Arbitrary Interaction Potential Research Articles

Equilibrium thermodynamics and thermodynamic processes in nonlinear systems

A general method for the calculation of thermodynamic quasi-equilibrium processes by molecular dynamics (MD) simulation in canonical ensemble is developed. The method is suitable for classical systems with arbitrary interaction potentials. Though this MD method does not allow to calculate directly partition function and entropy, it is possible to calculate necessary partial derivatives which enter into the expressions for the full derivatives dT/dV and d β/dV for adiabitic and isobaric processes. Namely the solutions of these ordinary differential equations define the corresponding thermodynamic processes. The adiabatic process for the 1D Toda lattice is analyzed in details. The usage of the Toda potential allows to perform all analytical calculus up to accurate answers and to compare numerical and analytical results. Exact analytical expressions for the thermodynamics of 1D lattices with few types of nearest neighbor interactions are obtained as a necessary interim solutions. MD-simulation of quasi equilibrium processes in canonical ensemble demands the achievement of thermodynamic equilibrium, thus the thermalization kinetics is briefly discussed.

Theory of peak coalescence in Fourier transform ion cyclotron resonance mass spectrometry

Peak coalescence, i.e. the merging of two close peaks in a Fourier transform ion cyclotron resonance (FTICR) mass spectrum at a high number of ions, plays an important role in various FTICR experiments. In order to describe the coalescence phenomenon we would like to propose a new theory of motion for ion clouds with close mass-to-charge ratios, driven by a uniform magnetic field and Coulomb interactions between the clouds. We describe the motion of the ion clouds in terms of their averaged drift motion in crossed magnetic and electric fields. The ion clouds are considered to be of constant size and their motion is studied in two dimensions. The theory deals with the first-order approximation of the equations of motion in relation to dm/m, where dm is the mass difference and m is the mass of a single ion. The analysis was done for an arbitrary inter-cloud interaction potential, which makes it possible to analyze finite-size ion clouds of any shape. The final analytical expression for the condition of the onset of coalescence is found for the case of uniformly charged spheres. An algorithm for finding this condition for an arbitrary interaction potential is proposed. The critical number of ions for the peak coalescence to take place is shown to depend quadratically on the magnetic field strength and to be proportional to the cyclotron radius and inversely proportional to the ion masses.

Binary collisions of charged particles in a magnetic field

Binary collisions between charged particles in an external magnetic field are considered within a classical second-order perturbation theory, i.e., up to contributions which are quadratic in the binary interaction, starting from the unperturbed helical motion of the particles. The calculations are done with the help of an improved binary collisions treatment which is valid for any strength of the magnetic field, where the second-order energy and velocity transfers are represented in Fourier space for arbitrary interaction potentials. The energy transfer is explicitly calculated for a regularized and screened potential which is both of finite range and nonsingular at the origin, and which involves as limiting cases the Debye (i.e., screened) and Coulomb potential. Two distinct cases are considered in detail: (i) the collision of two identical (e.g., electron-electron) particles, and (ii) the collision between a magnetized electron and a uniformly moving heavy ion. The energy transfer involves all harmonics of the electron cyclotron motion. The validity of the perturbation treatment is evaluated by comparing with classical trajectory Monte Carlo calculations which also allows to investigate the strong collisions with large energy and velocity transfer at low velocities. For large initial velocities on the other hand, only small velocity transfers occur. There the nonperturbative numerical classical trajectory Monte Carlo results agree excellently with the predictions of the perturbative treatment.

Long wavelength limit of evolution of nonlinear cosmological perturbations

In the general matter composition where the multiple scalar fields and the multiple perfect fluids coexist, in the leading order of the gradient expansion, we construct all of the solutions of the nonlinear evolutions of the locally homogeneous universe. From the momentum constraint, we derive the constraints which the solution constants of the locally homogeneous universe must satisfy. We construct the gauge invariant perturbation variables in the arbitrarily higher order nonlinear cosmological perturbation theory around the spatially flat Friedmann-Robertson-Walker universe. We construct the nonlinear long wavelength limit formula representing the long wavelength limit of the evolution of the nonlinear gauge invariant perturbation variables in terms of perturbations of the evolutions of the locally homogeneous universe. By using the long wavelength limit formula, we investigate the evolution of nonlinear cosmological perturbations in the universe dominated by the multiple slow rolling scalar fields with an arbitrary potential. The $\ensuremath{\tau}$ function and the $N$ potential introduced in this paper make it possible to write the evolution of the multiple slow rolling scalar fields with an arbitrary interaction potential and the arbitrarily higher order nonlinear Bardeen parameter at the end of the slow rolling phase analytically. It is shown that the nonlinear parameters such as ${f}_{\mathrm{NL}}$ and ${g}_{\mathrm{NL}}$ are suppressed by the slow rolling expansion parameters.

