Expression For Radial Distribution Function Research Articles

Simple and accurate expressions for radial distribution functions of hard disk and hard sphere fluids

Analytical expressions for radial distribution function (RDF) are of critical importance for various applications, such as development of the perturbation theories. Theoretically, RDF expressions for odd-dimensional fluids can be obtained by solving the Percus-Yevick integral equations. But for even-dimensional cases, such as the hard disk (2D) fluid, analytical expressions are infeasible. The only 2D RDF is a heuristic expression which provides acceptable estimations for an intermediate and low density range. In this work, we employ a simple and empirical expression for the 2D RDF and the 3D RDF based on an approach proposed for the 3D RDF. The parameters are determined in such a way that the final RDF expressions are thermodynamically consistent, namely the pressure and the isothermal compressibility constraints are both satisfied. The new RDFs for the 2D and 3D hard spheres are highly accurate for the entire density range up to the first-order phase transition points. The predictions of the first coordination numbers are consistent with simulation results for the 3D fluid. Finally, by using the 2D RDF with a primitive second-order perturbation theory, the pressure-volume-temperature relation and vapor-liquid equilibrium are calculated for the 2D Lennard-Jones fluid. Comparisons with the simulation data show promising results.

Radial Distribution Function of Hard Spheres at Different Densities

The model of hard sphere system has important part in modern theories of liquids. Hard sphere fluid is considered as the most intensively studied system among all model fluids. The thermodynamic and structural properties of real fluids can be studied with the help of hard spheres fluid. The exact solution of Ornstein-Zernike (OZ) equation with Percus-Yevick (PY) along with the equation of state provides an approximate analytical expression for radial distribution function (RDF). This work intends to study the radial distribution function of Hard sphere (HS) using the PY approximation for different densities ranging from low to high to find out their structural properties, packing behavior, phase transitions, and thermodynamics. We use FORTRAN program for this purpose. At low densities, the RDF has a single peak at a distance corresponding to the hard sphere diameter, indicating that the particles are well- separated and not interacting with each other like gaseous. As density increases, the peak becomes broader and shifts to smaller distances, indicating that the particles are coming into closer contact and interacting more strongly with each other. The height of the peak also increases, indicating that there is a greater probability of finding a particle at a certain distance from another particle.

Analytical accurate expressions for radial distribution function and equation of state of hard sphere fluid based on the Percus-Yevick exact solution and a new unifying closure

ABSTRACT New accurate analytic expressions for the radial distribution function and equation of state of hard sphere fluid are obtained by use of its analytic solution from Percus-Yevick closure and by a new unifying closure of the Ornstein-Zernike integral equation. This will be possible by linearisation of the closure to get a set of one-parameter structural functions incorporating both Percus-Yevick and Hyper-Netted Chain closures. This will enable us to deliver very general expressions for the structural functions and equation of state for a potential and fluid. For the most important case of hard sphere fluid the method can be complemented with its known virial coefficients to get accurate analytical expressions for its radial distribution function and equation of state. The expressions are then successfully compared with those of computer simulations for hard sphere fluid at all densities with great accuracy.

Analytic Equation of State for the Square-well Plus Sutherland Fluid from Perturbation Theory

Analytic expressions were derived for the compressibility factor and residual internal energy of the square-well plus Sutherland fluid. In this derivation, we used the second order Barker-Henderson perturbation theory based on the macroscopic compressibility approximation together with an analytical expression for radial distribution function of the reference hard sphere fluid. These properties are expressed in terms of density, temperature, and the potential parameters. Derived equations successfully applied to hard-core Lennard-Jones fluid. It is found that the agreement between theory and simulation is quite good for both the compressibility factor and the residual internal energy for a wide range of densities and temperature.

A radial distribution function-based open boundary force model for multi-centered molecules

We derive an expression for radial distribution function (RDF)-based open boundary forcing for molecules with multiple interaction sites. Due to the high-dimensionality of the molecule configuration space and missing rotational invariance, a computationally cheap, 1D approximation of the arising integral expressions as in the single-centered case is not possible anymore. We propose a simple, yet accurate model invoking standard molecule- and site-based RDFs to approximate the respective integral equation. The new open boundary force model is validated for ethane in different scenarios and shows very good agreement with data from periodic simulations.

N体问题复合区域解析函数近似计算

研究粒子模拟的计算问题，推导出单连通区域和复连通区域的粒子相互作用径向分布函数的解析解，得到了双球域的分析解，对计算复杂性进行了研究，结果表明，当粒子数M = 1800时，计算误差小于千分之一，分析解的计算时间非常少。对蛋白质的距离函数计算结果表明，分析解能显著减少计算量。 In this paper, proximately compute interacted N-body problem by analytic function, Analytic expression of radial distribution function of particles interacting is denoted by multiple integral, the analysis solution is obtained on double sphere domains. In the case of M = 1800, the error of analytic solution is less than 0.001, and numeric CPU time is much less than direct computation. The two polypeptide chains protein radial function is obtained, and points out that this method needs to be improved.

