The Pearcey Process

The extended Airy kernel describes the space-time correlation functions for the Airy process, which is the limiting process for a polynuclear growth model. The Airy functions themselves are given by integrals in which the exponents have a cubic singularity, arising from the coalescence of two saddle points in an asymptotic analysis. Pearcey functions are given by integrals in which the exponents have a quartic singularity, arising from the coalescence of three saddle points. A corresponding Pearcey kernel appears in a random matrix model and a Brownian motion model for a fixed time. This paper derives an extended Pearcey kernel by scaling the Brownian motion model at several times, and a system of partial differential equations whose solution determines associated distribution functions. We expect there to be a limiting nonstationary process consisting of infinitely many paths, which we call the Pearcey process, whose space-time correlation functions are expressible in terms of this extended kernel.     

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.     

Intermolecular potentials for ammonia based on the test particle model and the coupled pair functional method

The intermolecular interaction ΔE in (NH₃)₂ is investigated on the SCF level, with inclusion of correlation effects by means of the CPF method and within the simple test particle model. Whereas the linear hydrogen bonded structure is favoured on the SCF level, ΔE = -7·65 kJ mol^-1, the most stable geometry on the highest level of theory is a cyclic structure, ΔE = -12·96 kJ mol^-1. The minimum is very shallow and allows for appreciable angular motions. The test particle model reproduces the general features of ΔE but shows deviations in details. The computed potentials are used in MD simulations to compute static and dynamic properties of liquid NH₃. Good agreement with available experimental results is obtained throughout.     

Tagged-particle motion in glassy systems under shear: Comparison of mode coupling theory and Brownian dynamics simulations

We study the dynamics of a tagged particle in a glassy system under shear. The recently developed integration through transients approach, based on mode coupling theory, is continued to arrive at the equations for the tagged-particle correlators and the mean squared displacements. The equations are solved numerically for a two-dimensional system, including a nonlinear stability analysis of the glass solution, the so called β-analysis. We perform Brownian Dynamics simulations in 2D and compare with theory. After switch on, transient glassy correlation functions show strong fingerprints of the stress overshoot scenario, including, additionally to previously studied superexponential decay, a shoulder-like slowing down after the overshoot. We also find a new type of Taylor dispersion in glassy states which has intriguing similarity to the known low-density case. The theory qualitatively captures most features of the simulations with quantitative deviations concerning the shear-induced time scales. We attribute these deviations to an underestimation of the overshoot scenario in the theory.     

On the derivation of the generalized Langevin equation for interacting Brownian particles

The main result of this paper is a derivation of a generalized nonlinear Langevin equation (GLE) forn interacting particles in a bath. A consequence of the derivation is that the exact form of the (generalized) fluctuation-dissipation theorem is obtained. We discuss also the relation between the memory kernel of the GLE and some corresponding correlation functions which can be easily obtained in a molecular dynamics computer experiment. In the same spirit it is shown that the approach applies to a Brownian particle subjected to a stationary external field. The technique presented in a previous paper to simulate generalized Brownian dynamics can be easily extended to the present case. Our derivation intends to clarify the uses and (possibly) abuses of the Langevin equation in Brownian dynamics studies.     

Brownian motion in a critical fluid; A critical long time tail

The frictioncoefficient of a Brownian particle in a critical fluid was evaluated, taking into account the finite correlation length of density fluctuations. A $\text{t}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$ tail of the Brownian particle velocity autocorrelation function exists close to the critical point whose amplitude is 10²–10³ times the amplitude of the usual long time tail found far from the critical point.     

Quantum treatment of Brownian motion and influence of dissipation on diffusion in dynamically disordered systems

The time-dependent hamiltonian formulation of the Langevin equation is used as a starting point for a quantum treatment of the motion of a free Brownian particle. From an exact solution of the time-dependent Schrödinger equation for the density-matrix in one dimension we obtain the mean square displacement, x ²(t), of the Brownian particle, as well as the mean displacement induced by a uniform electric fieldE(t). While quantum effects are significant for time intervals up to the frictional relaxation time, the long time results are identical to those obtained directly from the solution of the Langevin equation. Next, we analyse in a similar way the motion of an electron in a dynamically disordered continuum where the effect of a classical friction force (dissipation) is taken into account. The friction effect is described using the phenomenological time-dependent hamiltonian inferred from the Langevin equation and the potential fluctuations are assumed to have a generalized gaussian white-noise form. The final result for x ² (t) shows a time-dependence similar to that obtained for the case of Brownian motion. In particular, it corresponds to diffusive behavior at long times, in contrast to thet ³-dependence obtained in a recent study for the case where friction is absent.     

Effect of the Noise Correlation Time on the Stationary Current of Brownian Particle in a Spatially Symmetric Periodic Potential

We investigate the noise-induced transport of Brownian particle in a deterministic spatial symmetrical periodic potential driven by colored cross correlation between a multiplicative white noise and an additive white noise. We derive the general formula of the stationary current. Based on numerical computation, we found that directed motion of the Brownian particles can be induced by the correlation time τ of cross correlation between the multiplicative noise and the additive noise and the current reversal and the direction of the current is controlled by the τ.     

