Two-particle Distribution Function Research Articles

Revised Enskog theory and molecular dynamics simulations of the viscosities and thermal conductivity of the hard-sphere fluid and crystal.

Hard-sphere (HS) shear, longitudinal, cross, and bulk viscosities and the thermal conductivity are obtained by molecular dynamics (MD) simulations, covering the entire density range from the dilute fluid to the solid crystal near close-packing. The transport coefficient data for the HS crystal are largely new and display, unlike for the fluid, a surprisingly simple behavior in that they can be represented well by a simple function of the density compressibility factor. In contrast to the other four transport coefficients (which diverge), the bulk viscosity in the solid is quite small and decreases rapidly with increasing density, tending to zero in the close-packed limit. The so-called cross viscosity exhibits a different behavior to the other viscosities, in being negative over the entire solid range, and changes sign from negative to positive on increasing the density in the fluid phase. The extent to which the viscosity tensor and thermal conductivity of the HS crystal can be represented by revised Enskog theory (RET) is investigated. The RET expressions are sums of an instantaneous (I), a kinetic (K), and a so-called α part. The I part of the transport coefficients evaluated directly by MD are statistically indistinguishable from those of the corresponding kinetic theory (Enskog and RET) expressions. For the K part the integral over the spatial two-particle distribution function at contact was determined and the α part was estimated using the direct correlation function and density functional theory approximations. All three parts were determined in this work which allowed the accuracy of RET for solid systems to be assessed rigorously. It is found that in the case of the thermal conductivity the predictions of RET are in excellent agreement with the MD results. Also, for the shear viscosity the agreement over the entire solid phase is quite good but is considerably worse for the three remaining viscosities in the solid phase.

О точности аппроксимации двухчастичной функции распределения для ферромагнетика

When constructing a system of equations of correlation magnetodynamics (a model of a continuous medium of a ferromagnet), the BBGKY hierarchy is used. To close the BBGKY hierarchy, instead of the traditional mean field approximation, an approximation of the two-particle distribution function is used, taking into account correlations between nearest neighbors. One of the goals of this work is to check the quality of this approximation, where «atom-to-atom» modeling using the Landau–Lifshitz equations is taken as a standard. It is shown that the approximation has an average absolute error of the order of 0.001 for the moments of the distribution function. The constructed approximation allows us to obtain the value of the spontaneous magnetization of a separate implementation based on the nonlinearity of the two-particle distribution function for an ensemble of implementations.

Reduced kinetic model of polyatomic gases

Kinetic models of polyatomic gas typically account for the internal degrees of freedom at the level of the two-particle distribution function. However, close to the hydrodynamic limit, the internal (rotational) degrees of freedom tend to be well represented just by rotational kinetic energy density. We account for the rotational energy by augmenting the ellipsoidal statistical Bhatnagar–Gross–Krook (ES–BGK) model, an extension of the BGK model, at the level of the single-particle distribution function with an advection–diffusion–relaxation equation for the rotational energy. This reduced model respects the$H$theorem and recovers the compressible hydrodynamics for polyatomic gases as its macroscopic limit. As required for a polyatomic gas model, this extension of the ES–BGK model not only has the correct specific heat ratio but also allows for three independent tunable transport coefficients: thermal conductivity, shear viscosity and bulk viscosity. We illustrate the effectiveness of the model via a lattice Boltzmann method implementation.

Time-dependent projection operator and nonlinear generalized master equations.

By introducing a special time-dependent projection operator P(t), the nonlinear version of the Nakajima-Zwanzig inhomogeneous generalized master equation(GME) for the relevant part of the N-particle distribution function containing the irrelevant initial correlation term is derived. For this projection operator the relevant part of the N-particle distribution function is a product of the distribution function of a group of s (s≤N) selected particles F_{s}(t) and the distribution function F_{N-s}(t) of N-s particles constituting the "environment" for a group of s selected particles. Thus, the linear projection operator approach results in the exact nonlinear GME for the relevant part of the distribution function, which is equivalent to the exact nonlinear GME for the distribution function of the complex of s particles of interest. This equationis further specified in the first approximation in the particle density, and then considered for one-particle and two-particle distribution functions. Given that there is no satisfactory way of dealing with the initial correlation term, the approach is suggested to rigorously include this term into consideration by converting the obtained inhomogeneous GME into homogeneous form containing initial correlations in the kernel governing the evolution of the relevant part of the distribution function. The homogeneous nonlinear GME for a one-particle distribution function is considered and the conditions for its equivalence to the nonlinear Boltzmann equationare discussed.

Two-particle kinetic equation: method and exact solution

In the paper (Saveliev, in: AIP conference proceedings, vol 1333, p 134, 2011), the kinetic equation for two-particle distribution function was obtained by making use of exactly the same physical assumptions as Ludwig Boltzmann did. Instead of the collision integral, there are the linear scattering operator and the chaos projector in the right part of this equation. The Boltzmann equation follows from this two-particle equation without any additional assumptions after a simple integration. The article presents the method of generalized functions and considers the properties of the obtained exact solution for the two-particle kinetic equation for Maxwell's molecules, which is an intermediate asymptotics for problems of spatial homogeneous relaxation. After reducing the two-particle distribution function to a one-particle distribution function, the solution is reduced to the well-known Bobylev-Krook-Wu (BKW) mode.