The changes in free energies and entropies from analytical radial distribution functions

The changes in Helmholtz free energies and entropies in dense fluids have been evaluated using three known analytical expressions for radial distribution functions (RDFs) of Lennard–Jones (L-J) fluid. This method provides a simpler and a more expeditious way for the calculation of free energy and entropy in L-J dense fluids through statistical mechanics. Previously, integral equations or perturbation theories were used for this purpose. Such approach not only tests the power of analytical distribution functions in predicting the changes in Helmholtz free energies and entropies, but also specifies better expressions in determining these properties. The results are compared with experimental data and an accurate analytic equation of state for the L-J fluid. It is shown if an expression properly presents RDFs as a function of interparticle distance, density and temperature, it is possible to calculate the changes in Helmholtz free energies and entropies from analytical distribution functions.

A New Expression for Radial Distribution Function of Nuclear Matter

A Variational Monte Carlo Method (VMC) is used for the calculations of radial distribution function of nuclear matter. Urbana 14 v potential is used for the nucleon-nucleon interactions in the calculations. The new expression for radial distribution function of nuclear matter is obtained by fitting of Monte Carlo simulations results to a function.

The sound velocities in dense fluids from distribution functions

The sound velocities and adiabatic compressibilities in dense fluids have been evaluated using three known analytical expressions for radial distribution functions (RDFs). Using such approach not only tests the power of distribution functions theory in predicting the sound velocities and adiabatic compressibilities, but also specifies better expressions in determining these properties. To calculate these quantities, the variation of RDF with density and temperature is required. Therefore, we should have analytical expressions which explicitly present RDF as a function of temperature, density and interparticle distance. It is shown that if an expression is used which properly presents RDFs as a function of interparticle distance, density and temperature, it is possible to calculate sound velocities and adiabatic compressibilities from distribution function theory.

THE HEAT CAPACITIES FROM ANALYTICAL DISTRIBUTION FUNCTIONS

The heat capacities and Joule–Thomson coefficients have been evaluated using three known analytical expressions for radial distribution functions (RDFs). Using such approach not only tests the power of distribution functions theory in predicting the heat capacities and Joule–Thomson coefficients, but also specifies better expressions in determining these properties. To calculate these quantities, the variation of RDF with density and temperature is required. Therefore, we should have analytical expressions which explicitly present RDF as a function of temperature, density, and interparticle distance. It is shown that if an expression is used which properly presents RDFs as a function of interparticle distance, density, and temperature, it is possible to calculate heat capacities and Joule–Thomson coefficients from distribution function theory.

Analytic equation of state for exponential-six fluid based on the Ross variational perturbation theory and the Percus–Yevick radial distribution function of hard spheres

The analytic expressions for equation of state and thermodynamic properties have been derived for the exp-6 fluids, by using the Ross variational perturbation theory and the analytic Percus–Yevick (PY) expression of radial distribution function of hard spheres. It is shown that the variational procedure is absolutely convergent and the calculations are fast. The comparison of the numerical results with the computer simulations shows that the precision of the analytic Ross theory is equivalent to the non-analytic modified Weeks–Chandler–Anderson (mWCA) theory, and is slightly better than the complicated optimized reference hypernetted chain (RHNC) theory.

Evaluation of P– V– T differential properties of the Lennard-Jones fluid using radial distribution functions and molecular dynamics

Evaluation of P– V– T differential properties such as internal pressure, thermal pressure and isothermal compressibility, is a critical test for any equation of state. Certainly, pressure and internal energy equations obtained from RDF theory are not excepted from this rule either. Internal pressure, thermal pressure and isothermal compressibility for a Lennard-Jones (LJ) fluid have been calculated using three known analytical expressions for radial distribution function (RDF) at different temperatures and densities and compared with the corresponding values of these properties obtained from molecular dynamics (MD) and two accurate equations of state for the LJ fluid. Until now, this approach has not been utilized to estimate these quantities. To calculate these quantities, the variation of radial distribution function (RDF) with density and temperature is required. Therefore, we should have analytical expressions which explicitly present RDF as a function of temperature, density and interparticle distance. It is shown that some of these expressions evaluate the values of these quantities well.

Evaluation of high-frequency elastic moduli and shear relaxation time of the Lennard-Jones fluid using three known analytical expressions for radial distribution function

High-frequency elastic moduli ( G ∞ and K ∞) for a Lennard-Jones (LJ) fluid have been calculated using three known analytical expressions for radial distribution function (RDF) at different temperatures and densities and compared with the corresponding values of these properties obtained from molecular dynamics (MD). The shear relaxation time of a LJ fluid at different temperatures and densities has been calculated for three expressions for RDF and using an analytical expression for viscosity and compared with those of obtained MD. The values of reduced shear relaxation time at different reduced temperatures and densities have been fitted to obtain an analytical equation.

