )}\) which decreases first from quite macroscopic values of \(\sigma _{0} =2^{12,\ldots ,25}\) to minimal one of \(\sim \!(1/\sqrt{2})\) at times \(\tau <\tau _0 =0.125\ln \!\left( {16\sigma _0^4 +1} \right) \) and then grows back unlimitedly. For certain phases \(\phi \) depending on the \(\Omega /\omega \) ratio and \(n=\log _2\!\sigma _0 \), the mass centre of the packet \(\xi _{\mathrm {av}}( \tau )= \overline{\hat{{x}}(\tau )} \cdot \sqrt{m\omega /\hbar }\) delays approximately two natural ‘periods’ \(\sim \!(4\pi /\omega )\) in the area of the stationary point and then escapes to ‘\(+\)’ or ‘?’ infinity in a bifurcating way.  For ‘resonant’ \(\Omega =\omega \), the bifurcation phases \(\phi \) fit well with the regression formula of Fermi–Dirac type of argument n with their asymptotic \(\phi ( {\Omega ,n\rightarrow \infty } )\) obeying the classical formula \(\phi _{\mathrm {cl}} ( \Omega )=-\hbox {arctg} \, \Omega \) for initial energy \(E = 0\) in the wide range of \(\Omega =2^{-4},...,2^{7}\).     

Foam droplet separation for nanoparticle synthesis

A novel approach to nanoparticle synthesis was developed whereby foam bubble bursting produced aerosol droplets, an approach patterned after the marine foam aerosol cycle. The droplets were dried to remove solvent, leaving nanometer-sized particles composed of precursor material. Nanoparticles composed of sodium chloride (mean diameter, 100 nm), phosphotungstic acid ( 55 nm), and bovine insulin ( 5–30 nm) were synthesized. Foam droplet separation can be carried out at ambient temperature and pressure. The ‘soft’ nature of the process makes it compatible with a wide range of materials.     

Antiferromagnetic Ordering in TmCu2

Intermetallic compound TmCu₂ crystalizes in the orthorhombic structure, orders antiferromagnetically (AF) and exhibits four different AF phases below T _N = 6.5 K. In all these phases the Tm magnetic moment is parallel to the b-axis. The ground-state AF1 (T < 3.4 K) corresponds to the squared-up AF structure with propagation vector ₀₁ = (5/8, 0, 0) and magnetic unit cell 8a × b × c. Narrow phases AF2 and AF3 (3.3 K < T < 3.7 K and 3.6 K < T < 4.6 K) are 'spin-slip'-derivatives from the AF1 structure, yielding propagation vectors ₀₂ = (0.633, 0, 0.02) and ₀₃ = (0.618, 0, 0), respectively. The phase AF4 (4.4 K < T < T _N) is then incommensurate transversal spin-wave with the propagation ₀₄ = (0.64, 0, 0). The co-existence of AF phases indicates the first-order transitions between them.     

5D generalized inflationary cosmology

We consider a 5D Kaluza-Klein type cosmological model with the fifth coordinate being a generalization of the invariant historical time of the covariant theory of Horwitz and Piron. We distinguish between vacuum-, off-shell matter-, and on-shell matter-dominated eras as the solutions of the corresponding 5D gravitational field equations, and build an inflationary scenario according to which passage from the off-shell matter-dominated era to the on-shell one occurs, probably as a phase transition. We study the effect of this phase transition on the expansion rate in both cases of localO(4,1) andO(3,2) invariance of the extended (x ^µ,) manifold and show that it does not change in either case. The expansion of the model we consider is not adiabatic; the thermodynamic entropy is a growing function of cosmic time for the closed universe, and can be a growing function of historical time for the open and the flat universe. A complete solution of the 5D gravitational field equations is obtained for the on-shell matter-dominated universe. The open and the closed universe are shown to tend asymptotically to the standard 4D cosmological models, in contrast to the flat universe which does not have the corresponding limit. Finally, possible cosmological implications are briefly discussed.     

