Two-Term and multi-term approximation of the nonstationary electron velocity distribution in an electric field in a gas

A very efficient, strict nonstationary multi-term approach has been developed as a generalization of the conventional and the strict nonstationary two-term approximations used for solving the nonstationary , electron Boltzmann equation. As a first application the temporal relaxation of electrons in a model plasma acted upon by a de electric field has been investigated. The results are compared with corresponding ones obtained by the conventional and the strict nonstationary two-term approach as well as with very accurate Monte Carlo simulations. Perfect agreement between nonstationary eight-term Boltzmann and Monte Carlo calculations is found.     

Conformational sampling with Poisson–Boltzmann forces and a stochastic dynamics/Monte Carlo method: Application to alanine dipeptide

We apply a combination of stochastic dynamics and Monte Carlo methods (MC/SD) to alanine dipeptide, with solvation forces derived from a Poisson–Boltzmann model supplemented with apolar terms. Our purpose is to study the effects of the model parameters, such as the friction constant and the size of the electrostatic finite difference grid, on the rate of conformational sampling and on the accuracy of the resulting free energy map. For dialanine, a converged Ramachandran map is produced in significantly less time than what is required by stochastic dynamics or Monte Carlo alone. MC/SD is also shown to be faster, per timestep, than explicit methods. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 1750–1759, 1997     

Langevin equation for the parametric oscillator: a model for ion confinement in a radio frequency trap

We have calculated 〈x ²〉 and 〈v ²〉 for an ion in a rf trap being cooled by a stochastic force. The results agree with earlier work (using a Fokker-Planck equation) as long as there is white noise. However, there is an appreciable reduction for both quantities, if non-Markovian processes are included.     

Boundary element solution of the linear Poisson-Boltzmann equation and a multipole method for the rapid calculation of forces on macromolecules in solution

The Poisson-Boltzmann equation is widely used to describe the electrostatic potential of molecules in an ionic solution that is treated as a continuous dielectric medium. The linearized form of this equation, applicable to many biologic macromolecules, may be solved using the boundary element method. A single-layer formulation of the boundary element method, which yields simpler integral equations than the direct formulations previously discussed in the literature, is given. It is shown that the electrostatic force and torque on a molecule may be calculated using its boundary element representation and also the polarization charge for two rigid molecules may be rapidly calculated using a noniterative scheme. An algorithm based on a fast adaptive multipole method is introduced to further increase the speed of the calculation. This method is particularly suited for Brownian dynamics or molecular dynamics simulations of large molecules, in which the electrostatic forces must be calculated for many different relative positions and orientations of the molecules. It has been implemented as a set of programs in C++, which are used to study the accuracy and speed of this method for two actin monomers.     

Multiple time scale dynamics of distance fluctuations in a semiflexible polymer: a one-dimensional generalized Langevin equation treatment

Time-dependent fluctuations in the distance x(t) between two segments along a polymer are one measure of its overall conformational dynamics. The dynamics of x(t), modeled as the coordinate of a particle moving in a one-dimensional potential well in thermal contact with a reservoir, is treated with a generalized Langevin equation whose memory kernel K(t) can be calculated from the time-correlation function of distance fluctuations C(t) identical with x(0)x(t). We compute C(t) for a semiflexible continuum model of the polymer and use it to determine K(t) via the GLE. The calculations demonstrate that C(t) is well approximated by a Mittag-Leffler function and K(t) by a power-law decay on time scales of several decades. Both functions depend on a number of parameters characterizing the polymer, including chain length, degree of stiffness, and the number of intervening residues between the two segments. The calculations are compared with the recent observation of a nonexponential C(t) and a power law K(t) in the conformational dynamics within single molecule proteins [Min et al., Phys. Rev. Lett. 94, 198302 (2005)].     

The conductivity of dilute electrolyte solutions: Expanded lee and wheaton equation for symmetrical,unsymmetrical and mixed electrolytes

The conductivity equation of Lee and Wheaton is expanded in a power series in the ionic strength I. Different levels of approximation are analyzed and it is shown that for unsymmetrical electrolytes it is necessary to keep the terms I^3/2lnI and I² to maintain the performance of the full equation.     

