Boltzmann Integral Research Articles

Assessing Guest Diffusion in Nanoporous Materials by Boltzmann's Integration Method

With the introduction of interference microscopy to diffusion studies in zeolites, the direct, space-resolved observation of transport diffusion in nanoporous materials has become possible for the first time. The method is based on recording transient concentration profiles during molecular uptake or release so that the local intracrystalline diffusivities directly result by a microscopic (“differential”) application of Fick's second law. We show that under certain conditions the thus accessible wealth of information on the transport characteristics of nanoporous materials may even be surpassed by the application of an integration method that Ludwig Boltzmann invented more than a century ago and that thus encounters a large new field of application. The procedure is illustrated by discussing its application to the adsorption process of methanol on FER-type zeolite crystallites. The advantages and limitations of the method in comparison with the differential application of Fick's second law are discussed.

Swell and Slanting-Fetch Effects on Wind Wave Growth

AbstractWind-sea generation was observed during two experiments off the coast of North Carolina. One event with offshore winds of 9–11 m s−1 directed 20° from shore normal was observed with eight directional stations recording simultaneously and spanning a fetch from 4 to 83 km. An opposing swell of 1-m height and 10-s period was also present. The wind-sea part of the wave spectrum conforms to established growth curves for significant wave height and peak period, except at inner-shelf stations where a large alongshore wind-sea component was observed. At these short fetches, the mean wave direction θm was observed to change abruptly across the wind-sea spectral peak, from alongshore at lower frequencies to downwind at higher frequencies. Waves from another event with offshore winds of 6–14 m s−1 directed 20°–30° from shore normal were observed with two instrument arrays. A significant amount of low-frequency wave energy was observed to propagate alongshore from the region where the wind was strongest. These measurements are used to assess the performance of some widely used parameterizations in wave models. The modeled transition of θm across the wind-sea spectrum is smoother than that in the observations and is reproduced very differently by different parameterizations, giving insights into the appropriate level of dissipation. Calculations with the full Boltzmann integral of quartet wave–wave interactions reveal that the discrete interaction approximation parameterization for these interactions is reasonably accurate at the peak of the wind sea but overpredicts the directional spread at high frequencies. This error is well compensated by parameterizations of the wind input source term that have a narrow directional distribution. Observations also highlight deficiencies in some parameterizations of wave dissipation processes in mixed swell–wind-sea conditions.

The WRT method for the computation of non-linear four-wave interactions in discrete spectral wave models

An overview is given of the WRT method for the computation of weakly resonant non-linear four-wave interactions in a gravity wave spectrum and its application in discrete spectral wave models. The WRT method is based on Webb's [Webb, D.J., 1978. Nonlinear transfer between sea waves. Deep-Sea Res., 25, 279–298.] transformation of the Boltzmann integral and the numerical method introduced by Tracy and Resio [Tracy, B.A., Resio, D.T., 1982. Theory and calculation of the nonlinear energy transfer between sea waves in deep water. WIS technical report 11. US Army Engineer Waterways Experiment Station, Vicksburg, Mississippi, USA, 47 pp.]. It is shown that Webb's method produces an attractive set of integrable equations. Moreover, the Jacobian term arising from the integration over the frequency delta-function in the Boltzmann integral has a singularity well outside the energy containing part of the wave spectrum. A description is given of methods for computing the integration space for a given discrete spectral grid, both for deep and finite depth water. Thereafter, the application of Webb's method to discrete spectral wave models is described, followed by a summary of techniques reducing the computational workload while retaining sufficient accuracy. Finally, some methods are presented for the optimal inclusion of the WRT method in operational discrete spectral wave prediction models.

Implementation of a microcanonical variational transition state theory for direct dynamics calculations of rate constants

An implementation of the variational quantum RRKM program is presented to utilize the direct ab initio dynamics approach for calculating k(tE, J), k(E) and k(T) within the framework of the microcanonical transition state (μTST) and microcanonical variational TST (μVT) theories. An algorithm including tunneling contributions in Beyer-Swinehart method for calculating microcanonical rate constants is also proposed. An efficient piece-wise interpolation method is developed to evaluate the Boltzmann integral in calculation of thermal rate constants. Calculations on several test reactions, namely the H(D)2CO→ H(D)2 + CO, CH2CO→CH2 + CO and CH4 + H→CH3 + H2 reactions, show that the results are in good agreement with the previous rate constants calculations. This approach would require much less computational resource.