The van Kampen expansion for the Fokker-Planck equation of a duffing oscillator excited by colored noise

In Rodríguez and van Kampen's 1976 paper a method of extracting information from the Fokker-Planck equation without having to solve the equation is outlined. The Fokker-Planck equation for a Duffing oscillator excited by white noise is expanded about the intensity of the forcing function. In Weinstein and Benaroya, the effect of the order of expansion is investigated by carrying the expansion to a higher order. The effect of varying the system parameters is also investigated. All results are verified by comparison to Monte Carlo experiments. In this paper, the van Kampen expansion is modified and applied to the case of a Duffing oscillator excited by colored noise. The effect of the correlation time is investigated. Again the results are compared to those of Monte Carlo experiments. It is found that the expansion compares closely with those of the Monte Carlo experiments as the correlation time _c is varied from 0.001 to 10 sec. Examination of the results reveals that the colored noise can be categorized in one of four ways: (1) for the noise can be considered as white for all intents and purposes, (2) for the noise can be considered white for some purposes, (3) for the correlated nature of the noise must be considered in an analysis, and (4) for the noise can be considered as deterministic.     

Self-similarity and pattern selection in the roughening of binary liquid films

Films of spinodally decomposing binary liquid mixtures show transient wetting of both confining interfaces by one of the phases, and rupture, with characteristic wavelength lambda(c) and time tau(rupture), leading to flat droplets of the nonwetting phase encapsulated by the wetting phase. Over the entire range of film thicknesses d approximately 100-3500 nm, we find tau(rupture)~d(1.01+/-0.08), indicating film structures that scale self-similarly with d, and find also that lambda(c) approximately 60d(0.97+/-0.03), the large prefactor suggesting a rupture wavelength which minimizes the interfacial tension of the roughened film.     

Initially linear echo-spacing dependence of 1/T2 measurements in many porous media with pore-scale inhomogeneous fields

For a liquid sample with unrestricted diffusion in a constant magnetic field gradient g, the increase R in R2=1/T2 for CPMG measurements is 1/3(taugammag)2D, where gamma is magnetogyric ratio, tau is the half the echo spacing TE, and D is the diffusion constant. For measurements on samples of porous media with pore fluids and without externally applied gradients there may still be significant pore-scale local inhomogeneous fields due to susceptibility differences, whose contributions to R2 depend on tau. Here, diffusion is not unrestricted nor is the field gradient constant. One class of approaches to this problem is to use an "effective gradient" or some kind of average gradient. Then, R2 is often plotted against tau2, with the effective gradient determined from the slope of some of the early points. In many cases, a replot of R2 against tau instead of tau2 shows a substantial straight-line interval, often including the earliest available points. In earlier work [G.C. Borgia, R.J.S. Brown, P. Fantazzini, Phys. Rev. E 51 (1995) 2104; R.J.S. Brown, P. Fantazzini, Phys. Rev. B 47 (1993) 14823] these features were noted, and attention was called to the fact that very large changes in field and gradient are likely for a small part of the pore fluid over distances very much smaller than pore dimensions. A truncated Cauchy-Lorentz (C-L) distribution of local fields in the pore space was used to explain observations, giving reduced effects of diffusion because of the averaging properties of the C-L distribution, the truncation being at approximately +/-1/2chiB0, where chi is the susceptibility difference. It was also noted that, when there is a narrow range of pore size a, over a range of about 40 of the parameter xi=1/3chinua2/D, where nu is the frequency, R2 does not depend much on pore size a nor on diffusion constant D. Examples are shown where plots of R2 vs tau show better linear fits to the data for small tau values than do plots vs tau2. The present work shows that, if both grain-scale and sample-scale gradients are present for samples with narrow ranges of T2, it may be possible to identify the separate effects with the linear and quadratic coefficients in a second-order polynomial fit to the early data points. Of course, many porous media have wide pore size and T2 distributions and hence wide ranges of xi. For some of these wide distributions we have plotted R2 vs tau for signal percentiles, normalized to total signal for shortest tau, again showing initially linear tau-dependence even when available data do not cover the longest and/or shortest T2 values for alltau values. For the examples presented, both the intercepts and the initial slopes of the plots of R2 vs tau increase systematically with signal percentile, starting at smallest R2.     