Free-Energy Functional Approach to Inverse Problems for Self-Assembly of Three-Dimensional Crystals

In this study, a variational method for the inverse problem of self-assembly, i.e., a reconstruction of the interparticle interaction potential of a given structure, is applied to three-dimensional crystals. According to the method, the interaction potential is derived as a function that maximizes the free-energy functional of the one- and two-particle density distribution functions. The interaction potentials of the target crystals, including those with face-centered cubic (fcc), body-centered cubic (bcc), and simple hexagonal (shx) lattices, are obtained by numerical maximization of the functional. Monte Carlo simulations for the systems of particles with these interactions were carried out, and the self-assembly of the target crystals was confirmed for the bcc and shx cases. However, in the many-particle system with the predicted interaction for the fcc lattice, the fcc lattice did not spontaneously form and was metastable.

Уравнения корреляционной магнитодинамики с учетом одноосной квадратичной поправки в аппроксимации одночастичной функции распределения

The system of equations for correlation magnetodynamics (CMD) is based on the Bogolyubov chain and approximation of the two-particle distribution function taking into account the correlations between the nearest neighbors. CMD provides good agreement with atom-for-atom simulation results (which are considered ab initio), but there is some discrepancy in the phase transition region. To solve this problem, a new system of CMD equations is constructed, which takes into account the quadratic correction in the approximation of the one-particle distribution function. The system can be simplified in a uniaxial case.

Possible Scenarios of a Phase Transition from Isotropic Liquid to a Hexatic Phase in the Theory of Melting in Two-Dimensional Systems

A two-stage process consisting of two continuous Berezinskii-Kosterlitz-Thouless-type transitions with an intermediate anisotropic liquid, a hexatic phase, is a well-known scenario of melting in two-dimensional systems. A direct first-order transition, similar to melting in three-dimensional systems, is another scenario variant. We prove the possibility in principle of the existence of a third scenario according to which melting occurs via two transitions, but in contrast to predictions of the Berezinskii-Kosterlitz-Thouless theory, the transition from an isotropic liquid to a hexatic phase is a first-order transition. Such a scenario was recently observed in a computer simulation of two-dimensional systems and then in a real experiment. Our proof is based on an analysis of branching solutions of an exact closed nonlinear integral equation for a two-particle conditional distribution function.

Influence of the structure of the boundary layer of an adsorbed viscoelastic fluid on the dielectric properties of a metamaterial

We study the change in the shear elasticity of a viscous fluid near a solid surface of a metamaterial. The interest in studying the shear mechanical properties of these fluids is caused by their wide use in engineering. Surface forces in boundary layers or thin films of classic molecular systems should be considered when describing different phenomena in near-surface layers. We obtained the solution for two-particle distribution function at low densities. It is shown that the solution describes the transition from axial to spherical symmetry while each particle goes to infinity.

Emergence of spatial order in highly interacting Rydberg gases

We describe the emergence of strong spatial correlations, akin to liquid-like behavior and crystallization effects, in low (one and two) dimensional gases of cold Rydberg atoms. The presence of an external electric field permanently polarizes the atoms, which became highly correlated due to the long-range dipole-dipole interaction. We describe a theoretical approach particularly suited for strongly coupled systems and numerically obtain both the two-particle distribution function and the static structure factor. The experimental implementation of such highly interacting systems is discussed, including detailed calculations of the interaction strength for different Rydberg states. The results provide new insights into many-body effects associated with strongly interacting Rydberg atoms, including the possibility of observing novel highly ordered phases.

Kinetic Theory beyond the Stosszahlansatz

In a recent paper (Chliamovitch, et al., 2015), we suggested using the principle of maximum entropy to generalize Boltzmann’s Stosszahlansatz to higher-order distribution functions. This conceptual shift of focus allowed us to derive an analog of the Boltzmann equation for the two-particle distribution function. While we only briefly mentioned there the possibility of a hydrodynamical treatment, we complete here a crucial step towards this program. We discuss bilocal collisional invariants, from which we deduce the two-particle stationary distribution. This allows for the existence of equilibrium states in which the momenta of particles are correlated, as well as for the existence of a fourth conserved quantity besides mass, momentum and kinetic energy.

