More efficient Brownian dynamics algorithms

We examine the relative efficiencies of three- algorithms for performing Brownian Dynamics simulations without many-body hydrodynamics. We compare the conventional Brownian Dynamics algorithm of Ermak (CBD), Smart Monte Carlo (SMC) which incorporates Boltzmann sampling into essentially a CBD procedure, and the Stochastic Runge Kutta (SRK) method. We show, using the repulsive potential φ(r) = ε(σ/r) ⁿ , where n = 36 and 72, that the SRK algorithm gives the most accurate short-time dynamics for the mean-square displacements. The SRK algorithm static and dynamical properties converge better with a reducing time step to the exact values, than those generated by the CBD algorithm; giving efficiency gains typically of a factor of 3–4. Both CBD and SMC have the incorrect sign for the first correction term to the mean square displacement in a time step, whereas the SRK algorithm gives essentially the exact solution to order Δt ², where Δt is the simulation time step. In fact, these correction terms are almost equal and opposite in sign. Expressions for these terms were derived in terms of the average interaction energy per particle. The force, shear and bulk stress autocorrelation functions were calculated. The average energy per particle and time correlation functions at short time have values in excess of the exact values, while the corresponding quantities for SRK are below this. This difference in behaviour can be traced back to the extent of compliance of the particle trajectories with the exact expansion of the Smoluchowski equation. The accuracy, at a given value of the time step, of the stochastic algorithms can significantly depend on the form of the interaction potential between particles. It is also demonstrated that the long time limits of various correlation functions are fairly insensitive to a particular scheme (SRK or CBD) used in the simulations. All the correlation functions have a stretched exponential region at intermediate to long times, and the values of the exponents on density and force law steepness have been determined.     

Lagrangian formulation of a generalized Lane-Emden equation and double reduction

Abstract

We classify the Noether point symmetries of the generalized Lane-Emden equation y″+ ny′/x+ f(y)?=?0 with respect to the standard Lagrangian L = xⁿy′²/2 — xⁿ ∫f(y)dy for various functions f(y). We obtain first integrals of the various cases which admit Noether point symmetry and find reduction to quadratures for these cases. Three new cases are found for the function f(y). One of them is f(y) = αy^r , where r ≠ 0,1. The case r?=?5 was considered previously and only a one-parameter family of solutions was presented. Here we provide a complete integration not only for r?= 5 but for other r values. We also give the Lie point symmetries for each case. In two of the new cases, the single Noether symmetry is also the only Lie point symmetry.     

Effective field theory approach to structure functions at small $x_{\mathrm{Bj}}$

We relate the structure functions of deep inelastic lepton-nucleon scattering to current-current correlation functions in a Euclidean field theory depending on a parameter r. The r-dependent Hamiltonian of the theory is P ⁰ -(1-r)P ³ , with P⁰ the usual Hamiltonian and P³ the third component of the momentum operator. We show that a small in the structure functions corresponds to the small r limit of the effective theory. We argue that for there is a critical regime of the theory where simple scaling relations should hold. We show that in this framework Regge behaviour of the structure functions obtained with the hard pomeron ansatz corresponds to a scaling behaviour of the matrix elements in the effective theory where the intercept of the hard pomeron appears as a critical index. Explicit expressions for various analytic continuations of the structure functions and matrix elements are given as well as path integral representations for the matrix elements in the effective theory. Our aim is to provide a framework for truly non-perturbative calculations of the structure functions at small for arbitrary Q². Received: 16 July 2002 / Published online: 9 December 2002 RID="a" ID="a" e-mail: O.Nachtmann@thphys.uni-heidelberg.de     

Computer simulation of cavity pair distribution functions of hard spheres in a hard slit pore

