Direct Correlation Function

The density-wave theory of Ramakrishnan and Yussouff is used to study phase transitions between liquid, liquid-crystalline, and crystalline phases. The different phases considered are liquid, nematic, smectic, discotic, bcc plastic crystal, orientationally ordered bcc, and a new incommensurate bcc crystal with orientational order. The direct correlation function, required as an input for the theory, is expressed approximately in terms of five generalized Fourier coefficients. The theory is then used to obtain sections through the phase diagram in the five-dimensional space of these coefficients. Simple approximations for the direct correlation function of hard ellipsoids of revolution are used to compare these phase diagrams with those obtained from experiments and numerical simulations. Molecular-field theories of smectic and discotic ordering are reexamined, and, given the potentials they use, it is shown that an orientationally ordered bcc crystal has a lower free energy than either the smectic or the discotic phase. The conditions required to stabilize smectic and discotic phases are examined.

We formulate a density-functional theory that is capable of describing simultaneously the solid, liquid, and gas phases of a simple classical material. The formalism can be reduced to the Ebner-Saam-Stroud theory for the liquid-gas case and to a generalized version of the Ramakrishnan-Youssouff theory for the liquid-solid case. The theory requires as input the direct correlation functions of a uniform fluid. As an example we apply the formalism to the calculation of the phase diagram of a system with Lennard-Jones intermolecular interactions. We obtain the correlation functions from a closure scheme proposed by Zerah and Hansen [J. Chem. Phys. 84, 2336 (1986)]. The calculated density-temperature phase diagram compares favorably with those obtained from numerical simulations of the same model system. We also compute the equations of state in the solid and fluid phases. 'Density-functional theories of simple classical systems are reviewed by R.

We develop a phase field crystal model for structural transformations in two-component alloys. In particular, the interactions between components are described by direct correlation functions that are an extension of those introduced by Greenwood et al. [Phys. Rev. Lett. 105, 045702 (2010)] for pure materials. These correlation functions result in broad density modulations that can be treated with high numerical efficiency, hence enabling simulations of phase transformations between a wide range of crystal structures. A simplified binary alloy model is shown to describe the equilibrium properties of eutectic and peritectic binary alloys in two and three dimensions. The robustness and versatility of this method is demonstrated by applying the model to the growth of structurally similar and dissimilar eutectic lamella and to segregation to defects.

We have systematically investigated the diffusion behavior of silica nanoparticles within supercritical ethanol, in terms of solvent properties by varying temperature ͑T͒ and pressure ͑P͒, to elucidate how the inhomogeneous solvent structures and density fluctuations in the solvent affect the diffusion behavior of solute particles. Results show that at a constant pressure, the diffusion coefficient ͑D͒ of the particles increases with increasing temperature, reaches the maximum ͑D max ͒ within the gaslike supercritical fluid ͑slightly below the ridge͒, and finally decreases abruptly at very low fluid density when temperature is increased further. Results reveal that D is appreciably larger than the theoretical prediction ͑Einstein-Stokes relationship͒ in the vicinity of the critical density ͑ c ͒ of the solvent. We interestingly observed that D becomes maximum ͑D max ͒ at a particular thermodynamic condition ͑T i , P i ͒, which is expressed by the empirical formula T ri = P ri 0.16 ͑for T ri Ͼ 1, P ri Ͼ 1͒. Here, T ri = T i / T c and P ri = P i / P c ; T c and P c are the temperature and the pressure at critical point, respectively. Results further reveal that D max increases significantly with decreasing solvent density within the gaslike supercritical fluid where the changes in viscosities are negligible. These findings are unique, novel, and intriguing. We suggest that the enhancement of the diffusion coefficient in the vicinity of the critical density and the abrupt decrease in the diffusion coefficient in very low density gaslike fluid are associated with the change in the solvent-solvent and solute-solvent direct correlation function ͑related to the effective interaction potential͒ upon density change when the fluid crosses the ridge of density fluctuations and within the gaslike fluid.