Simplification of the Boltzmann collision integral

Close collision events play an important role in situations where the Coulomb logarithm approaches unity. The five-fold Boltzmann collision integral takes into account close collision events. However, the five-fold Boltzmann collision integral is generally considered too complex for practical use. In a previous paper [Y. Chang, Phys. Fluids B 4, 313 (1992)], the Boltzmann integral with a Maxwellian velocity distribution function for the field particles was reduced to a three-fold integral with a kernel function. However, the kernel function was expressed in the form of an infinite series. In this brief communication, a finite form for the kernel function is obtained and the Boltzmann integral is finally reduced to a simple form exactly.

Numerical Computations of the Nonlinear Energy Transfer of Gravity-Wave Spectra in Finite Water Depths

A new computational scheme for calculating the nonlinear energy transfer in finite-depth gravity wave spectra has been developed by extending the methods established by Masuda (1980) and Komatsu et al. (1993). In this paper, the formulations for the numerical computation of the nonlinear energy transfer for finite-depth water waves are shown with the analytical solution around the singular point of the Boltzmann integral. The numerical computational procedure for the proposed method is described in detail. Some results of numerical examinations for finite water depth, assuming two different types of directional spectra, are shown with discussions on the validity of the proposed method.

A fully nonlinear regional wave model for the Bight of Abaco: 1. Nonlinear‐transfer computation

This paper is the first in a series of papers describing a fully nonlinear regional wave model for the Bight of Abaco, Bahamas. It discusses this model's hybrid representation for nonlinear transfer and the numerical errors associated with this representation. This discussion extends a number of results previously reported by Snyder et al. [1993], doubling both the Boltzmann integration resolution and the spectral resolution of the resulting nonlinear‐transfer estimates and evaluating the errors associated with both resolutions. It also better resolves the structure of the negative midfrequency lobe of the nonlinear transfer for JONSWAP input [Hasselmann et al, 1973], evaluates the errors associated with the diagnostic range of the nonlinear‐transfer computation, and extends the hybrid representation to various finite depths characteristic of the Abaco Bight. It also extends the previous discussion of truncations of the hybrid representation, defining some renormalized hybrid implementations of the discrete‐interaction approximation [Hasselmann et al., 1985] and generalizing its selection algorithm to improve the accuracy of this approximation (at some cost in efficiency). Finally, it develops a systematic scheme for recursively selecting hybrid coefficients to define a family of recursively optimized renormalized hybrid truncations. Because this scheme selects coefficient groups (the interactions for which are scaled versions of one another or of mirror images of one another) rather than individual coefficients and because it optimizes over multiple spectral inputs, the resulting truncations perform well over a full range of peak frequency. The resulting truncations for a nominal spectral resolution of 16 prognostic and four diagnostic wave‐number bands and 12 angle bands include some truncations that essentially trade a factor of 10 in efficiency for a factor of 10 in accuracy (relative to the discrete‐interaction approximation). Other truncations, running only 600 times slower than the discrete‐interaction approximation, give a very accurate representation of the full nonlinear transfer. These truncations, employed in sequential fashion and extended to multiple spectral resolution, enable a relatively accurate and efficient staged inverse modeling of the action‐balance equation. By first inverse modeling to convergence at nominal spectral resolution, then inverse modeling to convergence at double spectral resolution (starting from the converged results for nominal spectral resolution), it should be possible to extrapolate the results of the inverse modeling with an error no worse than a few percent (assuming that the errors in the results of the inverse modeling generated by errors in the nonlinear‐transfer computation are no worse than these generating errors and that the impact of other numerical errors can be similarly contained).