We describe efficient Monte Carlo computer simulation techniques to calculate conditional distribution functions for pairs of hard-sphere (HS) cavities in a hard slit pore of width L, n* (z ₁,z ₂,r), and use these as an efficient route to calculating the corresponding dimensionless excess chemical potentials μ ^e (z ₁,z ₂,r). z_i is the distance of an HS centre from a (specified) wall and r is the distance between the cavity centres. This is the first calculation of such functions, which are of interest in their own right and provide data for the testing of theories, in addition to providing data for a simple model for the infinite dilution behaviour of a polyatomic solute in a simple molecularly confined solvent. Results are presented for special cases for the cavity functions n* (z ₁,z ₂,r) which occur when the spheres are in the same plane, when the line of sphere centres is perpendicular to the walls, and when the spheres are in contact. We compare results obtained using the Kirkwood superposition approximation in conjunction with results obtained from the computer simulation data using the first member of the BGY integral equation hierarchy. The approximation is found to be exact in certain limiting geometrical situations, but in general is quantitatively poor.     

Elastic interaction between defects in thin and 2D films

Elastic interactions between defects is investigated at the surface of thin layers, a question on which we have given a brief account [P. Peyla et al. Phys. Rev. Lett. 82, 787 (1999)]. Two isotropic defects do not interact in an unlimited medium, regardless of the spatial dimension, a result which can be shown on the basis of the Gauss theorem in electrostatics. Within isotropic elasticity theory, defects interact only (i) if they are, for example, at a surface (or at least if they feel a boundary), or if their action on the material is anisotropic (e.g. they create a non central force distribution, though the material elasticity is isotropic). It is known that two identical isotropic defects on the surface of a semi-infinite material repel each other. The repulsion law behaves as 1/r ³(r = defects separation). We first revisit the Lau-Kohn theory and extend it to anisotropic defects. Anisotropy is found to lead to attraction. We show that in thin films defects may either attract or repel each other depending on the local geometric force distribution caused by the defect. It is shown that the force distribution (or more precisely the forces configuration symmetry) fixes the exponent in the power law 1/r ⁿ (e.g. for a four-fold symmetry n = 4). We discuss the implication of this behaviour in various situations. We treat the interactions in terms of the symmetries associated with the defect. We argue that if the defects are isotropic, then their effective interaction in an unlimited 2D (or a thin film) medium arises from the induced interaction, which behaves as 1/r ⁴ for any defect symmetry. We shall also comment on the contribution to the interaction which arises from flexion of thin films. Received 7 October 2002 Published online 4 June 2003 RID="a" ID="a"e-mail: chaouqi.misbah@ujf-grenoble.fr     

Decay of bcc-structure and of bond-orientational order in a fluid

The decay of an initially prepared bcc structure and the bond-orientational order (anisotropy of the first coordination shell) are studied in a non-equilibrium molecular dynamics simulation for a fluid of 1024 particles interacting with a repulsive r^-12 potential. Data are presented for pair-correlation functions and order parameters associated with an angle dependence described by cubic harmonics of the ranks 4, 6 and 8. These cubic pair-correlation functions and cubic order parameters are defined by the expansion of the pair-correlation function and of the bond orientational distribution function with respect to Cartesian tensors. The relaxation of the local anisotropy shows a pretransformational slowing down for densities approaching the freezing point.     

Solvation properties of Li+ and CI− in water: molecular dynamics simulation with a non-rigid model

Molecular dynamics (MD) simulations were carried out to investigate the solvation properties of Li⁺ and C1^? ions in water with a relatively accurate but rarely used non-rigid model, RWK2, in this study. A new set of ion-water interaction parameters was evaluated from the experimental data and first principles calculation results of stable clusters, Li⁺(H₂O) _n (n = 1- 6) and CI^?(H₂O) _n (n = 1–4). With the ion-water potential parameters evaluated from the data of the clusters and the water-water potential predetermined from the non-rigid RWK2 model, the structural (radial distribution functions, angular distribution functions, spatial distribution functions, coordination number), dynamical (residence time) and energetic properties of the ionic salvations in bulk water were studied through a comprehensive analysis of our MD simulation outputs. These results not only agree well with experimental data and first principles calculations, but also reveal some new insights into the microscopic ionic salvation processes.     