Definition of activity coefficients provides a convenient way of describing the nonidealities encountered in solution thermodynamics. H, i4 , is However, there exists very limited theoretical understanding through which one can derive expressions for activity coefficients using the principles of statistical mechanics. As a result, most of the available analytic expressions for activity coefficients are empirical which are based on classical thermodynamic definition of excess properties. Lattice theories of statistical mechanics were successful in providing us with analytic expressions for activity coefficients. 2 • 3 There exist numerous group-contribution expressions for activity coefficients which are based on the lattice theories of statistical mechanics. The high degree of combinatorial approximations which are used in the lattice theories have made their resulting activity coefficient expressions only a limited success. Another statistical mechanical route for the development of analytic expressions of activity coefficients is the use of Gibbs, or NPT, ensemble. 4 Gibbs ensemble, joined with the conformal solution theory of statistical mechanics, have provided us with the formalism in systematically deriving analytic expressions of activity coefficients. The success of this technique is also limited to simple molecular tluid mixture for which the intermolecular potential energy functions are available. In the case of polar mixtures and mixtures possessing hydrogen-bonding species the route through the Gibbs ensemble will be at a loss. The third statistical mechanical route to activity coefficients is the Kirkwood-Buff solution theory of statistical mechanics. 5 • 16 • 17 The activity coefficient, 'Y;, of component i in a mixture is defined by the following relation In (X;'Y;) = (µ,;-/.1,;p) / kT (I) where /.I,; is the chemical potential of component i in the mixture, /.1,;p is the standard-state chemical potential of i, k is the Boltzmann constant, and Tis the absolute temperature. By utilizing the above definition, activity coefficients in solutions are obtained from the relations between the chemical potentials, µ, 1 , and the tluctuation integrals, G ; / s. According to the Kirkwood-Buff solution theory 5 the expression for the chemical potential, /.l;, of component i in a binary mixture takes the following form: x;[aµ;/8x;)p,r = kT/[I + x 1 x 2 p(G 11 + G22-2G 1 2)) (2) where (3) X; is the mole fraction of component i, P is the pressure, p is the

Density functional theories such as the Poniewierski–Stecki theory relate the elastic properties of nematic liquid crystals with their local liquid structure, i.e., with the direct correlation function (DCF) of the particles. We propose a way to determine the DCF in the nematic state from simulations without any approximations, taking into account the dependence of pair correlations on the orientation of the director explicitly. Using this scheme, we evaluate the Frank elastic constants K11, K22, and K33 in a system of soft ellipsoids. The values are in good agreement with those obtained directly from an analysis of order fluctuations. Our method thus establishes a reliable way to calculate elastic constants from pair distributions in computer simulations.

By rescaling the exact low-density results practical expressions are obtained for the equation of state and the direct correlation function of a quid of hard D-spheres allowing a unified treatment of both the odd and even space dimensionalities D. The accuracy of these results is tested against those of the current literature and found to be excellent, in particular for hard disks (D =2), whereas the hard-rod results (D =1) are reproduced exactly. 'Source, from Eq. (2.6). Source, from Eq. (2.9).

The sensitivity of the results obtained within a recently proposed theory of freezing is tested against ail the underlying approximations: (1) the PercusYevick approximation for the direct corrélation function, (2) the choice of the effective fluid whose structure scales with that of the solid, (3) the path joining the solid to the fluid phase, and (4) the gaussian approx imation for the local density of the solid. The gênerai features of the freezing transition are stable with respect to the approximations while the quantitative agreement with the simulation results improves with the quality of the approximations. We also propose a semiempirical hardsphere direct corrél ation function which may be useful for other applications.

By rescaling the exact low-density results practical expressions are obtained for the equation of state and the direct correlation function of a quid of hard D-spheres allowing a unified treatment of both the odd and even space dimensionalities D. The accuracy of these results is tested against those of the current literature and found to be excellent, in particular for hard disks (D =2), whereas the hard-rod results (D =1) are reproduced exactly. 'Source, from Eq. (2.6). Source, from Eq. (2.9).

The sensitivity of the results obtained within a recently proposed theory of freezing is tested against ail the underlying approximations: (1) the PercusYevick approximation for the direct corrélation function, (2) the choice of the effective fluid whose structure scales with that of the solid, (3) the path joining the solid to the fluid phase, and (4) the gaussian approx imation for the local density of the solid. The gênerai features of the freezing transition are stable with respect to the approximations while the quantitative agreement with the simulation results improves with the quality of the approximations. We also propose a semiempirical hardsphere direct corrél ation function which may be useful for other applications.