Molecular theory of hydrodynamic boundary conditions in nanofluidics

Motivated by the fundamental questions raised by the most recent experimental achievements in nanofluidics, we propose the first-ever derivation and calculation of the hydrodynamic slip length from the first principles of statistical mechanics, namely, a combination of linear response theory and equilibrium molecular theory of solvation. The slip length derived is related to the fluid organization near the solid surface, as governed by the solid-liquid interaction. In the wide range of shear rates and surface-liquid interactions, the slip length is expressed in terms of the Green-Kubo-Nakano relations as a function of the anisotropic inhomogeneous time-correlation function of density fluctuations of the liquid in contact with the surface. The time dependence of the correlation function is factored out by treating it in the hydrodynamic limit. The spatially inhomogeneous two-body correlation function is represented in the Kirkwood-type approximation as a product of the three-dimensional density distributions of interaction sites of the liquid near the surface and the site-site pair correlations of the bulk liquid. The presented treatment generalizes the phenomenological definition of the friction coefficient (as well as the slip length) to a tensor quantity, which reflects an anisotropic nature of an ordered crystalline or nanopatterned surface. This enables theoretical prediction of friction forces acting aslant to the liquid flow direction for such surfaces. We derive generic analytical expressions for the liquid-surface friction coefficient (and slip length) for an arbitrary surface-liquid interaction potential. We further illustrate it by numerical calculations for a laminar flow of nine different molecular liquids, including water, at ambient conditions in contact with the (100) face-centered cubic cell surface of gold, copper, and nickel modeled by using optimized potential for liquid simulation models for liquids and the Steele potential for crystalline surfaces. The obtained values for slip length range from few to hundreds of nanometers and microns and are consistent with experimental measurements.

The Lieb-Liniger Model as a Limit of Dilute Bosons in Three Dimensions

We show that the Lieb-Liniger model for one-dimensional bosons with repulsive $\delta$-function interaction can be rigorously derived via a scaling limit from a dilute three-dimensional Bose gas with arbitrary repulsive interaction potential of finite scattering length. For this purpose, we prove bounds on both the eigenvalues and corresponding eigenfunctions of three-dimensional bosons in strongly elongated traps and relate them to the corresponding quantities in the Lieb-Liniger model. In particular, if both the scattering length $a$ and the radius $r$ of the cylindrical trap go to zero, the Lieb-Liniger model with coupling constant $g \sim a/r^2$ is derived. Our bounds are uniform in $g$ in the whole parameter range $0\leq g\leq \infty$, and apply to the Hamiltonian for three-dimensional bosons in a spectral window of size $\sim r^{-2}$ above the ground state energy.

Partial nearest neighbor PDF for multi-component material systems

We develop general expressions for partial nearest neighbor probability density functions (PNNPDF’s) for equilibrium multi-component systems, valid for arbitrary partial densities, temperatures, and interaction potentials; thus providing an alternative means of describing structure at the microscopic (atomic scale) level for multi-component material systems. This paper thus complements an earlier paper (U.F. Edgal and D.L. Huber, J. Phys. Chem. B 108, 13777–13788 2004) in the analytical investigation of the classical statistical thermodynamics of multi-component systems. The connection between PNNPDF’s and the commonly employed partial m-body distribution functions is detailed. Results for PNNPDF’s and partial m-body distribution functions applicable for the poisson-distributed multi-component system and the low density binary mixture of hard spheres are provided. The statistical geometry of the systems is further studied through a brief investigation of particle clustering. A major hallmark of the above investigation involves the several multiple integrals and multiple sums encountered, that were quite formidable to perform, even in the absence of particle interactions.

Group-theory classification of oscillations of plasma clusters

Normal oscillations of three-dimensional configurations of dust grains trapped in a spherically symmetric potential well are studied. All possible oscillations of a system consisting of 7–13 grains are classified in terms of group theory. For a Coulomb interaction potential, all the oscillation frequencies are calculated. The frequencies and polarizations of some oscillation modes are obtained for an arbitrary interaction potential.