Phenomenological models of cellulose pyrolysis

Using yields vs. residence time and temperature from 50 to 1000 ms and 650–900°C, measured with the ultra pyrolysis system at the University of Western Ontario (UWO) we establish an approximate total gaseous yield function Y(t, T). With UWO data, we also establish approximate correlations between individual gaseous yields (CO, CO 2, C 2H 4, CH 4, C 2H 2, C 2H 6 and C 3H 6) and the total gaseous yield that could be used to give Y i(t, T) for individual gases. We further extend Y(t, T) using shock tube pyrolysis measurements from 0.3 to 2 ms and 900 to 2100°C made at Kansas State University (KSU). In doing so, we develop a global decay model that gives analytical time and temperature dependencies for cellulose, activated cellulose, tar, prompt total gas and late total gas. We next examine the impact of heating rates and heat transfer upon pyrolysis of cellulose using slow pyrolysis data obtained by thermogravimetric analysis at the Colorado School of Mines (CSM). In this effort, we first develop an accurate general relationship for Boltzmann integrals. Then using an analytically convenient Arrhenius reaction rate (ARR) we examine data taken at varying heating rates and with three Biot numbers. We find some phenomenological analytical relationships giving ARR parameter dependencies on heating rate and particle size that appear indicative of heat transfer impacts. If adequate data becomes available these relationships might be applied to hemicellulose and lignin. Then the pyrolysis rates of any plant species might be predicted in terms of the pyrolytic characteristics of their cellulose, hemicellulose and lignin components.

Poisson Boltzmann Integral Equation in A Charged Capillary and Its Numerical Solutions

Poisson Boltzmann Integral Equation in A Charged Capillary and Its Numerical Solutions

The experimental nature of stress relaxation and recovery of high-frequency information

The Fourier transform of stress-relaxation curves allows the data to be examined as a continuous frequency spectrum of loss and storage moduli. Information up to kilohertz frequencies can be recorded and analysed, but the high-frequency information is affected by the finite time taken to strain the specimen. This paper examines the nature of the stress-relaxation experiment as it realistically appears; a finite loading period during which both stress and strain are functions of time, followed by the pure relaxation part of the experiment, during which the strain is held constant. The discrepancy between the forms of the relaxation function obtained after a finite loading period (given by the Boltzmann integral) and after an infinitely small loading period (defined by the Heaviside step function) is considered for a linear viscoelastic material, and the effect that the loading time has on the relaxation function is shown. A deconvolution relationship between stress and strain for the simple case of linear strain application is established, and the effect of this deconvolution over the loading part of the experiment to retrieve the higher frequency components of the relaxation function is examined. Deconvolution using the transform of the analytical strain function (as applied in the experiment) allows the quality of the high-frequency data to be improved considerably.

Finite representation of an infinite bulk system: Solvent boundary potential for computer simulations

An approach is developed to obtain statistical properties similar to those of an infinite bulk system from computer simulations of a finite cluster. A rigorous theoretical formulation is given for the solvent boundary potential which takes the influence of the surrounding bulk into account. The solvent boundary potential is the configuration-dependent solvation free energy of an effective cluster composed of an arbitrary solute and a finite number of explicit solvent molecules embedded inside a hard sphere of variable radius; the hard sphere does not act directly on the solute or the explicit solvent molecules, and its radius varies according to the instantaneous configurations. The formulation follows from an exact separation of the multidimensional configurational Boltzmann integral in terms of the solvent molecules nearest to the solute and the remaining bulk solvent molecules. An approximation to the solvent boundary potential is constructed for simulations of bulk water at constant pressure, including the influence of van der Waals and electrostatic interactions. The approximation is illustrated with calculations of the solvation free energy of a water molecule and of sodium and potassium ions. The influence of bulk solvent on the conformational equilibrium of molecular solutes is illustrated by performing umbrella sampling calculations of n-butane and alanine dipeptide in water. The boundary potential is tested to examine the dependence of the results on the number of water molecules included explicitly in the simulations. It is observed that bulk-like results are obtained, even when only the waters in the first hydration shell are included explicitly.