An accurate expression for radial distribution function of the Lennard-Jones fluid

A simple and accurate expression for radial distribution function (RDF) of the Lennard-Jones fluid is presented. The expression explicitly states the RDF as a continuous function of reduced interparticle distance, temperature, and density. It satisfies the limiting conditions of zero density and infinite distance imposed by statistical thermodynamics. The distance dependence of this expression is expressed by an equation which contains 11 adjustable parameters. These parameters are fitted to 353 RDF data, obtained by molecular dynamics calculations, and then expressed as functions of reduced distance, temperature and density. This expression, having a total of 65 constants, reproduces the RDF data with an average root-mean-squared deviation of 0.0152 for the range of state variables of 0.5 ⩽ T ∗ ⩽ 5.1 and 0.35 ⩽ ρ ∗ ⩽ 1.1 ( T ∗ = kT / ε and ρ ∗ = ρσ 3 are reduced temperature and density, respectively). The expression predicts the pressure and the internal energy of the Lennard-Jones fluid with an uncertainty that is comparable to that obtained directly from the molecular dynamics simulations.

Analytic equation of state for the exponential-six fluids based on the Ross variational perturbation theory

An analytic expression of radial distribution function of hard spheres is developed in terms of a polynomial expansion of nonlinear base functions and the Carnahan–Starling equation of state (EOS). The comparison with the Monte-Carlo data and the Percus-Yevick expression shows that the expression developed gives out better results. The expression is very simple that can make most perturbation theories become analytic ones, and a simple analytic EOS for the fluids with continuous exponential-six potential is established based on the Ross variational perturbation theory. The main thermodynamic quantities have been analytically derived, the resulting expressions are surprisingly simple, the variational procedure is greatly simplified and the calculations are absolutely convergent. The numerical results are compared with the Monte-Carlo data and the original non-analytic theory. It is shown that the precision of the analytic EOS is as good as the original non-analytic one.

Perturbation theory for the square-well dimer fluid

An analytical equation of state is presented for the square-well dimer fluid of variable well width (1 ≤ λ ≥ 2) based on Barker-Henderson perturbation theory using the recently developed analytical expression for radial distribution function of hard dimers. The integral in the first- and the second-order perturbation terms utilizes the Tang, Y and Lu, B. C.-Y., 1994, J. chem. Phys., 100, 6665 formula for the Hilbert transform. To test the equation of state, NVT and Gibbs ensemble Monte Carlo simulations for square-well dimer fluids are performed for three different well widths (λ = 1.3, 1.5 and 1.8). The prediction of the perturbation theory is also compared with that of thermodynamic perturbation theory in which the equation of state for the square-well dimer is written in terms of that of square-well monomers and the contact value of the radial distribution function.

Structure and property prediction of sub- and supercritical water

In this report two approaches are presented to predict the structure and PVT behavior of associating fluids with emphasis on water. One approach is the development of equations of state based on the analytic chain association theory (ACAT). An associating fluid is assumed to be a mixture of monomers, dimers, trimers, etc., for which the composition distribution is obtained. The resulting equations are simple enough to be used for PVT calculations. The second approach is the development of an effective Kihara pair potential for water which incorporates the hydrogen bonding using the ACAT. This potential function has been used in an analytical expression to predict the first shell of the radial distribution function (RDF) for water. The expression for RDF which was initially applied to simple potential energy functions, such as the Lennard–Jones and Kihara functions satisfies the general functionality of the RDF with respect to intermolecular potential, temperature and density as well as all the limiting values of RDF at high temperature and dilute gas, and infinite separation. The effective potential parameters are determined to predict the first shell of the RDF data for water at various subcritical temperatures and densities. The predicted results for water at near-critical and supercritical conditions are shown to be in agreement with the data obtained by neutron diffraction experiments and with the simulation data.

Ab InitioMolecular Dynamics for Simple Liquid Metals Based on the Hypernetted-Chain Approximation

Abstract We present an ab initio molecular dynamics (MD) method for simple liquid metals based on the quantal hypernetted-chain (QHNC) theory derived from exact expressions for radial distribution functions (RDF's) of the electron-ion model for liquid metals. In our method based on the QHNC equations, the classical MD is performed repeatedly to determine a self-consistent effective interionic potential, which depends on the ion-ion RDF of the system. This resultant effective ionic potential is obtained to be consistent with the density distribution of a pseudoatom and the electron-ion RDF, as well as the ion-ion RDF and the ion-ion bridge function, which are determined exactly as a result of the repeated MD simulation. We have applied this QHNC-MD method for Li, Na, K, Rb, and Cs near the melting temperature using upto 16,000 particles for the MD simulation. It is found that the convergence of the effective interionic potential is fast enough for practical applications; typically two MD runs are enough fo...