Spectroscopy of a Thomson atom

An analysis is made of the normal oscillations of three-dimensional configurations of dust grains trapped in a spherically symmetric potential well. The frequencies and polarizations of oscillations of a system consisting of four to six grains are calculated for an arbitrary intergrain interaction potential. The possibility of reconstructing the interaction potential from the experimentally measured coordinates of individual grains is discussed.

Simulation of the growth of nanoparticle aggregates reproducing their natural structure in disperse systems

A three-dimensional continuum model of the generation of aggregates of spherical particles is developed that allows us to reproduce natural conditions of structure formation in real disperse systems with an arbitrary viscosity of a dispersion medium. The exchange between translational and rotational degrees of freedom for aggregates, subaggregates, and single particles at an arbitrary interparticle interaction potential, as well as at the preset functions of the distribution over particle sizes and thicknesses of the adsorption layer, is taken into account in the model. The interaction potential of the particles includes electrostatic (among them, screened), elastic, and van der Waals interactions and allows for the effect of external electric, magnetic, and gravitational fields. Structural and statistical characteristics of aggregates formed at different stages of particle coagulation are studied, and the quasi-ordering of single subaggregates in the elastic adsorption layer of particles is revealed. The model can be applied to the study of optical absorption by large aggregates of nanoparticles with natural structure and the kinetics of sol aggregation as functions of the properties of the adsorption layer of particles and the action of external physical factors.

Vapor-liquid equilibrium and critical behavior of the square-well fluid of variable range: A theoretical study

The vapor-liquid phase behavior and the critical behavior of the square-well (SW) fluid are investigated as a function of the interaction range, lambdain [1.25, 3], by means of the self-consistent Ornstein-Zernike approximation (SCOZA) and analytical equations of state based on a perturbation theory [A. L. Benavides and F. del Rio, Mol. Phys. 68, 983 (1989); A. Gil-Villegas, F. del Rio, and A. L. Benavides, Fluid Phase Equilib. 119, 97 (1996)]. For this purpose the SCOZA, which has been restricted up to now to a few model systems, has been generalized to hard-core systems with arbitrary interaction potentials requiring a fully numerical solution of an integro-partial differential equation. Both approaches, in general, describe well the liquid-vapor phase diagram of the square-well fluid when compared with simulation data. SCOZA yields very precise predictions for the coexistence curves in the case of long ranged SW interaction (lambda>1.5), and the perturbation theory is able to predict the binodal curves and the saturated pressures, for all interaction ranges considered if one stays away from the critical region. In all cases, the SCOZA gives very good predictions for the critical temperatures and the critical pressures, while the perturbation theory approach tends to slightly overestimate these quantities. Furthermore, we propose analytical expressions for the critical temperatures and pressures as a function of the square-well range.

Derivation of 13 Moment Equations for Rarefied Gas Flow to Second Order Accuracy for Arbitrary Interaction Potentials

A recent approach to derive transport equations for rarefied gases from the Boltzmann equation within higher orders of the Knudsen number [H. Struchtrup, Phys. Fluids, 16 (2004), pp. 3921--3934] is used to derive a set of 13 moment equations for arbitrary molecular interaction potentials. It is shown that the new set of equations is accurate to second order, while Grad's original 13 moment equations are of second order accuracy only for Maxwell molecules and Bhatnagar--Gross--Krook models.

Free Energy of Multicomponent Systems Employing Partial NNPDF's

A new scheme analogous to that employed for the determination of the free energy of classical single-component systems, with arbitrary many-body interaction potentials at arbitrary densities and temperatures, is extended to classical multicomponent systems. Self-consistency of the scheme has been shown for some limiting conditions, including the noninteraction limit. Thermodynamic expressions are obtained which are natural extensions of those for the single-component case and readily reduce to their correct forms in the single-component limit. Application to the simple case of low-density mixtures of hard particle gases in 3-D is outlined. It is expected (as in the single-component case) that progressive application to more complex systems (such as quantum mixed systems) will be done under a single mathematical framework, as provided in this paper.