Viscoelastic behaviour of fluids with steeply repulsive potentials

We analyse the shear stress, C _s(t) and pressure or ‘bulk’, C _b(t) time-correlation functions for steeply repulsive inverse power fluids (SRP) in which the particles interact via a pair potential with the analytic form, φ(r) = ε(σ/r) ⁿ , in a new approach to the understanding of their viscoelastic properties. We show analytically, and confirm by molecular dynamics simulations, that close to the hard-sphere limit both these time-correlation functions have the analytic form, C _s(t)/C _s(0) and C _b(t)/C _b(0) = 1 – T*(nt*)²+ O((nt*)⁴), where T* = k _B T/ε, is the reduced temperature, k _B is Boltzmann's constant and t* = (ε/mσ²)^½ t is the reduced time. This leads to an alternative and much simpler derivation of formulae for the shear and bulk viscosities which for the limiting case of hard spheres are numerically very close to the traditional Enskog relations. These simple relations for the finite and continuous SRP interaction are in satisfactory agreement with the essentially exact molecular dynamics simulation results for ca. n ≥ 18.     

Optical properties and radiation effects in layered crystals of (CnH2n+nNH3)2MCl4: n = 1, 2, 3; M = Cd,Cu, Mn

Abstract

A new absorption band has been found at 5.10 eV in (C _n H_{2n + 1}NH₃)₂CdCl₄: n = 1, 2, 3 in addition to the absorption bands of CdCl₂ whose electronic structure resembles the former crystals. The energy of the additional peak shifts with temperature by as much as 0.38 eV from 5.10eV at room temperature (RT) to 5.48 eV at liquid nitrogen temperature. This large peak shift is attributed to a structural phase transition between these two temperatures. A new type of electron center has been found in these crystals (M = Cd, Mn; n = 1, 2, 3) irradiated with X-rays at 15 K in addition to the Cl₂ ^?. This shows optical absorption bands (IR bands) in the infrared region of 10 ～ 20 kcm ^?1. The IR bands are assigned to an electron center where an electron is trapped at an ammonium site in the neighborhood of a Cl^? vacancy.     

Lattice energy of alkali halide crystals

A new repulsive term in the ionic interaction potential ψ(r) = Ar^?ne^?r/gl, is suggested and the three unknown parameters A, λ and n are evaluated. Lattice energies of alkali halide crystals are calculated using this form. The results agree fairly well with the experimental values.     

Equation of state of inverse power fluids

A new equation of state for the inverse power, r ^{? n} potential, fluid is proposed. It is derived on the basis of the local scaling behaviour of its structural properties, without referring to any perturbative scheme, and therefore recourse to an effective hard sphere diameter. It is shown that the general formula for the compressibility factor can be expressed as the product of three functions. The first represents the hard-sphere equation of state at the same packing fraction, and the other two incorporate the effects of the potential softness, again as functions of density. Using computer simulation results, explicit forms for these soft parts have been established, to give an approximate analytic expression for the r ^{? n} fluid equation of state. Two different regions, characterized by positive and negative softness ‘compressibility’ have been found.     

Comparative molecular field analysis and comparative molecular similarity indices analysis of hydroxyethylamine derivatives as selective human BACE-1 inhibitor