Implementation of an efficient scheme for calculating nonlinear transfer from wave‐wave interactions

Nonlinear transfer from wave‐wave interactions is an important term in the action‐balance equation governing the evolution of the surface‐gravity‐wave field. Computation of this term, however, has hitherto been so consuming of computer resources that its full representation has not been feasible in nonparametric two‐dimensional computer models of this equation. This paper describes the implementation of a hybrid computational scheme, incorporating a simplification first proposed by Thacker into the EXACT‐NL Boltzmann integration scheme of Hasselmann and Hasselmann. This hybrid scheme retains EXACT‐NL's symmetry, precision, and two‐stage structure, but, by transferring a spectrum‐independent preintegration from the second stage to the first, dramatically accelerates the resulting second‐stage computation, enabling a relatively efficient and precise determination of nonlinear transfer in two‐dimensional wave models. Physically, this preintegration collects together in single hybrid interactions multiple interactions belonging to identical spectral‐band quadruplets. Thus all possible interactions are represented, and these interactions are represented in a uniquely efficient manner consistent with the spectral representation. We compute the coefficients in the resulting second‐stage hybrid sum by essentially sorting and pre‐summing the coefficients generated by a piecewise‐constant first‐stage EXACT‐NL computation, using a variant of EXACT‐NL that replaces the gather‐scatter operations with a simpler bin‐assignment procedure and employs a somewhat simpler set of integration variables. By exploiting the natural scaling of the integrand and partially pre‐summing prior to sorting, we are able to further improve the efficiency of this computation for the deep‐water case and to refine its integration‐grid resolution almost to convergence. In wave‐model computations of nonlinear transfer, vectorization on the spatial grid points of the model and selective truncation of the hybrid sum potentially reduce the working computation time for a single model time step to well under one Cray Y‐MP single‐processor CPU second per hundred grid points, while preserving a remarkably faithful representation of the full transfer.

A numerical study of nonlinear energy fluxes due to wave-wave interactions Part 1. Methodology and basic results

Nonlinear transfer due to wave-wave interactions was first described by the Boltzmann integrals of Hasselmann (1961) and has been the subject of modelling ever since. We present an economical method to evaluate the complete integral, which uses selected scaling properties and symmetries of the nonlinear energy transfer integrals to construct the integration grid. An important aspect of this integration is the inherent smoothness and stability of the computed nonlinear energy transfer. Energy fluxes associated with the nonlinear energy transfers and their behaviour within the equilibrium range are investigated with respect to high-frequency power law, peak frequency, peakedness, spectral sharpness and angular spreading. We also compute the time evolution of the spectral energy and the nonlinear energy transfers in the absence of energy input by wind or dissipated by wave breaking. The response of nonlinear iterations to perturbations is given and a formulation of relaxation time in the equilibrium range is suggested in terms of total equilibrium range energy and the nonlinear energy fluxes within the equilibrium range.

Failure of the Fokker–Planck approximation to the Boltzmann integral for (1/r) potentials

It is shown that the widely used Landau–RMJ collision term provides an inadequate description of collisional relaxation caused by Coulomb or gravitational interactions because of its inability to correctly describe large angle scattering events. This is established by extending the Taylor series expansion of the Boltzmann integral to fourth order thereby obtaining an approximation of the form JB[ fa, fb]ΔΩχ =ln(xmax/xmin)L2[ fa, fb] +(x2max−x2min)L4[ fa, fb] +O(x4max), where xmax=sin(χmax/2), xmin=sin(χmin/2), and χ is the scattering angle in the center of mass frame. For χmax =180°, the first term in the series is the familiar Landau–RMJ collision term. The higher-order collision term L4 is derived and its properties are discussed. In particular it is shown that for like-particle and light–heavy particle collision, L4 can exceed L2 ln Λ over large regions of velocity space whenever the particle distribution functions fa, fb differ significantly from common equilibrium distributions. The conclusion drawn from this result is that large angle scattering events play a more significant role in the evolution of such distributions than currently believed.

Shallow‐Water Waves. I: Theory

Based on a generalized treatment of geometric scaling laws inherent in the Boltzmann integral for nonlinear wave-wave interactions, a theoretical framework for the characteristic form of equilibrium spectra in water of arbitrary depth is developed. A theoretical consequence of this spectral equilibrium range formulation is that a strong, constant flux of wave energy exists through the equilibrium range of the spectrum, analogous to the Komolgorov range in a turbulence spectrum. One possibly important conclusion from the theoretical material presented here is that the equilibrium range of a wind wave spectrum should be a depth-independent constant in terms of a representation of the spectral action density, rather than the spectral energy density. Another point of particular interest to wave generation in shallow water is the apparent existence of a natural limit to the evolution of the frequency of the spectral peak into lower frequencies. This removes the necessity of requiring that only bottom friction can limit wave growth in shallow water.