The mean squared torque in pure and mixed dense fluids

The mean squared torque on a molecule can be obtained from infra-red or Raman band moments, and provides a direct measure of the strength of the anisotropic intermolecular forces. The expression for the mean squared torque on a molecule of species α in a fluid mixture is given in terms of the intermolecular potential and the angular pair correlation functions. This relation is made tractable by introducing a perturbation expansion in powers of the anisotropic potential strength for the angular pair correlation functions. Monte Carlo calculations of the mean squared torque are presented for a liquid of linear molecules having dipolar, quadrupolar and anisotropic overlap interactions. Comparison of the perturbation expansion to second order with these ‘exact’ results shows good agreement for μ*=μ/(εσ³)^1/2 and Q*=Q/(εσ⁵)^1/2 values less than about 0·5, and for values of the dimensionless overlap constant |δ| less than about 0·2. Finally, experimental values of the mean squared torque for both pure and mixed liquids are compared to the Monte Carlo and to the perturbation theory results.     

Onsager-Zerlegung und generalisierte Langevin-Theorie für ein Brownsches Teilchen in der harmonischen Kette

To illustrate the so-called Onsager separability of dynamic correlation functions (which is fundamental to the Statistical Mechanics of phenomenological processes) we consider the exact generalized Langevin equation for the momentum of a Brownian particle in a harmonic chain. The time dependence of the Langevin force and its dynamic autocorrelation (the after effect kernel) are calculated and shown to have no hydrodynamic part—in contrast to the particle's momentum correlation.     

Chaotic properties of multipoint correlation functions of an Ising model with long-range interactions on the Sierpiński—Gasket lattice

A class of multispin correlation functions of an Ising model with ferromagnetic nearest neighbor interactionsK and constant (distance-independent) long-range interactionsQ ₁=Q,l=1,2,..., on the Sierpiski-gasket lattice is considered. Using an exact method for calculating thermodynamic functions of hierarchically constructed Ising systems, it is shown that, for a set of values ofQ and for almost all values ofK, someM _k-spin correlation functions, whereM _k=3 ^k +3 withk=1,2,...,n andn=1,2,... being the order of lattice construction, change chaotically asn, k, and therebyM _k increase to infinity. Accordingly, in the thermodynamic limit, these correlation functions prove to be nonanalytic for appropriate values ofQ andK. SinceM _k-point correlation functions withk being finite, i.e., correlation functions involving finite numbers of spins, remain analytic asn tends to infinity, there is a smooth crossover between analytic properties of correlation functions of the two types.     

Diffusion in a bistable potential at intermediate and high friction

We study the motion of a Brownian particle in a bistable potential for intermediate and high-friction . Following ideas of Titulaer we perform a high-friction expansion of the distribution functionP(v,x,t) in velocity and space. We show (for arbitrary potential) that the expansion coefficients obey simple recursion relations, which allow them to be calculated easily. When terms of order ^–5 are neglected the resulting differential equations can be transformed into Hermitian Schrödinger-type equations. Using the WKB technique we solve these equations analytically for the case of the bistable potential and discuss the various time regimes involved in the system, in particular we show that the final approach to equilibrium is governed by the Kramers rate. Our results become exact in the limit of low temperatures.     

Different methods of calculating the pinning energy of plane vortices in a 3D Josephson medium: A comparative study

The pinning energy of plane (laminar) vortices in a 3D Josephson medium is calculated within a continuous vortex model considering functions of two types: V=1−cosϕ and V= 2/π⁴ϕ²(2π−ϕ)². The shape and energy of the stable and unstable vortices are found with an algorithm for the exact numerical solution of a set of difference equations. The vortex magnetic and Josephson energies diverge. The magnetic and Josephson components of the pinning energy are close in magnitude but differ in sign; as a result, the total pinning energy is smaller than its components by one order of magnitude. This result is confirmed analytically. An analytical computing method within the continuous vortex model is suggested. This method preserves the difference terms in the energy expression. The magnetic energy found by this method differs from the Josephson energy in magnitude, and the magnetic component of the pinning energy is opposite in sign to the Josephson component. Comparative analysis of the approximate approaches to energy calculation within the continuous vortex model when the difference terms are retained and when they are replaced by derivatives is performed. It is shown that the continuous vortex model gives incorrect values of the Josephson and magnetic components of the pinning energy. The actual values are several tens or several hundreds of times higher than those obtained with the continuous vortex model. Yet, since the Josephson and magnetic components of the pinning energy have different signs, the exact value of the total pinning energy and the approximate value obtained within the continuous vortex model differ insignificantly.     

Perturbation allowed transitions in the infrared spectrum of 14ND3: determination of the K-dependent parameters in the ground state

ABSTRACT

Accurate values of the K-dependent constants ^{( i )} C, ^{( i )} D_K and ^{( i )} H_K in the ground state of ¹⁴ND₃, with i = s, a, have been determined for the first time thanks to the detections of ‘perturbation allowed’ transitions in the ν₁, ν₂, ν₃, ν₄ and 2ν₄ infrared bands. The rotation–inversion and inversion transitions from the literature, together with 7289 ground state combination differences from the infrared vibration–rotation–inversion transitions have been simultaneously analysed. The adopted rotation–inversion Hamiltonian includes distortion constants up to the eighth power and the Δk = ±3 and Δk = ±6 interaction terms. Precise values of the diagonal constants and of the Δk = ±3 interaction coefficients have been obtained. Accurate values of the ground state term values have been calculated for both s and a levels up to J = 21.     