We present methodologies for calculating the direct correlation function, c(1, 2), the cavity function, y(1, 2), and the bridge function, b(1, 2), for molecular liquids, from Monte Carlo simulations. As an example we present results for the isotropic hard spheroid fluid with elongation e = 3. The simulation data are compared with the results from integral equation theory. In particular, we solve the Percus-Yevick and Hypernetted Chain equations. In addition, we calculate the first two terms in the virial expansion of the bridge function and incorporate this into the closure. At low densities, the bridge functions calculated by theory and from simulation are in good agreement, lending support to the correctness of our numerical procedures. At higher densities, the hypernetted chain results are brought into closer agreement with simulation by incorporating the approximate bridge function, but significant discrepancies remain.

The exchange-correlation hole density of the infinitely stretched (dissociated) hydrogen molecule can be cast into a closed analytical form by using its exact wave function. This permits to obtain an explicit exchange-correlation energy functional of the electron density which allows for its functional derivation to yield the corresponding Kohh-Sham effective exchange-correlation potential. We have shown that this exchangecorrelation functional is exact for the dissociated hydrogen molecule, yields its dissociation energy correctly, and its corresponding exchange-correlation potential has the correct −1/r asymptotic behavior.

Self-consistent Polymer Reference Interaction Site Model (PRISM) calculations and molecular dynamics (MD) simulations were performed on athermal solutions of linear polymers. Unlike most previous treatments of polymer solutions, we explicitly included the solvent molecules. The polymers were modeled as tangent site chains and the solvent molecules were taken to be spherical sites having the same intermolecular potential as the polymer sites. The PRISM theory was solved self-consistently for both the single chain structure and intermolecular correlations as a function of chain length and concentration. The rms end-to-end distance from PRISM theory was found to be in agreement with corresponding MD simulations, and exhibited molecular weight dependence in accordance with scaling predictions in the dilute and concentrated solution limits. The presence of explicit solvent molecules had a significant effect on the packing of the polymer by inducing additional structure in the intermolecu...

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Results for the density profile for Yukawa molecules near a hard wall and an exponential attractive wall are presented for Grand Canonical Monte Carlo ͑GCMC͒ simulations, for the singlet hypernetted chain ͑HNC͒ integral equation and for a modified version of the Lovett-Mou-Buff-Wertheim ͑LMBW-1͒ which uses the exact contact value theorem. The results of the standard singlet HNC are quite poor. If the LMBW equation is modified ͑but still using the bulk direct correlation function͒ the results at high temperature become reasonable. However, the results at low temperatures, close to the bulk coexistence curve, are only a partial improvement. The contact value of the density profile is now quite good but the profile has the incorrect oscillatory behavior predicted by the unmodified approximate equation. The GCMC simulation results show a clear effect of drying at low temperatures.

In the linear-response regime, a normal Bose or Fermi fluid can be described by an exact kinetic equation whose kernel is nonlocal in space and time. We derive a general expression for this kernel and evaluate it explicitly to second order in the interparticle potential. The result is a wave-vectorand frequency-dependent generalization of the linear Uehling-Uhlenbeck kernel with the Born-approximation cross section. Our theory is developed in terms of a second-quantized form of the Wigner representation. Convenient expressions are obtained for the commutators and anticommutators of the phase-space density operators, and the equilibrium averages of these operators are analyzed in terms of momentum-dependent generalizations of the classical pair distribution function h(k} and direct correlation function c(k). The central quantity in this study is a twoparticle equilibrium correlation function, the phase-space density-density anticommutator, whose Fourier transform S{kmpp') gives the symmetrized scattering function S(kco} by integration over the momenta. The kinetic equation is obtained by a formal closure of the quantum BBGKY hierarchy, with the nonlocal kernel expressed in terms of correlation functions involving two, three, and four particles. We show that our method for approximating the kernel and initial condition by a second-order expansion preserves all the sum rules of S(kcopp') to the same order and that the result satisfies the appropriate positivity and symmetry conditions.

