Time-dependent Pair Research Articles

Dynamics of boson quantum films.

We employ a quantitative microscopic theory of nonuniform quantum liquids to explore the excitations in thin films of $^{4}\mathrm{He}$ adsorbed onto a substrate. These liquid films studied undergo a series of structural phase transitions coinciding with the completion of individual atomic layers. A generalized Feynman ansatz is used for the wave function of the excited states; multiphonon effects are included by generalizing the Feynman theory to allow for time-dependent pair correlations. We study the dispersion relation, excitation mechanisms, transition densities, and particle currents, as a function of the surface coverage, including coverages near the phase transitions. Because of the film's layered growth, the sound velocity exhibits a series of minima and maxima. A pronounced long-wavelength softening of the lowest-energy mode is observed near the transitions. In the monolayer, the nature of the excitations undergoes a noticeable change at the coverage where the velocity of sound starts to decrease. This is a crossover from ``essentially two-dimensional'' to ``essentially three-dimensional'' behavior. At long wavelengths, below and above the crossover coverage, the lowest-energy excitation is a longitudinal phonon (propagating within the monolayer) and a surface excitation, respectively. At shorter wavelengths, a layer-phonon propagating within the liquid layers, level crosses with a surface excitation to become the lowest-energy mode. For double- and higher-layer films the excitations are complicated by multiple (layer phonon with layer phonon and layer phonon with surface excitation) level crossings. At higher coverages, a mode is identifiable that will evolve into the bulk phonon-maxon roton. Our results agree qualitatively with the available spectra obtained by neutron scattering experiments.

Novel dynamical scaling in kinetic growth phenomena

A novel dynamical scaling characterized by infinitely-many growth exponents is presented. This new behavior is manifested in the time-dependent pair correlation function and therefore is amenable to direct experimental observation. An exact analytical solution on the time-dependent Ginzburg-Landau model is obtained in the large- N limit for a quench at zero temperature, which exhibits such multiscaling behavior. This new form of behavior reflects a morphological pattern in the growing domains and is expected in other growth phenomena, as indicated by preliminary analysis on DLA clusters. A connection between multiscaling and multifractality is discussed.

Reduced Density Matrices and Two-Fluid Model for Fermi Superfluid

From the assumption of the off-diagonal long-range order (ODLRO) in the reduced density matrices (RDM) and the neglect of long range three-particle correlation, the equation of motion for the time dependent macroscopic pair wave function ϕ (x,y) in Fermi superfluid has been derived, clearly showing the gauge invariance. The normal (noncondensate) density and super (condensate) density have thus been defined. The wave equation then leads to the setting up a “two-fluid” hydrodynamics for Fermi superfluid, probably in superconductors.

On the approach of the stationary state in Kauffman's random Boolean network

Kauffman's model is a Boolean network having k quenched random connections per node, and evolving deterministically in time according to a quenched random rule. For a systems of N nodes we consider a time domain 0≤t ≤ tN, with $$ tN = ∞. It is shown (i) that in this limit the average overlap between two arbitrary initial states rigorously obeys an equation first derived by Derrida and Pomeau, and (ii) that for k ≤ 2 each state evolves, in a weak sense, to a fixed point state. Also, (iii) upper bounds on the finite size corrections are obtained. Lastly, (iv) for k =1 the exact time-dependent pair overlap is found and compared to the results (i)-(iii).

A simple model for the dynamics towards metastable states

Circumstances under which a quenched system will “freeze” in a metastable state are studied in simple systems with long-range order. The model used is the time-dependent pair approximation, based on the most probable path (MPP) method. The time dependence of the solution is shown by means of flow diagrams. The fixed points and other features of the differential equations in time are independent of the choice of the rate constants. It is explained qualitatively how the system behaves under varying descending temperatures: the role of the initial conditions, the dependence on the quenching rate, and the response to precooling.

Surface correlations in a quantum mechanical one-component plasma

The time-dependent pair correlation function of a quantum mechanical onecomponent plasma bounded by a plane hard wall is studied near that wall. Along the wall, this function has an algebraic asymptotic form: it decays only as the inverse cube (square) of the distance for a three (two)-dimensional system (the case of fermions at zero temperature is excluded from the present study). The amplitude of the asymptotic form obeys a universal sum rule. Similar results hold at the plane interface between two different one-component plasmas.

Quasihydrodynamic theory of ionic conductance in spatially homogeneous solutions of strong electrolytes

Correlation function expressions are derived for the contributions of the frequency-dependent relaxation effect to the conductivity and dielectric permittivity of the electrolyte solution. The evolution equation for the time-dependent irreducible ionic pair correlation function is derived in the mean field approximation by the application of the generating functional method.