耦合生物细胞体系中内噪声对检测弱刺激的积极作用

By constructing a mesoscopic stochastic model for intracellular calcium oscillations in coupled cell system, we investigated the influence of internal noise on the detection of weak stimulation using the chemical Langevin equation (CLE). We found that an optimal cell size V existed for a coupled cell chain length N and an optimal value of N existed for a given cell size V. At these values, the collective calcium oscillations showed the best performance, indicating the occurrence of“system size resonance (SSR)”or“internal noise stochastic resonance (INSR)”. And such a phenomenon was robust to the coupling strength. Living cells may have learned to exploit the internal noise to detect weak stimulation via the mechanism of INSR, and then encode information to specifically regulate distinct cellular functions. It is interesting to note that the optimal cell size is always present at V抑103 μm3, which is close to the real living cell size in vivo. Since the internal noise in living systems can not be ignored and the systems may often encounter weak stimulations, our findings might have significance for stimulation detecting processes in living systems.     

Conductance equation of dilute mixed strong electrolytes. II. Hydrodynamic and osmotic terms in relaxation field

The hydrodynamic and osmotic terms in the relaxation field are computed up to order linear in concentration for a dilute solution of mixed strong electrolytes using the primitive model. The computation is based on the relative function μ_ji computed in part earlier for the purely electrostatic interaction and in part in this work for the hydrodynamic interaction assuming the adequacy of the Fuoss velocity field. The results are compared with the earlier computations of Fuoss and Onsager, Murphy and Cohen, and Falkenhagen, Ebeling, and Kraeft. It is found that some terms of the new results are in agreement with the earlier computations and the other terms represent an improved computation.     

General equation to describe the dependence of the gas holdup time on the initial temperature and program rate in temperature-programmed gas chromatography

Summary The influence of the program rate and initial temperature on the holdup time in linear temperature-programmed gas chromatography was studied. An equation describing the dependence of the holdup time on the program rate and the initial temperature is given. Experimental verification of the equation was carried out on three capillary columns coated with stationary phases having different polarities, at three program rates and four initial temperatures. Enlarged text of the paper presented at the Eighth International Symposium on Capillary Chromatography, Riva del Garda, Italy, May 19–21, 1987.     

The role of acoustic phonon scattering in charge transport in organic semiconductors: a first-principles deformation-potential study

The electron-acoustic phonon scattering for charge transport in organic semiconductors has been studied by first-principles density functional theory and the Boltzmann transport equation with relaxation time approximation. Within the framework of deformation-potential theory, the electron-longitudinal acoustic phonon scattering probability and the corresponding relaxation time have been obtained for oligoacene single crystals (naphthalene, anthracene, tetracene and pentacene). Previously, the electron-optic...     

Attachment and ionization in a self-consistent treatment of the electron kinetics of a collision-dominated rf plasma

This paper deals with the self-consistent determination of the rf field amplitude for sustaining the steady-state collision-dominated weakley ionized plasmas in the bulk of the rf discharge and of the time-resolved behavior of the isotropic part of the distribution function as well as of relevant macroscopic quantities in plasmas whose particle loss is dominantly determined by electron attachment. The strict timeresolved treatment is based on the nonstationary Boltzmann equation of the electrons and its numerical solution including, apart from electron number conservative collision processes, the electron attachment and ionization. The investigations are related to an rf plasma in a model gas and in SF₆ and are performed for reduced rf field frequencies around 10 MHz Torr^–1 which are of particular interest from the point of application of rf discharges for plasma processing. The numerical results show that a large field amplitude of around 160 V cm^–1 Torr^–1 is necessary to maintain the discharge and that the isotropic distribution, the relevant collision frequencies for attachment and ionization, and the electron density undergo a large modulation during a period of the rf field.     