An analysis of the asymmetric part of electron–electron Boltzmann integral

A numerical analysis of the asymmetric part of electron–electron collision integral is presented. The results are given in the form of graphs for two commonly considered plasma situations: the collision-dominated case (symmetric part of electron distribution is Maxwellian) and the field-dominated case (symmetric part of electron distribution is Druyvesteynian). The importance of the asymmetric part of e-e collision integral in the Boltzmann equation is also discussed.

The Estimation of Wind-Wave Generation in a Discrete Spectral Model

The estimation of wind-wave generation using a new discrete spectral model is compared to Hasselmann et al.'s (1976) parametric model and to models driven primarily by direct transfer of energy from the atmosphere into the surface waves. The main source term in this new model is a new parameterization of the net energy transfer due to nonlinear wave-wave interactions. After calibration of the wave-wave interaction source term to resemble the form of the solution to the complete Boltzmann integrals, the discrete spectral model is able to reproduce the fetch-limited results of Hasselmann et al. and fits well within the envelope of duration-limited growth curves from recent investigations. Since this model is discretized into frequency and direction components, a finite-difference scheme is used to model propagation effects. The formulation of this model allows simulations in oceanic conditions to consider both wave growth under local winds and swell decay of waves passing through a region simultaneously.

Transformation of the viscoelastic functions of calcified tissues and interfacial biomaterials into a common representation

Since both connective and calcified tissues are markedly viscoelastic in nature, an understanding of the behavior of these tissues intrinsically as materials on their own, as well as in composite formation with synthetic implants, is of prime importance in order to predict and anticipate materials' design and function. Thus considerable interest has developed in recent years with respect to measurements of the viscoelastic properties of biological materials. However, attempts to characterize the viscoelasticity of calcified tissues have involved many different experimental procedures; hence results appear in terms of different functions, e.g. relaxation modulus, creep compliance. Since this diversity precludes a simple useful comparison of the results, the present study was initiated so that measured functions could be cast into a common representation, and thus compared. Linear viscoelasticity theory implies definite exact relationships between the functions. Using these relations, experimental results on bone, dentin and implant materials presently used to interface to the natural tissues, e.g. polymethyl methacrylate and high density linear polyethylene, were transformed into the complex dynamic modulus representation. Analysis shows that the results of experiments on bone are not in agreement as to dispersion (i.e. change of modulus with frequency) and its variation with strain. Further, analysis of the internal consistency of some experiments demonstrates a violation of the Boltzmann integral which indicates that linear viscoelasticity (almost invariably assumed by workers in the field) fails for bone in compression. It is concluded that the dynamic behavior of bone is not as well understood as has been thought heretofore; direction is given for future experiments.

Verification of the nature of the approximation of the Boltzmann integral by Krook's kinetic relaxation model

Up to now a significant number of aerodynamic problems have been solved with the aid of Krook's kinetic relaxation model. However, because of the absence of reliable solutions of boundary problems for the Boltzmann equation, the correctness of the assumed model of the collision integral remains unclarified. In the present paper, in order to verify the nature of the approximation of the collision operator by the given model, a machine experiment is undertaken. The Boltzmann collision operator is computed for a variety of test functions characteristic of the motion of a rarefied gas and the values obtained according to it are compared to the Krook model. Some physical hypotheses embedded in the relaxation model are also examined.

Evaluation of Boltzmann Integrals for Inelastic Collision and Radiative Processes in Monatomic Plasmas

Within the scope of a moment method collision integrals of the Boltzmann type are formulated and evaluated for the different excited particle states of a stationary monatomic plasma macroscopically at rest. All collision and radiative processes except the three body recombination are considered. The calculations are performed within Grad’s 8-moment approximation of the momentum distribution functions of the material particle components. Assuming axial symmetry of the differential collision cross sections the evaluation leads to linear functions of generalized Chapman-Cowling integrals for binary inelastic particle interactions and to Einstein and photoionization coefficients for the radiative processes. Simplifications are given in special and limiting cases.