Relative motions in atomic fluids: A molecular dynamics investigation

The generalized time-dependent pair distribution function G 2( R, R '; t) is examined for the first time by two molecular dynamics experiments in Lennard-Jones argon at room temperature and different densities. This function gives insight into the relative motion of atoms and plays an important role in all collision-induced phenomena. The results stress the features of G 2 at intermediate and long times; in the latter case they are found to be in agreement with a simple theoretical model. The behavior of some physical quantities dependent on the relative motion of atoms is also discussed.

A computer simulation for a simple model of liquid hydrogen chloride-time correlation functions

The results of a previous simulation for a fluid of rigid heteronuclear diatomic molecules using a shifted force, site-site potential [1] are extended to evaluate several time correlation functions for four state points for the orthobaric liquid. Time autocorrelation functions are reported for the centre of mass velocity, the force, the torque, the angular velocity and the orientational functions P 1 (cos ϑ) and P 2 (cos ϑ). We also report the self and distinct parts of the Van Hove time dependent pair correlation function. The orientational correlation times and functions are compared with experimental results from nuclear magnetic resonance, Raman scattering and far infrared absorption and also with two popular simple models of molecular reorientation; the simulation results are in only moderate agreement with experiment.

Time-dependent pair distribution in the mean-field approximation

The time-dependent pair-distribution function, which plays a central role in the description of nuclear magnetic relaxation and collision-induced absorption in classical fluids, is evaluated in a mean-field approximation proposed earlier. In this scheme the binary dynamics of the colliding pair is treated without approximation, and the interaction with the background fluid is taken into account via a renormalization of the pair potential. The result is expressed in terms of orbits having specified end points and specified elapsed times.

Time-dependent pair distribution function: The constant acceleration approximation revisited

The constant-acceleration approximation for the time-dependent pair distribution function as introduced by Bloom and Oppenheim is critically examined. The previously noted inconsistencies for the initial behaviour of the correlation functions are eliminated by a proper treatment of the short time dynamics.

Nuclear quadrupolar relaxation in molten salts

From the straightforward extension of a general theory for nuclear quadrupolar spin–lattice relaxation in monatomic liquids the rate RQ,α for an α ion in a simple uni-univalent ionic melt is expressed in terms of the local electric field gradients (e.f.g.) which originate from unlike or like charges external to the ion of interest, the van Hove time-dependent pair distribution function, and the static pair distribution function between unlike or like ions. RQ,α may be approximated in terms of the partial elastic limit Sγδ(k, 0) of the dynamic liquid structure factor and the partial weighting factor Iαγ, αδ(k) involving the e.f.g. contribution. For a sodium cation in molten NaCl the terms of INaγ,Naδ(k)(γ, δ; Na or Cl) are estimated under an assumption of an electrostatic quadrupolar interaction. These numerical values are 10–100 times greater than those of I(k) for liquid Ne calculated from exchange–van der Waals e.f.g. The experimental results of the partial static-structure factor Sγδ(k) and the static pair radial distribution function gNaγ(r)(γ; Na or Cl) for molten NaCl are of great consequence in calculations of both Sγδ(k, 0) and Iαγ,αδ(k). Numerical calculation of RO,Na for the sodium cation in the molten NaCl is in rough agreement with experiment. The relaxation process of 23Na in molten NaNO3 is caused by both translation and reorientational motions. The rotation-like motion gives rise to the time dependent e.f.g. at the Na nucleus and contributes mainly to the observed relaxation rate. Numerical calculations of the temperature dependence of RQ,Na in molten NaNO3 are in no good agreement with experiment.

Renormalized theory of the time-dependent pair distribution function. I. General formulation

A classical renormalized theory of a time-dependent pair-distribution function (TDPDF), previously introduced by Oppenheim and Bloom, is presented. An equation of motion for the TDPDF is derived in which the memory function of the system appears. This is then split into a part which contains only static correlation functions and a part which describes the dynamics. The mean field approximation is discussed in some detail and contact is made witn the theory of Oppenheim and Bloom.

Interpretation of quasi-elastic light scattering measurements from moderately concentrated solutions

The autocorrelation functions of elastic and quasi-elastic light scattering measurements of polystyrene ( M w = 670,000 ) in toluene at 20° and cyclohexane at 35° at concentrations up to 0·07 g cm −3 are examined employing a scattering law that includes both the self and distinct part of the time-dependent pair correlation function. By taking the convolution approximation of Vineyard, it is shown that within experimental accuracy both terms are governed by the translational diffusion coefficient.