Continuous development of schemes for parallel computing of the electrostatics in biological systems: Implementation in DelPhi

Due to the enormous importance of electrostatics in molecular biology, calculating the electrostatic potential and corresponding energies has become a standard computational approach for the study of biomolecules and nano‐objects immersed in water and salt phase or other media. However, the electrostatics of large macromolecules and macromolecular complexes, including nano‐objects, may not be obtainable via explicit methods and even the standard continuum electrostatics methods may not be applicable due to high computational time and memory requirements. Here, we report further development of the parallelization scheme reported in our previous work (Li, et al., J. Comput. Chem. 2012, 33, 1960) to include parallelization of the molecular surface and energy calculations components of the algorithm. The parallelization scheme utilizes different approaches such as space domain parallelization, algorithmic parallelization, multithreading, and task scheduling, depending on the quantity being calculated. This allows for efficient use of the computing resources of the corresponding computer cluster. The parallelization scheme is implemented in the popular software DelPhi and results in speedup of several folds. As a demonstration of the efficiency and capability of this methodology, the electrostatic potential, and electric field distributions are calculated for the bovine mitochondrial supercomplex illustrating their complex topology, which cannot be obtained by modeling the supercomplex components alone. © 2013 Wiley Periodicals, Inc.     

Comment on "The surface potential of a spherical colloid particle: functional theoretical approach"

In a work published in this journal by Z.W. Wang, G.Z. Li, D.R. Guan, X.Z. Yi, and A.J. Lou [J. Colloid Interface Sci. 246 (2002) 302], an iterative method for the determination of the potential around a colloidal particle is presented. It is claimed that successive terms of the iteration series converge to the exact solution of the Poisson-Boltzmann equation. This claim seems to be unfounded when the analytical expressions of the iteration terms are compared with well established numerical data.     

On the Kinetic Study of the Electron Velocity Distribution Function in the Cathode Region of a Glow Discharge

The recently developed multi-term approach for solving the space-dependent electron Boltzmann equation is adopted to study, under the extreme field conditions occurring in the cathode region of the dc glow discharge, the energy distribution and all important macroscopic quantities of the electrons. It is shown that even under the difficult conditions in the cathode region an efficient determination of significant components of the electron velocity distribution by the multi-term treatment of the kinetic equation is possible. These significant components can be calculated with remarkable accuracy and with relatively less computational expenditure by a multi-term approach with the inclusion of about ten coefficients of the Legendre polynomial expansion of the velocity distribution. Based on the results for a typical example, main features of the electrons in the cathode region are presented and discussed.     

pKA in proteins solving the Poisson–Boltzmann equation with finite elements

Knowledge on pK_A values is an eminent factor to understand the function of proteins in living systems. We present a novel approach demonstrating that the finite element (FE) method of solving the linearized Poisson–Boltzmann equation (lPBE) can successfully be used to compute pK_A values in proteins with high accuracy as a possible replacement to finite difference (FD) method. For this purpose, we implemented the software molecular Finite Element Solver (mFES) in the framework of the Karlsberg+ program to compute pK_A values. This work focuses on a comparison between pK_A computations obtained with the well‐established FD method and with the new developed FE method mFES, solving the lPBE using protein crystal structures without conformational changes. Accurate and coarse model systems are set up with mFES using a similar number of unknowns compared with the FD method. Our FE method delivers results for computations of pK_A values and interaction energies of titratable groups, which are comparable in accuracy. We introduce different thermodynamic cycles to evaluate pK_A values and we show for the FE method how different parameters influence the accuracy of computed pK_A values. © 2015 Wiley Periodicals, Inc.     