Free Energy of Multicomponent Systems Employing Partial NNPDF'S

A new scheme analogous to that employed for the determination of the free energy of classical single-component systems, with arbitrary many-body interaction potentials at arbitrary densities and temperatures, is extended to classical multicomponent systems. Self-consistency of the scheme has been shown for some limiting conditions, including the noninteraction limit. Thermodynamic expressions are obtained which are natural extensions of those for the single-component case and readily reduce to their correct forms in the single-component limit. Application to the simple case of low-density mixtures of hard particle gases in 3-D is outlined. It is expected (as in the single-component case) that progressive application to more complex systems (such as quantum mixed systems) will be done under a single mathematical framework, as provided in this paper.

Analytical Evaluation of Multicenter Multielectron Integrals of Central and Noncentral Interaction Potentials over Slater Orbitals Using Overlap Integrals and Auxiliary Functions

Using expansion formulas for central and noncentral interaction potentials (CIPs and NCIPs, respectively) in terms of Slater type orbitals (STOs) obtained by the author (I.I. Guseinov, J. Mol. Model., in press), the multicenter multielectron integrals of arbitrary interaction potentials (AIPs) are expressed through the products of overlap integrals with the same screening parameters and new auxiliary functions. For auxiliary functions, the analytic and recurrence relations are derived. The relationships obtained for multicenter multielectron integrals of AIDs are valid for the arbitrary quantum numbers, screening parameters and location of orbitals.

Comment on “Energy partitioning and particle spectra in multicomponent collision cascades”

Four equations proposed by Vicanek et al. have been studied based on the pertinent transport equation. Several difficulties have been found at using them and three corresponding revised ones have been derived rigorously for arbitrary particle interaction potentials.

The Condition of Invariance of the Energy of an Atomic-Molecular System with Respect to Rotations of the Jacobi Coordinates

An additional condition for the wave functions of the ground and excited states of atomic-molecular systems, which follows from the energy invariance with respect to rotations of the Jacobi coordinates, is studied. This condition makes a substantial contribution to the conventional criterion of the quality of wave functions, based on the variational principle for the average energy value. The explicit form of this condition is found for an arbitrary interaction potential dependent on the particle-particle distances and the requirements for realization of this condition are ascertained. Variational calculations of 3He2+ μ−e− and 4He2+ μ−e− mesoatoms in the basis of correlated exponential functions dependent on the particle-particle distances are performed with a detailed optimization of all nonlinear parameters for each individual energy level. A high sensitivity of the condition under study to the quality of approximate wave functions and the efficiency of the detailed optimization of the nonlinear variational parameters in calculation of each individual energy level are demonstrated.

Addition theorems for Slater-type orbitals and their application to multicenter multielectron integrals of central and noncentral interaction potentials.

By the use of complete orthonormal sets of psi(alpha)-ETOs (alpha=1, 0, m1, m2,...) introduced by the author, new addition theorems are derived for STOs and arbitrary central and noncentral interaction potentials (CIPs and NCIPs). The expansion coefficients in these addition theorems are expressed through the Gaunt and Gegenbauer coefficients. Using the addition theorems obtained for STOs and potentials, general formulae in terms of three-center overlap integrals are established for the multicenter t-electron integrals of CIPs and NCIPs that arise in the solution of the N-electron atomic and molecular problem (2hthN) when a Hylleraas approximation in Hartree-Fock-Roothaan theory is employed. With the help of expansion formulae for translation of STOs, the three-center overlap integrals are expressed through the two-center overlap integrals. The formulae obtained are valid for arbitrary quantum numbers, screening constants and location of orbitals.

The Interaction between Two Fluctuating Phospholipid Bilayers

The dependence of the interaction force between two undulating phospholipid bilayers and of the root-mean-square fluctuation of their separation distances on the average separation can be determined once the distribution of the intermembrane separation is known as a function of the applied pressure. However, most of the present theories for interacting membranes start by assuming that the distance distribution is symmetric, a hypothesis invalidated by Monte Carlo simulations. Here we present an approach to calculate the distribution of the intermembrane separation for any arbitrary interaction potential and applied pressure. The procedure is applied to a realistic interaction potential between neutral lipid bilayers in water, involving the hydration repulsion and van der Waals attraction. A comparison with existing experiments is provided.

The concept of elementary excitations in classical statistical mechanics

Exact representations of the generating functional of one-component classical systems with a central interatomic interaction potential are obtained in terms of the functional integrals over auxiliary fields determined in the wavevector space. It is proved that the problem of calculating a classical partition function for a system of particles with an arbitrary interatomic interaction potential is equivalent to the problem of calculating a statistical integral for a system of interacting oscillators with a temperature-dependent Hamiltonian.