Measurement of the Lambda b lifetime in the exclusive decay Lambda b --> J/psi Lambda

We have measured the Lambda b lifetime using the exclusive decay Lambda b --> J/psi Lambda, based on 1.2 fb(-1) of data collected with the D0 detector during 2002-2006. From 171 reconstructed Lambda b decays, where the J/psi and Lambda are identified via the decays J/psi --> mu+ mu- and Lambda --> ppi, we measured the Lambda b lifetime to be tau(Lambda b)=1.218 (+0.130)/(-0.115) (stat) +/- 0.042(syst) ps. We also measured the B0 lifetime in the decay B0 --> J/psi(mu+ mu-)K(0)/(S)(pi+ pi-) to be tau(B0)=1.501 (+0.078)/(-0.074) (stat) +/- 0.050(syst) ps, yielding a lifetime ratio of tau(Lambda b)/tau(B0)=0.811 (+0.096)/(-0.087) (stat) +/- 0.034(syst).     

Narrow Escape, Part III: Non-Smooth Domains and Riemann Surfaces

We consider the narrow escape problem in two-dimensional Riemannian manifolds (with a metric g) with corners and cusps, in an annulus, and on a sphere. Specifically, we calculate the mean time it takes a Brownian particle diffusing in a domain Ω to reach an absorbing window when the ratio between the absorbing window and the otherwise reflecting boundary is small. If the boundary is smooth, as in the cases of the annulus and the sphere, the leading term in the expansion is the same as that given in part I of the present series of papers, however, when it is not smooth, the leading order term is different. If the absorbing window is located at a corner of angle α, then if near a cusp, then grows algebraically, rather than logarithmically. Thus, in the domain bounded between two tangent circles, the expected lifetime is , where is the ratio of the radii. For the smooth boundary case, we calculate the next term of the expansion for the annulus and the sphere. It can also be evaluated for domains that can be mapped conformally onto an annulus. This term is needed in real life applications, such as trafficking of receptors on neuronal spines, because is not necessarily large, even when is small. In these two problems there are additional parameters that can be small, such as the ratio δ of the radii of the annulus. The contributions of these parameters to the expansion of the mean escape time are also logarithmic. In the case of the annulus the mean escape time is .     

A stochastic model for metabolizing systems with computer simulation

A stochastic model for a first-order metabolizing system which was studied in the deterministic sense by Branson and others is formulated and a detailed study of the random integral equation arising in the probabilistic model is presented. The equation is used to describe the evolution in time of the amount of metabolite present in the system. Specifically we present a study of the random integral equation of the Volterra type given by $$M\left( {t; \omega } \right) = M\left( {0; \omega } \right)e^{ - et} + \int_0^t {R\left( {\tau ; \omega } \right) e^{ - e\left( {t - \tau } \right)} d\tau , } t \geqslant 0$$ whereM(t; ω) is an unknown random function giving the amount of metabolite in the system at time t ≥ 0. This equation can be expressed in the general form $$x\left( {t; \omega } \right) = h\left( {t; \omega } \right) + \int_0^t {k\left( {t, \tau ; \omega } \right) f\left( {\tau , x\left( {\tau ; \omega } \right)} \right) d\tau } t \geqslant 0$$ which is of a type whose theoretical aspects have recently been studied by the present authors using as a basis the techniques of probabilistic functional analysis. Conditions are derived under which there exists a unique random solution to the above equation. The usefulness of the model is illustrated using computer simulation by considering a one-organ model, an organ-heart model, and a multicompartment model.     