SMPBS: Web server for computing biomolecular electrostatics using finite element solvers of size modified Poisson‐Boltzmann equation

SMPBS (Size Modified Poisson‐Boltzmann Solvers) is a web server for computing biomolecular electrostatics using finite element solvers of the size modified Poisson‐Boltzmann equation (SMPBE). SMPBE not only reflects ionic size effects but also includes the classic Poisson‐Boltzmann equation (PBE) as a special case. Thus, its web server is expected to have a broader range of applications than a PBE web server. SMPBS is designed with a dynamic, mobile‐friendly user interface, and features easily accessible help text, asynchronous data submission, and an interactive, hardware‐accelerated molecular visualization viewer based on the 3Dmol.js library. In particular, the viewer allows computed electrostatics to be directly mapped onto an irregular triangular mesh of a molecular surface. Due to this functionality and the fast SMPBE finite element solvers, the web server is very efficient in the calculation and visualization of electrostatics. In addition, SMPBE is reconstructed using a new objective electrostatic free energy, clearly showing that the electrostatics and ionic concentrations predicted by SMPBE are optimal in the sense of minimizing the objective electrostatic free energy. SMPBS is available at the URL: smpbs.math.uwm.edu © 2017 Wiley Periodicals, Inc.     

The use of a nonlinear equation for calculation of the retention indices of polar substances in gas chromatography with linear temperature programming

New possibilities for using the equation that takes into account the nonlinear variation of parameters of the reference n-alkane scale for the calculation of retention indices of polar substances at different modes of temperature programming were considered. The advantages of this equation over the linear scale used traditionally were demonstrated in relation to C₃—C₁₁ alkan-1-ols. The equation appears to have considerable promise regarding the search for the equivalent isothermal index.     

Adaptive multilevel finite element solution of the Poisson–Boltzmann equation I. Algorithms and examples

This article is the first of two articles on the adaptive multilevel finite element treatment of the nonlinear Poisson–Boltzmann equation (PBE), a nonlinear eliptic equation arising in biomolecular modeling. Fast and accurate numerical solution of the PBE is usually difficult to accomplish, due to the presence of discontinuous coefficients, delta functions, three spatial dimensions, unbounded domain, and rapid (exponential) nonlinearity. In this first article, we explain how adaptive multilevel finite element methods can be used to obtain extremely accurate solutions to the PBE with very modest computational resources, and we present some illustrative examples using two well‐known test problems. The PBE is first discretized with piece‐wise linear finite elements over a very coarse simplex triangulation of the domain. The resulting nonlinear algebraic equations are solved with global inexact Newton methods, which we have described in an article appearing previously in this journal. A posteriori error estimates are then computed from this discrete solution, which then drives a simplex subdivision algorithm for performing adaptive mesh refinement. The discretize–solve–estimate–refine procedure is then repeated, until a nearly uniform solution quality is obtained. The sequence of unstructured meshes is used to apply multilevel methods in conjunction with global inexact Newton methods, so that the cost of solving the nonlinear algebraic equations at each step approaches optimal O(N) linear complexity. All of the numerical procedures are implemented in MANIFOLD CODE (MC), a computer program designed and built by the first author over several years at Caltech and UC San Diego. MC is designed to solve a very general class of nonlinear elliptic equations on complicated domains in two and three dimensions. We describe some of the key features of MC, and give a detailed analysis of its performance for two model PBE problems, with comparisons to the alternative methods. It is shown that the best available uniform mesh‐based finite difference or box‐method algorithms, including multilevel methods, require substantially more time to reach a target PBE solution accuracy than the adaptive multilevel methods in MC. In the second article, we develop an error estimator based on geometric solvent accessibility, and present a series of detailed numerical experiments for several complex biomolecules. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 1319–1342, 2000     

Self-Consistent Analysis of a Helium Plasma in a Cylindrical Hollow Cathode

The helium plasma in a cylindrical, axially symmetric direct current hollow cathode discharge is theoretically investigated. A self-consistent hybrid method is used to describe the radial behaviour of the plasma components and the electric field around the axial centre of the discharge. The hybrid method includes the solution of an equation system consisting of Poissons equation and fluid equations for electrons, ions and excited helium atoms. Using the electric field and excited atom densities obtained in this system, the space-dependent transport and collision rate coefficients of the electrons are obtained by a kinetic treatment of the electrons. This treatment is based on a powerful multi-term method for solving the inhomogeneous Boltzmann equation in cylindrical coordinates. The theoretical results obtained for a discharge current of some mA and a pressure of few Torr are compared with available experimental ones.