A minimal coupling model exhibiting isotropic, uniaxial, and biaxial nematic phases is analyzed in detail and its relation to existing models known in the literature is clarified. Its intrinsic symmetry properties are exploited to restrict the relevant ranges of coupling constants. Further on, properties of the model are thoroughly investigated by means of bifurcation theory as proposed by Kayser and Raveché ͓Phys. Rev. A 17, 2067 ͑1978͔͒ and Mulder ͓Phys. Rev. A 39, 360 ͑1989͔͒. As a first step toward this goal, the bifurcation theory is applied to a general formulation of density functional theory in terms of direct correlation functions. On a general formal level, the theory is then analyzed to show that the bifurcation points from the reference, high-symmetry equilibrium phase to a low-symmetry structure depend only on the properties of the oneparticle distribution function and the direct pair correlation function of the reference phase. The character of the bifurcation ͑whether spinodal, critical, tricritical, isolated Landau point, etc.͒ depends, in addition, on a few higher-order direct correlation functions. Explicit analytical results are derived for the case when only the leading L = 2 terms of the potential ͑mean-field analysis͒ or of the direct pair correlation function expansion in the symmetry-adapted basis are retained. Formulas are compared with the numerical calculations for the mean-field, momentum L = 2 potential model, in which case they are exact. In particular, bifurcations from the isotropic and uniaxial nematic to the biaxial nematic phases are discussed. The possibility of the recently reported nematic uniaxial-nematic biaxial tricritical point ͓A. M. Sonnet, E. G. Virga, and G. E. Durand, Phys. Rev. E 67, 061701 ͑2003͔͒ is analyzed as well.

The success of Density Functional Theory (DFT) is partly due to that of simple approximations, such as the Local Density Approximation (LDA), which uses results of a model, the homogeneous electron gas, to simulate exchange-correlation effects in real materials. We turn this intuitive approximation into a general and in principle exact theory by introducing the concept of a connector: a prescription how to use results of a model system in order to simulate a given quantity in a real system. In this framework, the LDA can be understood as one particular approximation for a connector that is designed to link the exchange-correlation potentials in the real material to that of the model. Formulating the in principle exact connector equations allows us to go beyond the LDA in a systematic way. Moreover, connector theory is not bound to DFT, and it suggests approximations also for other functionals and other observables. We explain why this very general approach is indeed a convenient sta...

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Density functional theories such as the Poniewierski-Stecki theory relate the elastic properties of nematic liquid crystals with their local liquid structure, i.e., with the direct correlation function (DCF) of the particles. We propose a way to determine the DCF in the nematic state from simulations without any approximations, taking into account the dependence of pair correlations on the orientation of the director explicitly. Using this scheme, we evaluate the Frank elastic constants K11, K22 and K33 in a system of soft ellipsoids. The values are in good agreement with those obtained directly from an analysis of order fluctuations. Our method thus establishes a reliable way to calculate elastic constants from pair distributions in computer simulations.

By computer simulations of systems of ellipsoids, we study the influence of the isotropic/nematic phase transition on the direct correlation functions (DCF) in anisotropic fluids. The DCF is determined from the pair distribution function by solving the full Ornstein-Zernike equation, without any approximations. Using a suitable molecular-fixed reference frame, we can distinguish between two qualitatively different contributions to the DCF: One which preserves rotational invariance, and one which breaks it and vanishes in the isotropic phase. We find that the symmetry preserving contribution is barely affected by the phase transition. However, symmetry breaking contributions emerge in the nematic phase and may become quite substantial. Thus the DCF in a nematic fluid is not rotationally invariant. In the isotropic fluid, the DCF is in good agreement with the prediction of the Percus-Yevick theory.

We present large scale molecular dynamics simulations of liquid crystals, which are modeled as fluids of soft repulsive ellipsoidal molecules. In the first part of the paper, we discuss the bulk structure of nematic liquid crystals. The direct correlation function (DCF) has been determined for the first time in a nematic fluid without any approximations. We demonstrate that it can be used to calculate the Frank elastic constants, which are important phenomenological parameters in the continuum theory of liquid crystals. In the second ...

The sensitivity of the results obtained within a recently proposed theory of freezing is tested against ail the underlying approximations: (1) the PercusYevick approximation for the direct corrélation function, (2) the choice of the effective fluid whose structure scales with that of the solid, (3) the path joining the solid to the fluid phase, and (4) the gaussian approx imation for the local density of the solid. The gênerai features of the freezing transition are stable with respect to the approximations while the quantitative agreement with the simulation results improves with the quality of the approximations. We also propose a semiempirical hardsphere direct corrél ation function which may be useful for other applications.