NMR line-shape calculation for a linear dipolar chain

Numerical calculations have been performed to obtain exact results for the high-temperature space- and time-dependent transverse spin pair correlation functions ${G}_{r}^{x}(t)$, where $\mathcal{r}$ is the spatial separation of the spins, and the free-induction-decay function $F(t)$, for a linear chain of $N$ dipolar coupled spins ($S=\frac{1}{2}$) for $N=7, 9, \mathrm{and} 11$ with periodic boundary conditions. The range $\ensuremath{\Delta}$ of the dipolar interaction was allowed to take separately the values 1, 2, and 3. The moments of the NMR line shape (${M}_{2}$ through ${M}_{20}$) were obtained for the infinite chain with $\ensuremath{\Delta}=1$ and $\ensuremath{\Delta}=2$ and were found to be in excellent agreement with previous reliable moment calculations (${M}_{2}$ through ${M}_{8}$). By an evaluation of the contribution of the four-particle terms to ${M}_{6}$, it is shown that the Bersohn-Das theorem fails in one dimension. We find that ${G}_{r}^{x}(t)$ for even (odd) values of $\mathcal{r}$ are predominantly positive (negative). This gives rise to a damped beat structure for $F(t)$. A study of $F(t)$ for $N=7, 9, \mathrm{and} 11$ shows that our 11-spin solution reproduces that of the infinite chain at least up through the fifth node, and is also in good agreement with recent $^{19}\mathrm{F}$ free-induction-decay measurement in fluoroapatite. The nodes of $F(t)$ were found to shift towards shorter values of time as $\ensuremath{\Delta}$ was allowed to increase. Further, for $N=11$, a comparison of our exact results for $F(t)$ for $\ensuremath{\Delta}=3$, with a Gaussian broadening of $F(t)$ for $\ensuremath{\Delta}=2$, shows that the effect of the weak interactions is not well approximated by a simple Gaussian broadening for this case.

Theory of inelastic neutron scattering from molecular fluids

The time-dependent angular pair correlation function is discussed and its use in the analysis of inelastic neutron scattering experiments from polyatomic fluids is described, including both the coherent and incoherent spectra. The set of formal results given here permits a systematic interpretation of neutron inelastic scattering spectra on simple molecular liquids. Neutron spectra second moments are reviewed, and a new result for the fourth moment is given for the incoherent spectrum. Numerical results for the moments are obtained. The fourth moment depends on the mean squared torque and the mean squared force acting on a molecule in the fluid, and may provide a means for studying intermolecular forces in dense fluids. In addition, a method of calculating the correlation function for weak anisotropic forces is outlined.

A theory of the diffusion coefficient in a liquid metal : II

Abstract Calculations are reported of the diffusion coefficient and frequency spectrum of the velocity autocorrelation function for a model of liquid sodium using long-range oscillatory interionic potentials. They are based on a theory of the associated memory function which employs a previously known expression for the timedependent pair distribution function. The latter is introduced to try to solve the difficult problem of describing the relative motion of two atoms, but the results are not consistent with the information obtained from molecular dynamics calculations. On the basis of a physical argument, a modified theory of the pair distribution is then employed and shown to produce results which are in much better agreement with those obtained from computer experiments. Some information is thereby obtained about the relative motion of atoms in a liquid.

Relativistic quantum theory for charged spinless particles in external vector fields

Guided by a diagonalized form of the classical field-energy we construct a time-dependent canonical pair of Schrödinger fields Φ t (x) and Π t (x) which diagonalizes the field-HamiltonianH t . These Schrödinger fields in general belong to inequivalent representations of the canonical commutation relations for differentt's. The Heisenberg field is constructed by solving the Heisenberg equation of motion and its time-evolution turns out to be governed by a unitary operator, i.e. the Heisenberg fields at different times are unitarily equivalent. Scattering theory (including eventual incoming and/or outgoing bound-states) is finally constructed.

Time-Dependent Ginzburg-Landau Theory of Nonequilibrium Relaxation

The system is considered which is brought into a state far from thermal equilibrium by a sudden change of independent variables. Its approach towards the equilibrium state is studied introducing nonequilibrium relaxation functions. Near the equilibrium state the lowest-order relaxation functions reduce to the time-dependent pair correlation functions. The relaxation process is especially interesting near a first-order phase transition, where metastable states can occur. Characterization of these states in terms of the nonequilibrium relaxation functions is discussed, and a "constructive" definition of the metastable states in terms of a "flatness" property of the relaxation curve is proposed. In order to give explicit results, the time-dependent Ginzburg-Landau theory is treated in detail, and exact solutions for the relaxation functions are given. If an equilibrium state is described by a real-valued order parameter, the inverse lifetime of a metastable state is determined from the imaginary part of its order parameter. Transition from the "metastable" to the stable state is characterized by fluctuations which increase with time for wave vectors up to some critical value, and later decrease again because of the nonlinearity of the Ginzburg-Landau equations. The validity of these results is discussed with respect to systems with long-range interactions, and also a few remarks are given concerning systems with short-range interactions and the nucleation picture.

A Theory of the Diffusion Coefficient in a Liquid Metal: I

Abstract An expression has been derived for the diffusion coefficient in terms of the interatomic potential, the pair distribution function and the selfcorrelation function in a liquid. By making use of an earlier result for the time-dependent pair distribution function, which includes correlation in the relative motion of two atoms, the expression has been made applicable to any type of interatomic core potential. In particular it will be useful in a liquid metal where a relatively soft core is a generally accepted feature of the effective interionic potential.