Бета-аппроксимация двухчастичной функции распределения при описании цепочек фазовых осцилляторов

Бета-аппроксимация двухчастичной функции распределения при описании цепочек фазовых осцилляторов

Close order in the molecular system near hard surface

Surface forces in boundary layers or thin films of classic molecular systems should be considered when describing different phenomena in near-surface layers. In this case a molecular system has axial symmetry. We obtained the solution for two-particle distribution function at low densities. It is shown that the solution describes the transition from axial to spherical symmetry while each particle goes to infinity

Quantum canonical ensemble and correlation femtoscopy at fixed multiplicities

Identical particle correlations at fixed multiplicity are considered by means of quantum canonical ensemble of finite systems. We calculate one-particle momentum spectra and two-particle Bose-Einstein correlation functions in the ideal gas by using a recurrence relation for the partition function. Within such a model we investigate the validity of the thermal Wick's theorem and its applicability for decomposition of the two-particle distribution function. The dependence of the Bose-Einstein correlation parameters on the average momentum of the particle pair is also investigated. Specifically, we present the analytical formulas that allow one to estimate the effect of suppressing the correlation functions in a finite canonical system. The results can be used for the femtoscopy analysis of the A+A and p+p collisions with selected (fixed) multiplicity.

A Truncation Scheme for the BBGKY2 Equation

In recent years, the maximum entropy principle has been applied to a wide range of different fields, often successfully. While these works are usually focussed on cross-disciplinary applications, the point of this letter is instead to reconsider a fundamental point of kinetic theory. Namely, we shall re-examine the Stosszahlansatz leading to the irreversible Boltzmann equation at the light of the MaxEnt principle. We assert that this way of thinking allows to move one step further than the factorization hypothesis and provides a coherent—though implicit—closure scheme for the two-particle distribution function. Such higher-order dependences are believed to open the way to a deeper understanding of fluctuating phenomena.

Generalized kinetic equations for dense gases and liquids in the Zubarev nonequilibrium statistical operator method and Renyi statistics

To describe kinetic processes in gases and liquids far from equilibrium, we obtain the nonequilibrium statistical operator (NSO) and the generalized kinetic equations for the nonequilibrium one- and two-particle distribution functions based on the Zubarev NSO method and the maximum entropy principle for the Renyi entropy. We consider a q-generalization of the Liouville equation and obtain its general solution using the NSO method.

Successive Scaling of Quasiparticle Velocities on a Time Step in the Kinetic Force Method with a Two-Particle Distribution Function

As is well known, the algorithm of the kinetic force method, based on the kinetic equation for the two-particle distribution function, guarantees conservation of energy and momentum during a “quasicollision” of each separate pair of quasiparticles. But in the modeling of relaxation of the entire gas an energy deficit nevertheless appears at the step of reconstructing the one-particle distribution function. A new variant of the algorithm of the kinetic force method is proposed in which the energy of the system of molecules is conserved at any step of the modeling of relaxation without additional corrections. By virtue of this, the advantages of modeling on the basis of the two-particle distribution function of pairs of quasiparticles are used more efficiently.

Two-particle Distribution Function of a Non-ideal Molecular System near a Hard Surface

Surface forces in boundary layers or thin films of classic molecular systems should be considered when describing different physical-chemical phenomena (adsorption, wetting). In this case a molecular system has axial symmetry. We obtained the solution for two-particle distribution function at low densities. It is shown that the solution describes the transition from axial to spherical symmetry while each particle goes to infinity

Creation of the set of kinetic equations for a nonideal gas near critical temperatures

In this article, the derivation of the Boltzmann equation from BBKIY of the chain is generalized for the case when the intermolecular interaction potential has both repulsive and attractive components. In this case, the application of the Bogolyubov method leads to situation when the term taking into account the transfer of molecules from the region where the hypothesis of the molecular chaos occurs into the region where molecules are arranged in a bound state is added to the usual collision integral. A two-particle distribution function of molecules in the bound state is assumed to be quasi-equilibrium with parameters depending on the variables that characterize the Boltzmann gas. Kinetic equations are written for these parameters performing the corresponding averaging over the region of bound states. Thus, this resulted in a closed set of kinetic equations describing nonideal gas. After introducing the corresponding macroparameters, all conservation laws and their consequences invariant relative to the Galileo transform follow from the corresponding set. The equation of state derived for such gas resembles the van der Waals equation by form. When considering the relaxation problem, the H-theorem is proven.

Clausius inequality and [formula omitted]-theorems for some models of random wealth exchange

We discuss a possibility of deriving an H-theorem for nonlinear discrete time evolution equation that describes random wealth exchanges. In such kinetic models economical agents exchange wealth in pairwise collisions just as particles in a gas exchange their energy. It appears useful to reformulate the problem and represent the dynamics as a combination of two processes. The first is a linear transformation of a two-particle distribution function during the act of exchange while the second one corresponds to new random pairing of agents and plays a role of some kind of feedback control. This representation leads to a Clausius-type inequality which suggests a new interpretation of the exchange process as an irreversible relaxation due to a contact with a reservoir of a special type. Only in some special cases when equilibrium distribution is exactly a gamma distribution, this inequality results in the H-theorem with monotonically growing ‘entropy’ functional which differs from the Boltzmann entropy by an additional term. But for arbitrary exchange rule the evolution has some features of relaxation to a non-equilibrium steady state and it is still unclear if any general H-theorem could exist.