Diffusion measurements with the aid of nutation spin echoes appearing after two inhomogeneous radiofrequency pulses in inhomogeneous magnetic fields

Nutation echoes are generated by radiofrequency (RF) pulses with an inhomogeneous amplitude, B(1) = B(1)(r), in inhomogeneous magnetic fields, B(0) = B(0)(r). The two gradients of strengths G(1) and G(0), respectively, must be aligned in parallel for a maximum echo signal. After two RF pulses, two echoes appear at times tau(a) = 2 tau(1) + tau(2) + (G(1)/G(0))tau(1) and tau(b) = 2 tau(1) + tau(2) + 2(G(1)/G(0))tau(1), where tau(1) is the RF pulse duration and tau(2) the interpulse interval. It is shown that these echoes can favorably be employed for the determination of self-diffusion coefficients even in the poor experimental situation one often faces in low-resolution or low-field NMR. The signal intensity is comparable to that of ordinary Hahn echoes. Diffusion coefficients and spin-lattice relaxation times can be evaluated from the same experimental data set if both nutation echoes are recorded. Test experiments are in good agreement with literature data. Applications of the technique to "inside out" NMR, well logging NMR, surface coil NMR, toroid cavity NMR, etc., are suggested.     

Ehrenfest time and the coherent backscattering off ballistic cavities

If the Ehrenfest time tau(E) of a ballistic cavity is not negligible in comparison to its dwell time tau(D), the weak localization correction to the cavity's transmission is suppressed proportional to exp(-tau(E/tau(D). At the same time, quantum interference enhances the probability of reflection into the mode of incidence by a factor two. This "enhanced backscattering" does not depend on the Ehrenfest time. We show that, in addition to the diagonal enhanced backscattering, there are off-diagonal contributions to coherent backscattering that become relevant if tau(E) > or = tau(D).     

Surface wetting of liquid nanodroplets: droplet-size effects

The spreading of liquid nanodroplets of different initial radii R0 is studied using molecular dynamics simulation. Results for two distinct systems, Pb on Cu(111), which is nonwetting, and a coarse-grained polymer model, which wets the surface, are presented for Pb droplets ranging in size from approximately 55,000 to 220,000 atoms and polymer droplets ranging in size from approximately 200,000 to 780 000 monomers. In both cases, a precursor foot precedes the spreading of the main droplet. This precursor foot spreads as r(2)(f)(t) = 2D(eff)t with an effective diffusion constant that exhibits a droplet-size dependence D(eff) approximately R(1/2)(0). The radius of the main droplet r(b)(t) approximately R(4/5)(0) is in agreement with kinetic models for the cylindrical geometry studied.     

Splitting of dislocations in the Peierls-Nabarro model

A numerical method of solution of the Peierls-Nabarro integro-differential eqation for a given force law(f) is proposed. The solution, i.e., the disregistryf(x) or the dislocation densityq(x)=df/dx is found in a form which describes the splitting of a dislocation into the chosen number of partial dislocations. The method is applied to the study of planar cores of 1/2111 dislocation in b.c.c. metals on {112} and on {110} planes. The force laws(f) are derived from the dependence of the stacking fault energy on disregistryf; the(f) curves calculated by Vítek (1969) for-Fe for two different interatomic potentials are used. In all cases, the solution is well represented by splitting into three partials.Na Slovance 2, Praha 8, Czechoslovakia.The authors are indebted to Dr. J. Moudrý for his help in programming for the computer Minsk 22.     

Measurement of the absolute branching fraction of Ds+ --> tau+ nutau decay

Using a sample of tagged D(s)(+) decays collected near the D(s)(*+/-)D(s)(-/+) peak production energy in e(+)e(-) collisions with the CLEO-c detector, we study the leptonic decay D(s)(+)-->tau(+)nu(tau) via the decay channel tau(+)-->e(+)nu(e)nu(tau). We measure B(D(s)(+)-->tau(+)nu(tau))=(6.17+/-0.71+/-0.34)%, where the first error is statistical and the second systematic. Combining this result with our measurements of D(s)(+)-->mu(+)nu(mu) and D(s)(+)-->tau(+)nu(tau) (via tau(+)-->pi(+)nu(tau)), we determine f(D(s))=(274+/-10+/-5) MeV.     

