On the validity of the Landau-Lifshitz method of deriving a higher order gravitational potential

The Landau-Lifshitz test particle method of deriving a higher order gravitational potential for many-particle systems is not valid in general coordinate systems. This is shown in the most simple case of the post-newtonian two-particle potential.     

Generation of exact analytic bound state solutions from solvable non-powerlaw potentials by a transformation method

A transformation method has been applied to the exactly solvable Hulthen problem to generate a hierarchy of exactly solved quantum systems in any chosen dimension. The generated quantum systems are, in general, energy-dependent with a single normalized eigenfunction, as the Hulthen potential is a non-powerlaw potential. A method has been devised to convert a subset of the generated quantum systems with energy-dependent potentials to a single normal system with an energy-independent potential that behaves like a potential qualitatively similar to the Poschl-Teller potential. A second-order application of the transformation method on the Hulthen system produces another Sturmian quantum system and a different method is given to regroup them into a normal quantum system which resembles the Morse potential. Existence of normalizable eigenfunctions for these systems are found to be dependent on the local and asymptotic behaviour of the transformation function. Received 30 August 2000 and Received in final form 16 March 2001     

Fluctuations and stability of stationary non-equilibrium systems in detailed balance

A generalized thermodynamic potential for Markoffian systems with detailed balance and far from thermal equilibrium has been derived in a previous paper. It was shown that the principle of detailed balance is equivalent to a set of conditions fulfilled by this potential (“potential conditions”). The properties of this potential allow us to extend the validity of a number of thermodynamic concepts well known for systems in or near thermal equilibrium to stationary states far from thermal equilibrium. The concept of symmetry breaking phase transitions for these systems is introduced in analogy to thermal equilibrium systems by considering the dependence of the stationary probability density of the system on a set of externally controlled parameters {λ}. A functional of the time dependent probability density of the system is defined in close analogy to the Gibb's definition of entropy. This functional has the properties of a Ljapunov functional of the governing Fokker-Planck equation showing the stability of the stationary probability density. The Langevin equations connected with the Fokker-Planck equation are considered. It is shown that, by means of the potential conditions, generalized “thermodynamic” fluxes and forces may be defined in such a way that the smoothly varying part of the Langevin equations (kinetic equations) constitutes a linear relation between fluxes and forces. The matrix of coefficients is given by the diffusion matrix of the Fokker-Planck equation. The symmetry relations which hold for this matrix due to the potential conditions then lead to the Onsager-Casimir symmetry relations extended to systems with detailed balance near stationary states far from thermal equilibrium. Finally it is shown that under certain additional assumptions the generalized thermodynamic potential may be used as a Ljapunov function of the kinetic equations.     

Solution of the Dirichlet problem for the Laplace equation for a multiply connected region with point symmetry

A method is proposed which uses an expansion of the potential in irreducible representations of the symmetry group of the field-defining elements of a system. A boundary-value problem is solved for multipole systems with planar plate electrodes for the C _nv symmetry group. A quadrature expression is obtained for the field potential of these systems. Constraints imposed on the electrode potentials, under which such a solution is possible, are determined. Results of calculations of the potential distribution are presented for various specific systems. Zh. Tekh. Fiz. 69, 1–9 (March 1999)     

Yukawa potentials in systems with partial periodic boundary conditions. I. Ewald sums for quasi-two-dimensional systems

Yukawa potentials are often used as effective potentials for systems such as colloids, plasmas, etc. When the Debye screening length is large, the Yukawa potential tends to the non-screened Coulomb potential; in this small screening limit, or Coulomb limit, the potential is long-ranged. As is well known in computer simulation, a simple truncation of the long-ranged potential and the minimum image convention are insufficient to obtain accurate numerical data on systems. The Ewald method for bulk systems, i.e. with periodic boundary conditions in all three directions of space, has already been derived for the Yukawa potential [Molec. Phys. 88, 1357 (1996); J. Chem. Phys. 113, 10459 (2000)], but for systems with partial periodic boundary conditions, the Ewald sums have only recently been obtained [J. Chem. Phys. 126, 056101 (2007)]. In this paper, we provide a closed derivation of the Ewald sums for Yukawa potentials in systems with periodic boundary conditions in only two directions and for any value of the Debye length. Special attention is paid to the Coulomb limit and its relation to the electroneutrality of systems.     