Three-dimensional quantitative structure–activity relationship (3D-QSAR) models were developed based on comparative molecular field analysis (CoMFA) and comparative molecular similarity indices analysis (CoMSIA), on a series of 43 hydroxyethylamine derivatives, acting as potent inhibitors of β-site amyloid precursor protein (APP) cleavage enzyme (BACE-1). The crystal structure of the BACE-1 enzyme (PDB ID: 2HM1) with one of the most active compound 28 was available, and we assumed it to be the bioactive conformation of the studied series, for 3D-QSAR analysis. Statistically significant 3D-QSAR model was established on a training set of 34 compounds, which were validated by a test set of 9 compounds. For the best CoMFA model, the statistics are, r ² =  0.998, r²_cv = 0.810{r^{2}_{\rm cv} = 0.810} , n =  34 for the training set and r²_pred = 0.934{r^{2}_{\rm pred} = 0.934} , n = 9 for the test set. For the best CoMSIA model (combined steric, electrostatic, hydrophobic, and hydrogen bond donor fields), the statistics are r ² =  0.978, r²_cv = 0.754{r^{2}_{\rm cv} = 0.754} , n =  34 for the training set and r²_pred = 0.750{r^{2}_{\rm pred} = 0.750} , n =  9 for the test set. The resulting contour maps, produced by the best CoMFA and CoMSIA models, were used to identify the structural features relevant to the biological activity in this series of analogs. The data generated from the present study will further help to design novel, potent, and selective BACE-1 inhibitors.     

Poisson Homology of r-Matrix Type Orbits I: Example of Computation

Abstract

In this paper we consider the Poisson algebraic structure associated with a classical r-matrix, i.e. with a solution of the modified classical Yang–Baxter equation. In Section 1 we recall the concept and basic facts of the r-matrix type Poisson orbits. Then we describe the r-matrix Poisson pencil (i.e the pair of compatible Poisson structures) of rank 1 or CP ⁿ-type orbits of SL(n, C). Here we calculate symplectic leaves and the integrable foliation associated with the pencil. We also describe the algebra of functions on CP ⁿ-type orbits. In Section 2 we calculate the Poisson homology of Drinfeld–Sklyanin Poisson brackets which belong to the r-matrix Poisson family.     

Noncrossing Normal Ordering for Functions of Boson Operators

Normally ordered forms of functions of boson operators are important in many contexts in particular concerning Quantum Field Theory and Quantum Optics. Beginning with the seminal work of Katriel (Lett. Nuovo Cimento 10(13):565–567, 1974), in the last few years, normally ordered forms have been shown to have a rich combinatorial structure, mainly in virtue of a link with the theory of partitions. In this paper, we attempt to enrich this link. By considering linear representations of noncrossing partitions, we define the notion of noncrossing normal ordering. Given the growing interest in noncrossing partitions, because of their many unexpected connections (like, for example, with free probability), noncrossing normal ordering appears to be an intriguing notion. We explicitly give the noncrossing normally ordered form of the functions (a ^r (a ^† ) ^s ) ⁿ ) and (a ^r +(a ^† ) ^s ) ⁿ , plus various special cases. We are able to establish for the first time bijections between noncrossing contractions of these functions, k-ary trees and sets of lattice paths.     

End-to-end distribution functions for polymers having a most probable molecular weight distribution

End-to-end chain distribution functions for a polymer with a most probable molecular weight distribution are obtained by assuming the end-to-end distribution for each molecular weight species is Gaussian and the degree of polymerization is large. The resulting radial distribution functions using the most probable number and weight species distributions are:

W_n (r,p) = k²re-^kr and W_w (r,p) = k³r²/2 e^?kr

respectively, where k is a function of the short-range effects and the extent of the reaction is p. The first and second moments of these functions are identical with those obtained by taking the number and weight averages of the moments for the monodisperse case. The most probable values of r, which cannot be obtained by the latter method, are found to be 1/k and 2/k, respectively, using the above distributions.     

Integral representation for non-maxwell models and the approach to equilibrium

An integral representation is proved for the nonequilibrium distribution function of molecules interaction through potentials which are repulsive near r = 0 and diverge less rapidly than r^?4 as r→ 0. The asymptotic approach to the absolute-equilibrium distribution is studied for general initial states.     

