Density expansion of the correlation function of a hard sphere gas

1979 ◽  
Vol 57 (3) ◽  
pp. 466-476 ◽  
Author(s):  
D. G. Blair ◽  
N. K. Pope ◽  
S. Ranganathan
Keyword(s):  
Correlation Function ◽  
Fourier Transforms ◽  
Hard Spheres ◽  
Kinetic Equations ◽  
Ideal Gas ◽  
Zero Density ◽  
Classical Gas ◽  
Density Expansions ◽  
The Moment ◽  
Derivatives Of

Using the grand canonical ensemble, the classical Van Hove correlation function G(r, t) is expanded in a power series in density. The zero density limit is the ideal gas result. We have derived, for a classical gas of hard spheres, exact expressions for [Formula: see text], the zero density derivative of the correlation function, and its Fourier transforms. These involve only two particle dynamics. The first two terms in the density expansions provide representation of the correlation functions for appropriate ranges of density and correlation function arguments. We also show that the same result can be obtained from generalized kinetic equations. To this order in density, the moment relations and the time derivatives of I(q, t) at t = 0+ are satisfied. Numerical results are compared with those of Mazenko, Wei, and Yip and with those of the Boltzmann equation and they show the expected behavior.

Low Density Density Derivative of the Self-Correlation Function of a Hard Sphere Gas

1974 ◽  
Vol 52 (10) ◽  
pp. 902-916
Author(s):  
D. G. Blair ◽  
N. K. Pope
Keyword(s):  
Correlation Functions ◽  
Hard Sphere ◽  
Hard Spheres ◽  
Time Model ◽  
Direct Derivation ◽  
Linearized Boltzmann Equation ◽  
Langevin Diffusion ◽  
Classical Gas ◽  
Density Derivative ◽  
Derivatives Of

For the classical gas of hard spheres, exact expressions are derived for [∂Is(r,t)/∂n]n = 0, [∂Is(q,t)/∂n]n = 0, and [∂Ss(q,ω)/∂n]n = 0, the density derivatives of the Van Hove self-correlation functions. The relationships between the direct derivation using the activity expansion, and the derivations based on the generalized kinetic equation and the linearized Boltzmann equation are discussed. Properties of these density derivatives and of the corresponding self-correlation functions, as given by the first two terms of the density expansion, are discussed in detail. The expressions are compared with the hard sphere results of Desai and Nelkin. of Sears and of Mazenko et al.; and also with the predictions of the single relaxation time model and the Langevin diffusion model.

The combined effect of lattice’s uniaxial strains and electron–phonon interaction’s screening on TBEC of the intersite bipolarons

2016 ◽  
Vol 30 (26) ◽  
pp. 1650186
Author(s):  
B. Yavidov ◽  
SH. Djumanov ◽  
T. Saparbaev ◽  
O. Ganiyev ◽  
S. Zholdassova ◽  
...  
Keyword(s):  
Ideal Gas ◽  
Einstein Condensation ◽  
Chain Model ◽  
One Dimensional ◽  
Electron Phonon Interaction ◽  
Model Lattice ◽  
Experimental Values ◽  
Bose Einstein ◽  
Derivatives Of ◽  
The Ideal

Having accepted a more generalized form for density-displacement type electron–phonon interaction (EPI) force we studied the simultaneous effect of uniaxial strains and EPI’s screening on the temperature of Bose–Einstein condensation [Formula: see text] of the ideal gas of intersite bipolarons. [Formula: see text] of the ideal gas of intersite bipolarons is calculated as a function of both strain and screening radius for a one-dimensional chain model of cuprates within the framework of Extended Holstein–Hubbard model. It is shown that the chain model lattice comprises the essential features of cuprates regarding of strain and screening effects on transition temperature [Formula: see text] of superconductivity. The obtained values of strain derivatives of [Formula: see text] [Formula: see text] are in qualitative agreement with the experimental values of [Formula: see text] [Formula: see text] of La[Formula: see text]Sr[Formula: see text]CuO4 under moderate screening regimes.