On the Brownian motion in a double-well potential in the overdamped limit

The Brownian motion of a particle in a double-well potential is considered and the position correlation function (excluding inertial effects) for the potential V(x)=ax²/2+bx⁴/4 is evaluated first by averaging the governing Langevin equation over its realizations. The exact solution is then obtained via matrix continued fractions by using a representation, which symmetrizes the recurrence relations for the observables generated by the averaging procedure leading to convergence of these recurrence relations unlike the previous approaches to the problem. A reliable approximate solution based on the exponential separation of the time scales of the fast intrawell and low overbarrier relaxation processes associated with the bistable potential is also given. It is shown that a knowledge of the three characteristic relaxation times (the integral, effective and the longest relaxation times) of the position correlation function allows one to accurately predict the relaxation behavior of the system in the overdamped limit for all time scales of interest.     

Fokker–Planck Equation,Molecular Friction,and Molecular Dynamics for Brownian Particle Transport near External Solid Surfaces

The Fokker–Planck (FP) equation describing the dynamics of a single Brownian particle near a fixed external surface is derived using the multiple-time-scales perturbation method, previously used by Cukier and Deutch and Nienhuis in the absence of any external surfaces, and Piasecki et al. for two Brownian spheres in a hard fluid. The FP equation includes an explicit expression for the (time-independent) particle friction tensor in terms of the force autocorrelation function and equilibrium average force on the particle by the surrounding fluid and in the presence of a fixed external surface, such as an adsorbate. The scaling and perturbation analysis given here also shows that the force autocorrelation function must decay rapidly on the zeroth-order time scale ₀, which physically requires N _Kn1, where N _Kn is the Knudsen number (ratio of the length scale for fluid intermolecular interactions to the Brownian particle length scale). This restricts the theory given here to liquid systems where N _Kn1. For a specified particle configuration with respect to the external surface, equilibrium canonical molecular dynamics (MD) calculations are conducted, as shown here, in order to obtain numerical values of the friction tensor from the force autocorrelation expression. Molecular dynamics computations of the friction tensor for a single spherical particle in the absence of a fixed external surface are shown to recover Stokes' law for various types of fluid molecule–particle interaction potentials. Analytical studies of the static force correlation function also demonstrate the remarkable principle of force-time parity whereby the particle friction coefficient is nearly independent of the fluid molecule–particle interaction potential. Molecular dynamics computations of the friction tensor for a single spherical particle near a fixed external spherical surface (adsorbate) demonstrate a breakdown in continuum hydrodynamic results at close particle–surface separation distances on the order of several molecular diameters.     

Ligand‐site exchange in intramolecular complexes of silicon: substituent effects

The intramolecular complexes containing coordination bonds Si←N or Si←O are distinguished for their stereochemical nonrigidity resulting in interconversion between isomers, that is, ligand‐site exchange. The influence of the substituents bound to the silicon atom on the free energies of activation for ligand exchange ΔG^≠ of specific interest is poorly understood. In this work, the literature data on substituent influence on the energies ΔG^≠ for 13 series of the complexes have been considered, using the correlation analysis. On the basis of the obedience of the energies ΔG^≠ to the linear free energy relationship, it has been established for the first time that the ΔG^≠ values depend not only on the inductive and resonance effects but also on the polarizability and steric effects of substituents. The reason for the occurrence of the polarizability effect is the appearance of excess charges on Si and N (or O) atoms as a result of intramolecular coordination consisting in the charge transfer from the donor center (N or O atom) to the acceptor one (Si atom). In some series the contribution of the polarizability or steric effect to the overall change in ΔG^≠ because of the influence of substituents is a maximum. An understanding of these effects may give a better insight into the mechanism of nucleophilic substitution, involving organoelement compounds. Copyright © 2012 John Wiley & Sons, Ltd.     

On the asymptotic exactness of the Bogoliubov approximation for many boson systems

The Bogoliubov approximation for many boson systems consists in replacing the field operatorsa ₀ anda ₀ ⁺ byc-numbers, to be determined by an extremum condition. Here we formulate the approximation in terms of coherent states of the condensed particles, and prove that for reasonable interactions it gives the exact values of the thermodynamical functions in the infinite volume limit.Research supported in part by the National Science Foundation.     

Semi-Integral Scheme for Simulation of Langevin Equation with Weak Inertia

A stable and accurate algorithm for simulating massive damped Brownian motion is proposed and discussed. The algorithm, being fully integral for the friction and noise terms and predictor-corrector for the potential force in the Langevin equations, is stable upon changing time step and for various masses of the particle. In particular, the limit of zero inertia can be safely taken, and the algorithm yields naturally the corresponding overdamped case. The steady velocity of a particle moving in a titled periodic potential is calculated and three algorithms are compared.