Static and dynamic structure factors with account of the ion structure for high-temperature alkali and alkaline earth plasmas*

The electron-electron, electron-ion, ion-ion and charge-charge static structure factors are calculated for alkali (at T = 30 000 K, 60 000 K, n _e = 0.7 × 10²¹ ÷ 1.1 × 10²² cm^-3) and Be²⁺ (at T = 20 eV, n _e = 2.5 × 10²³ cm^-3) plasmas using the method described by Gregori et al. The dynamic structure factors for alkali plasmas are calculated at T = 30 000 K, n _e = 1.74 × 10²⁰, 1.11 × 10²² cm^-3 using the method of moments developed by Adamjan et al. In both methods the screened Hellmann-Gurskii-Krasko potential, obtained on the basis of Bogolyubov's method, has been used taking into account not only the quantum-mechanical effects but also the repulsion due to the Pauli exclusion principle. The repulsive part of the Hellmann-Gurskii-Krasko (HGK) potential reflects important features of the ion structure. Our results on the static structure factors for Be²⁺ plasma deviate from the data obtained by Gregori et al., while our dynamic structure factors are in a reasonable agreement with those of Adamyan et al.: at higher values of k and with increasing k the curves damp down while at lower values of k, and especially at higher electron coupling, we observe sharp peaks also reported in the mentioned work. For lower electron coupling the dynamic structure factors of Li⁺, Na⁺, K⁺, Rb⁺ and Cs⁺ do not differ while at higher electron coupling these curves split. As the number of shell electrons increases from Li⁺ to Cs⁺ the curves shift in the direction of low absolute value of ω and their heights diminish. We conclude that the short range forces, which we take into account by means of the HGK model potential, which deviates from the Coulomb and Deutsch ones, influence the static and dynamic structure factors significantly.     

Possible improvements in the derivation of δ<<Emphasis Type="Italic">r</Emphasis><Superscript>2</Superscript>> from optical isotope shifts

It is shown that writing the correction factor K in the proper form, the value of δ‹r²› derived from the optical isotope shift may be made more reliable. Instead of the generally used parabolic Z dependence, the quantity 1 - K depends primarily on the actual slope δ‹r²›/δA, which may vary considerably from one isotope pair to another. This slope dependence is illustrated first for a spherical uniform nuclear charge distribution. Then, the case of the more realistic two-parameter Fermi charge distribution is investigated. Practical formulae of 1 - K are given for both model distributions. The improved evaluation is especially important in cases where δ‹r²›values are much lower or higher than the liquid drop value. In these cases the improvement in δ‹r²› values may lead to substantial changes in nuclear structure characteristics, e.g. in deformation parameters. Results are compared to those obtained by an alternative approach, the “two-parameter model” based on a deformed uniform distribution.     

Dielectric properties of a mixture of dipolar hard spheres

A theory for the dielectric constant, ε, of a fluid mixture of dipolar hard spheres is formulated by generalizing the methods developed by Ramshaw and Wertheim for the pure fluid case. The resulting expression for ε depends on the pair distribution functions, g _αβ(r ₁, θ₁, r ₂, θ₂) for a dipolar mixture. Due to the unavailability of exact representations for these dipolar pair distribution functions, the results of the mean spherical approximation are employed in the formalism developed. Numerical results are given for ε as calculated from the pair distribution functions for a spherical volume of macroscopic dimensions. The compositional dependence of the ε obtained in this way for a specific mixture is compared with the corresponding properties of the well established theories of Clausius-Mossotti-Debye and Onsager. In addition, the relative importance of the dipole moment and size of the hard sphere parameters in determining ε for a dipolar mixture (the correlative behaviour of which is described by the mean spherical approximation) is evaluated. It is found that the differences in hard core diameters can be largely ignored, in that ε for an ‘effective’ single component fluid can be given to within 2–5 per cent relative error (at worst) of the mean spherical approximation's result. Such an ‘effective pure fluid’ is described as having the same polarization content as the actual mixture being considered. Thereby, the properties of the effective fluid are determined by the quantity y = 4πβ(m ₁ ² ρ₁ + m ₂ ² ρ₂)/9 where m_i and ρ _i are the dipole moment and number density of component i in the binary mixture, with β = (kT)^-1.