On the isothermal density derivative ofG(r) and a new theory of the pair correlation function of hard spheres

1991 ◽  
Vol 64 (3-4) ◽  
pp. 481-500 ◽  
Author(s):  
L. Reatto ◽  
G. Stell ◽  
M. Tau
Keyword(s):  
Correlation Function ◽  
Pair Correlation Function ◽  
Hard Spheres ◽  
Pair Correlation ◽  
Density Derivative

scholarly journals A new entropy-variable-based discretization method for minimum entropy moment approximations of linear kinetic equations

2021 ◽  
Author(s):  
Tobias Leibner ◽  
Mario Ohlberger
Keyword(s):  
Optimization Problems ◽  
Hessian Matrix ◽  
Computational Cost ◽  
Kinetic Equations ◽  
New Method ◽  
Minimum Entropy ◽  
Reduced Basis ◽  
Simple Task ◽  
Wide Range ◽  
The Moment

In this contribution we derive and analyze a new numerical method for kinetic equations based on a variable transformation of the moment approximation. Classical minimum-entropy moment closures are a class of reduced models for kinetic equations that conserve many of the fundamental physical properties of solutions. However, their practical use is limited by their high computational cost, as an optimization problem has to be solved for every cell in the space-time grid. In addition, implementation of numerical solvers for these models is hampered by the fact that the optimization problems are only well-defined if the moment vectors stay within the realizable set. For the same reason, further reducing these models by, e.g., reduced-basis methods is not a simple task. Our new method overcomes these disadvantages of classical approaches. The transformation is performed on the semi-discretized level which makes them applicable to a wide range of kinetic schemes and replaces the nonlinear optimization problems by inversion of the positive-definite Hessian matrix. As a result, the new scheme gets rid of the realizability-related problems. Moreover, a discrete entropy law can be enforced by modifying the time stepping scheme. Our numerical experiments demonstrate that our new method is often several times faster than the standard optimization-based scheme.

scholarly journals KERNEL CONVERGENCE ESTIMATES FOR DIFFUSIONS WITH CONTINUOUS COEFFICIENTS

2011 ◽  
Vol 14 (07) ◽  
pp. 979-1004
Author(s):  
CLAUDIO ALBANESE
Keyword(s):  
Convergence Rate ◽  
Convergence Rates ◽  
Fourier Transforms ◽  
Uniform Continuity ◽  
Time Step ◽  
Uniform Bounds ◽  
Kernel Convergence ◽  
Discretization Schemes ◽  
A New Technique ◽  
Derivatives Of

Bidirectional valuation models are based on numerical methods to obtain kernels of parabolic equations. Here we address the problem of robustness of kernel calculations vis a vis floating point errors from a theoretical standpoint. We are interested in kernels of one-dimensional diffusion equations with continuous coefficients as evaluated by means of explicit discretization schemes of uniform step h > 0 in the limit as h → 0. We consider both semidiscrete triangulations with continuous time and explicit Euler schemes with time step so small that the Courant condition is satisfied. We find uniform bounds for the convergence rate as a function of the degree of smoothness. We conjecture these bounds are indeed sharp. The bounds also apply to the time derivatives of the kernel and its first two space derivatives. The proof is constructive and is based on a new technique of path conditioning for Markov chains and a renormalization group argument. We make the simplifying assumption of time-independence and use longitudinal Fourier transforms in the time direction. Convergence rates depend on the degree of smoothness and Hölder differentiability of the coefficients. We find that the fastest convergence rate is of order O(h2) and is achieved if the coefficients have a bounded second derivative. Otherwise, explicit schemes still converge for any degree of Hölder differentiability except that the convergence rate is slower. Hölder continuity itself is not strictly necessary and can be relaxed by an hypothesis of uniform continuity.

On the pair correlation function for hard spheres at low density and temperature

1968 ◽  
Vol 46 (7) ◽  
pp. 879-888 ◽  
Author(s):  
M. S. Miller ◽  
J. D. Poll
Keyword(s):  
Correlation Function ◽  
Pair Correlation Function ◽  
Hard Spheres ◽  
Pair Correlation ◽  
Quantum Mechanical Calculation ◽  
Quantum Mechanical ◽  
Low Density ◽  
Free Particles ◽  
Low Density Limit ◽  
Hard Sphere Diameters

A quantum-mechanical calculation of the pair correlation function for hard spheres in the low-density limit has been made. This calculation is, therefore, valid at low temperatures, where quantum-mechanical diffraction and symmetry effects are important. Results are given for various temperatures and hard-sphere diameters. The pair correlation function is presented in the form g = gB + gS, where gB is the correlation function for Boltzmann particles and gS describes the symmetry effects. It is found that gS(R) for any value of the separation R is always smaller than the corresponding value for free particles.

scholarly journals Microwave scattering coefficient of snow in MEMLS and DMRT-ML revisited: the relevance of sticky hard spheres and tomography-based estimates of stickiness