The recently proposed theory of freezing is reformulated entirely in real (direct lattice) space. This allows high précision calculations to be performed easily and also leads to a better understanding of the underlying freezing mechanism. When applied to the freezing of hard sphères into f.c.c. or h.c.p. lattices it is found that the free energy différences involved are extremely small. A diflferentiation between the compact structures will be possible only if the range of the true direct corrélation function appreciably exceeds its Percus-Yevick value used here.

Density functional theories such as the Poniewierski-Stecki theory relate the elastic properties of nematic liquid crystals with their local liquid structure, i.e., with the direct correlation function (DCF) of the particles. We propose a way to determine the DCF in the nematic state from simulations without any approximations, taking into account the dependence of pair correlations on the orientation of the director explicitly. Using this scheme, we evaluate the Frank elastic constants K11, K22 and K33 in a system of soft ellipsoids. The values are in good agreement with those obtained directly from an analysis of order fluctuations. Our method thus establishes a reliable way to calculate elastic constants from pair distributions in computer simulations.

By computer simulations of systems of ellipsoids, we study the influence of the isotropic/nematic phase transition on the direct correlation functions (DCF) in anisotropic fluids. The DCF is determined from the pair distribution function by solving the full Ornstein-Zernike equation, without any approximations. Using a suitable molecular-fixed reference frame, we can distinguish between two qualitatively different contributions to the DCF: One which preserves rotational invariance, and one which breaks it and vanishes in the isotropic phase. We find that the symmetry preserving contribution is barely affected by the phase transition. However, symmetry breaking contributions emerge in the nematic phase and may become quite substantial. Thus the DCF in a nematic fluid is not rotationally invariant. In the isotropic fluid, the DCF is in good agreement with the prediction of the Percus-Yevick theory.

Completely general expressions are given for the equilibrium composition fluctuation properties of systems of components that can undergo reactions (dissociation, association, ionization, solvation) to form new species. Concentration derivatives of the component chemical potentials obtained for non-reacting systems by Kirkwood and Buff are expressed for reacting systems in terms of integrals of the total correlation functions and of the direct correlation functions between the species. This formalism leads to rigorous expressions necessary for modelling fluctuation properties in electrolyte solutions, and in solution theories such as the chemical theory and the solution of groups method for activity coefficients. It is shown that the total correlation ~unction integrals implicitly contain the long range effects of the reactions, but that the direct correlation function integrals (DCFI) do not. Rather, the projection operators for the species DCFI contain the reaction thermodynamics explicitly.

The success of Density Functional Theory (DFT) is partly due to that of simple approximations, such as the Local Density Approximation (LDA), which uses results of a model, the homogeneous electron gas, to simulate exchange-correlation effects in real materials. We turn this intuitive approximation into a general and in principle exact theory by introducing the concept of a connector: a prescription how to use results of a model system in order to simulate a given quantity in a real system. In this framework, the LDA can be understood as one particular approximation for a connector that is designed to link the exchange-correlation potentials in the real material to that of the model. Formulating the in principle exact connector equations allows us to go beyond the LDA in a systematic way. Moreover, connector theory is not bound to DFT, and it suggests approximations also for other functionals and other observables. We explain why this very general approach is indeed a convenient sta...

X iv :1 81 0. 11 24 9v 1 [ ph ys ic s. ch em -p h] 2 6 O ct 2 01 8 Strong-interaction limit of an adiabatic connection in Hartree-Fock theory Michael Seidl, Sara Giarrusso, Stefan Vuckovic, Eduardo Fabiano, 3 and Paola Gori-Giorgi Department of Theoretical Chemistry and Amsterdam Center for Multiscale Modeling, Faculty of Science, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands Institute for Microelectronics and Microsystems (CNR-IMM), Via Monteroni, Campus Unisalento, 73100 Lecce, Italy Center for Biomolecular Nanotechnologies @UNILE, Istituto Italiano di Tecnologia, Via Barsanti, I-73010 Arnesano, Italy

Density functional theories such as the Poniewierski-Stecki theory relate the elastic properties of nematic liquid crystals with their local liquid structure, i.e., with the direct correlation function (DCF) of the particles. We propose a way to determine the DCF in the nematic state from simulations without any approximations, taking into account the dependence of pair correlations on the orientation of the director explicitly. Using this scheme, we evaluate the Frank elastic constants K11, K22 and K33 in a system of soft ellipsoids. The values are in good agreement with those obtained directly from an analysis of order fluctuations. Our method thus establishes a reliable way to calculate elastic constants from pair distributions in computer simulations.