The magnetic structures of NdCu2 determined by single crystal neutron diffraction

We investigated the magnetic structure of NdCu₂ by means of neutron diffraction as a function of temperature between 1.5 K and 8 K in zero external field. The diffraction data were obtained on two single crystals with different orientations using the triple-axis-spectrometer TAS6 at the DR-3 reactor at Risø. Two magnetic phases were observed between 1.5K andT _N =6.5K. From 1.5 K to 4.1 K the magnetic reflections can be described by the commensurate wave vector =(3/5 0 0) and its higher harmonics 3 and 5. Below 2.5K the structure is completely squared-up. For 4.1 KT6.5 K the magnetic structure is incommensurate with the chemical lattice and can be described by the wave vector=(3/5 0 0) and its higher harmonies 3 and 5M. Below 2.5 K the structure is completely squared-up. For 4.1 K T 6.5 K the magnetic structure is incommensurate with the chemical lattice and can be described by the wave vector ^*=(0.62 0.044 0). In both phases the Nd-moments are oriented along the easyb-direction.     

Measurement of the B0s lifetime in the exclusive decay channel B0s-->J/psiphi

Using the exclusive decay B0s-->J/psi(mu+mu-)phi(K+K-), we report the most precise single measurement of the B0s lifetime. The data sample corresponds to an integrated luminosity of approximately 220 pb(-1) collected with the D0 detector at the Fermilab Tevatron Collider in 2002-2004. We reconstruct 337 signal candidates, from which we extract the B0s lifetime, tau(B0s)=1.444(+0.098)(-0.090)(stat)+/-0.020(sys) ps. We also report a measurement for the lifetime of the B0 meson using the exclusive decay B0-->J/psi(mu+mu-)K*0(892)(K+pi-). We reconstruct 1370 signal candidates, obtaining tau(B0)=1.473(+0.052)(-0.050)(stat)+/-0.023(sys) ps, and the ratio of lifetimes, tau(B0s)/tau(B0)=0.980(+0.076)(-0.071)(stat)+/-0.003(sys).     

Pattern formation by growing droplets: The touch-and-stop model of growth

We investigate a novel model of pattern formation phenomena. In this model spherical droplets are nucleated on a substrate and grow at constant velocity; when two droplets touch each other they stop their growth. We examine the heterogeneous process in which the droplet formation is initiated on randomly distributed centers of nucleation and the homogeneous process in which droplets are nucleated spontaneously at constant rate. For the former process, we find that in arbitrary dimensiond the system reaches a jamming state where further growth becomes impossible. For the latter process, we observe the appearance of fractal structures. We develop mean-field theories that predict that the fraction of uncovered material (t) approaches to the jamming limit as (t)–()exp(Ct ^d ) for the heterogeneous process and as a power law for the homogeneous process. Exact solutions in one dimension are obtained and numerical simulations ford=1–3 are performed and compared with mean-field predictions.     

Spin-glass dynamics determined from muon spin relaxation and neutron spin echo measurements

Muon spin relaxation (SR) and neutron spin echo (NSE) measurements of magnetic ion correlation times and correlation functions in the spin glass systemsCuMn,AgMn, andAuFe are compared. It is found that theSR and NSE measurements are in excellent agreement both above and below the spin-glass freezing temperatures. The experimental results are compared to recent theories of spin-glass dynamics.We are grateful to D.L. Huber and R.E. Walstedt for stimulating discussions. This work was performed under the auspices of the U.S. Department of Energy, and was also supported by the U.S. National Science Foundation, Grant No. DMR-8115543.