2015 ◽  
Vol 9 (6) ◽  
pp. 2101-2117 ◽  
Author(s):  
H. Löwe ◽  
G. Picard
Keyword(s):  
Correlation Function ◽  
Volume Fraction ◽  
Hard Spheres ◽  
Microwave Emission ◽  
Scattering Coefficient ◽  
Equivalent Diameter ◽  
Emission Model ◽  
Data Set ◽  
Point Correlation ◽  
Point Correlation Function

Abstract. The description of snow microstructure in microwave models is often simplified to facilitate electromagnetic calculations. Within dense media radiative transfer (DMRT), the microstructure is commonly described by sticky hard spheres (SHS). An objective mapping of real snow onto SHS is however missing which prevents measured input parameters from being used for DMRT. In contrast, the microwave emission model of layered snowpacks (MEMLS) employs a conceptually different approach, based on the two-point correlation function which is accessible by tomography. Here we show the equivalence of both electromagnetic approaches by reformulating their microstructural models in a common framework. Using analytical results for the two-point correlation function of hard spheres, we show that the scattering coefficient in both models only differs by a factor which is close to unity, weakly dependent on ice volume fraction and independent of other microstructural details. Additionally, our analysis provides an objective retrieval method for the SHS parameters (diameter and stickiness) from tomography images. For a comprehensive data set we demonstrate the variability of stickiness and compare the SHS diameter to the optical equivalent diameter. Our results confirm the necessity of a large grain-size scaling when relating both diameters in the non-sticky case, as previously suggested by several authors.

scholarly journals Product rule for vector fractional derivatives

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
Diogo Bolster ◽  
Mark Meerschaert ◽  
Alla Sikorskii
Keyword(s):  
Vector Space ◽  
Fractional Derivatives ◽  
Fourier Transforms ◽  
Product Rule ◽  
Finite Dimensional ◽  
Euclidean Vector Space ◽  
Derivatives Of

AbstractThis paper establishes a product rule for fractional derivatives of a realvalued function defined on a finite dimensional Euclidean vector space. The proof uses Fourier transforms.

Thio derivatives of b-diketones and their metal chelates. XXII. Dipole moments of some bivalent and trivalent metal complexes of fluorinated monothio-b-diketones

1976 ◽  
Vol 29 (4) ◽  
pp. 767 ◽  
Author(s):  
M Das ◽  
SE Livingstone ◽  
JH Mayfield ◽  
DS Moore ◽  
N Saha
Keyword(s):  
Copper Complexes ◽  
Dipole Moments ◽  
Platinum Complexes ◽  
Mean Value ◽  
Tetrahedral Configuration ◽  
Square Planar ◽  
Palladium And Platinum Complexes ◽  
Static Polarization ◽  
The Moment ◽  
Derivatives Of

Dipole moments have been determined by static polarization measurements for some iron(111), ruthenium(111), rhodium(111), nickel(11), palladium(11), platinum(11), copper(11) and zinc(11) complexes of fluorinated monothio-β-diketones RC(SH)=CHCOCF3. The moments indicate a facial-octahedral configuration for the iron, ruthenium and rhodium complexes, a cis-square-planar configuration for the nickel, palladium and platinum complexes, and a tetrahedral configuration for the zinc complexes. The copper complexes have moments 0.5-1.0 D lower than the mean value for the corresponding nickel, palladium and platinum complexes; this lowering of the moment is attributed to significant distortion from the square-planar towards the tetrahedral configuration. The dipole moments of the square-planar and octahedral complexes decrease if the R groups are arranged in the order: p-MeC6H4 ≥ 2-thienyl > β-naphthyl > m-MeC6H4 > Ph > Pr? > Bui > Me > m-ClC6H4 > m-BrC6H4 > p-FC6H4 > p-ClC6H4 > p-BrC6H4 > m,p-Cl2C6H3.

scholarly journals Ultrarelativistic Gas with Zero Chemical Potential

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 249
Author(s):  
Daniel Mata-Pacheco ◽  
Gonzalo Parga ◽  
Fernando Angulo-Brown
Keyword(s):  
Chemical Potential ◽  
Ideal Gas ◽  
Cavity Radiation ◽  
Classical Gas ◽  
Photon Gas ◽  
Spin Degeneracy ◽  
Classical Particles

In this work, we propose a set of conditions such that an ultrarelativistic classical gas can present a photon-like behavior. This is achieved by assigning a zero chemical potential to the ultrarelativistic ideal gas. The resulting behavior is similar to that of a Wien photon gas. It is found to be possible only for gases made of very lightweight particles such as neutrinos, as long as we treat them as classical particles, and it depends on the spin degeneracy factor. This procedure allows establishing an analogy between an evaporating gas and the cavity radiation.

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