By computer simulations of systems of ellipsoids, we study the influence of the isotropic/nematic phase transition on the direct correlation functions (DCF) in anisotropic fluids. The DCF is determined from the pair distribution function by solving the full Ornstein-Zernike equation, without any approximations. Using a suitable molecular-fixed reference frame, we can distinguish between two qualitatively different contributions to the DCF: One which preserves rotational invariance, and one which breaks it and vanishes in the isotropic phase. We find that the symmetry preserving contribution is barely affected by the phase transition. However, symmetry breaking contributions emerge in the nematic phase and may become quite substantial. Thus the DCF in a nematic fluid is not rotationally invariant. In the isotropic fluid, the DCF is in good agreement with the prediction of the Percus-Yevick theory.

A free-energy density functional for a system of hard spheres is derived on a semiempirical basis. It is constructed to reproduce the thermodynamics and direct correlation function of a homogeneous fluid and then is tested in two highly inhomogeneous situations: the hard-wallhard-sphere interface and the hard-sphere solid. The results are very good in both cases, showing that this densityfunctional model may be used with advant'age in the study of the hard-sphere model by itself, or used as a reference system in a perturbative analysis.

In this work we present structure factors and triplet direct correlation functions extracted from extensive Monte Carlo simulations for a binary mixture of hard spheres. The results are compared with the predictions of two integral equation theories, namely, a recently proposed extension to mixtures of Attard's inhomogeneous integral equation approach, and Barrat, Hansen, and Pastore's factorization ansatz. In general, both theories yield acceptable estimates for the triplet structure functions, though, by construction, the inhomogeneous integral equation theory is more suited to furnish triplet distribution function results, whereas the factorization ansatz provides a more handy approach to triplet direct correlation functions.

A new theory to study isotropic-nematic transition is proposed. This theory requires a good knowledge of the thermodynamic properties of the isotropic phase. It allows to study nematic formation in systems possessing attractive forces. We determine isotropic-nematic equilibria for a number of hard linear models as hard spherocylinders, hard ellipsoids, and hard tangent spheres in a linear configuration. The theory predicts quite nicely the transitions when compared to simulation results. We also study the effect of an ideal dipole or quadrupole on nematic formation. Dipolar or quadrupolar forces favor the presence of a nematic phase although the effect is moderate. However, for large multipole moments no stable nematic phase was found.

By computer simulations of systems of ellipsoids, we study the influence of the isotropic/nematic phase transition on the direct correlation functions (DCF) in anisotropic fluids. The DCF is determined from the pair distribution function by solving the full Ornstein-Zernike equation, without any approximations. Using a suitable molecular-fixed reference frame, we can distinguish between two qualitatively different contributions to the DCF: One which preserves rotational invariance, and one which breaks it and vanishes in the isotropic phase. We find that the symmetry preserving contribution is barely affected by the phase transition. However, symmetry breaking contributions emerge in the nematic phase and may become quite substantial. Thus the DCF in a nematic fluid is not rotationally invariant. In the isotropic fluid, the DCF is in good agreement with the prediction of the Percus-Yevick theory.

Small maps of the Rees-Sciama (RS) effect are simulated by using an appropriate N-body code and a certain ray-tracing procedure. A method designed for the statistical analysis of cosmic microwave background (CMB) maps is applied to study the resulting simulations. These techniques, recently proposed-by our team-to consider lens deformations of the CMB, are adapted to deal with the RS effect. This effect and the deviations from Gaussianity associated to it seem to be too small to be detected in the near future. This conclusion follows from our estimation of both the RS angular power spectrum and the RS reduced n-direction correlation functions for n 6.

By rescaling the exact low-density results practical expressions are obtained for the equation of state and the direct correlation function of a quid of hard D-spheres allowing a unified treatment of both the odd and even space dimensionalities D. The accuracy of these results is tested against those of the current literature and found to be excellent, in particular for hard disks (D =2), whereas the hard-rod results (D =1) are reproduced exactly. 'Source, from Eq. (2.6). Source, from Eq. (2.9).

Using a general expression for the second term in the density expansion of the hard core potentials, we obtained an analytical expression for the direct correlation function of the hard core double Yukawa potential. The results of this calculation using the proposed direct correlation function and its related radial distribution function through the Ornstein-Zernike equation show good agreements with those obtained by the computer simulation of the same model.

Marine port as one important aspect of transportation network set particularly in marine field, should be able to support transportation activities or human and good transport from a place (hinterland) to another (foreland). Selection of port that could give best service would be one of important decision issues which would be giving positive impact toward regional economy surrounding the port in particular, and the nation in general. This study has taken object of Tanjung Perak port in Surabaya, particularly in container terminal. Policy development for regional economic growth in surrounding areas of the port was modeled by using dynamic system approach. Dynamic system approach could solve complex problems which cannot be solve by mathematical approach. Based on causal loop diagram, it is draw a dynamic system model such as stock and flow diagram to found out impact of changes in each variables affecting the system. Analysis result showed that economic growth rate in port was affected by several variables, in which port management has the authority to decide policies relevant with those variables. Thus, to increase regional economic growth in port surroundings, appropriate policy scenario to be initiated was to increase pier capacity and by adjusting tariff or retribution.

We present a method for calculating the thermodynamic and structural properties of a polydisperse liquid by means of a thermodynamic perturbation theory: the optimized random phase approximation ͑ORPA͒. The approach is an extension of a method proposed recently by one of us for an integral equation application ͓Phys. Rev. E 54, 4411 ͑1996͔͒. The method is based on expansions of all-dependent functions in the orthogonal polynomials p i () associated with the weight function f ⌺ (), where is a random variable ͑in our case the size of the particles͒ with distribution f ⌺ (). As in the one-component or general N-component case, one can show that the solution of the ORPA is equivalent to the minimization of a suitably chosen functional with respect to variations of the direct correlation functions. To illustrate the method, we study a polydisperse system of square-well particles; extension to other hard-core or soft-core systems is straightforward. ͓S1063-651X͑99͒06806-3͔

The density-wave theory of Ramakrishnan and Yussouff is used to study phase transitions between liquid, liquid-crystalline, and crystalline phases. The different phases considered are liquid, nematic, smectic, discotic, bcc plastic crystal, orientationally ordered bcc, and a new incommensurate bcc crystal with orientational order. The direct correlation function, required as an input for the theory, is expressed approximately in terms of five generalized Fourier coefficients. The theory is then used to obtain sections through the phase diagram in the five-dimensional space of these coefficients. Simple approximations for the direct correlation function of hard ellipsoids of revolution are used to compare these phase diagrams with those obtained from experiments and numerical simulations. Molecular-field theories of smectic and discotic ordering are reexamined, and, given the potentials they use, it is shown that an orientationally ordered bcc crystal has a lower free energy than either the smectic or the discotic phase. The conditions required to stabilize smectic and discotic phases are examined.

We have studied the ordering of a two dimensional charge stabilised colloidal system in the presence of a stationary one dimensionally modulated laser field formed by the superposition of two modulations with wavevectors q0τ and q0/τ , where q0 is the wavevector corresponding to the first peak of the direct correlation function of the unperturbed liquid, and τ is the golden mean. In the framework of the Landau-Alexander-McTague theory we find that a decagonal quasicrystalline phase is stabler than the liquid or the triangular lattice in certain regions of the phase diagram. Our study also shows a reentrant melting phenomena for larger laser field strengths. We find that the transition from the modulated liquid to the quasicrystalline phase is continuous in contrast to a first order transition from the modulated liquid to the triangular crystalline phase.

We formulate a density-functional theory that is capable of describing simultaneously the solid, liquid, and gas phases of a simple classical material. The formalism can be reduced to the Ebner-Saam-Stroud theory for the liquid-gas case and to a generalized version of the Ramakrishnan-Youssouff theory for the liquid-solid case. The theory requires as input the direct correlation functions of a uniform fluid. As an example we apply the formalism to the calculation of the phase diagram of a system with Lennard-Jones intermolecular interactions. We obtain the correlation functions from a closure scheme proposed by Zerah and Hansen [J. Chem. Phys. 84, 2336 (1986)]. The calculated density-temperature phase diagram compares favorably with those obtained from numerical simulations of the same model system. We also compute the equations of state in the solid and fluid phases.

Approximating the exchange-correlation energy in density functional theory (DFT) is a crucial task. As the only missing element in the Kohn-Sham DFT, the search for better exchange-correlation functionals has been an active field of research for fifty years. Many models and approximations are known and they can be summarized in what is known as the Jacob’s ladder. All the functionals