Conservation laws and hierarchies of potential symmetries for certain diffusion equations

We show that the so-called hidden potential symmetries considered in a recent paper [M.L. Gandarias, New potential symmetries for some evolution equations, Physica A 387 (2008) 2234-2242] are ordinary potential symmetries that can be obtained using the method introduced by Bluman and collaborators [G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, New York, 1989; G.W. Bluman, G.J. Reid, S. Kumei, New classes of symmetries for partial differential equations, J. Math. Phys. 29 (1988) 806-811]. In fact, these are simplest potential symmetries associated with potential systems which are constructed with single conservation laws having no constant characteristics. Furthermore we classify the conservation laws for classes of porous medium equations, and then using the corresponding conserved (potential) systems we search for potential symmetries. This is the approach one needs to adopt in order to determine the complete list of potential symmetries. The provenance of potential symmetries is explained for the porous medium equations by using potential equivalence transformations. Point and potential equivalence transformations are also applied to deriving new results on potential symmetries and corresponding invariant solutions from known ones. In particular, in this way the potential systems, potential conservation laws and potential symmetries of linearizable equations from the classes of differential equations under consideration are exhaustively described. Infinite series of infinite-dimensional algebras of potential symmetries are constructed for such equations.     

Molecular dynamics of binary metal nitrides and ternary oxynitrides

Ternary systems incorporating metals with oxygen and nitrogen are examined using Tersoff potentials. The apparent success of treating some binary nitride systems using the Tersoff potential is used as a way forward to obtain a new parameter set incorporates atomic features into a series of Tersoff potential for binary nitrides and ternary oxynitrides.     

Theory of slightly fluctuating ratchets

We consider a Brownian particle moving in a slightly fluctuating potential. Using the perturbation theory on small potential fluctuations, we derive a general analytical expression for the average particle velocity valid for both flashing and rocking ratchets with arbitrary, stochastic or deterministic, time dependence of potential energy fluctuations. The result is determined by the Green’s function for diffusion in the time-independent part of the potential and by the features of correlations in the fluctuating part of the potential. The generality of the result allows describing complex ratchet systems with competing characteristic times; these systems are exemplified by the model of a Brownian photomotor with relaxation processes of finite duration.     

Method for construction of a biased potential for hyperdynamic simulation of atomic systems

An approach to constructing a biased potential for hyperdynamic simulation of atomic systems is considered. Using this approach, the diffusion of an atom adsorbed on the surface of a two-dimensional crystal and a vacancy in the bulk of the crystal are simulated. The influence of the variation in the potential barriers due to thermal vibrations of atoms on the results of calculations is discussed. It is shown that the bias of the potential in the hyperdynamic simulation makes it possible to obtain statistical samples of transitions of atomic systems between states, similar to those given by classical molecular dynamics. However, hyperdynamics significantly accelerates computations in comparison with molecular dynamics in the case of temperature-activated transitions and the associated processes in atomic systems.     

Collective modes of one-dimensional lennard-jones systems

The density fluctuations of one-dimensional Lennard-Jones systems are investigated by molecular dynamics simulation. The full Lennard-Jones potential is compared to the repulsive Lennard-Jones potential. It is found that the behavior of the density fluctuations at small wave vectors is determined by the repulsive portion of the potential. The variation of the fluctuations with density is explained. It is shown that these systems do not display hydrodynamics.     

Perturbation theory for mixtures of discrete potential fluids

A thermodynamic perturbation theory for mixtures of fluids composed of particles interacting via discrete potentials is presented, based on previous work for pure component systems. Square-well and square-shoulder mixtures are accurately described by this theory, giving the necessary information for studying a wide range of discrete potential fluids. As an example of this, the theory is applied to a discrete Lennard-Jones mixture, obtaining very good results when compared against computer simulation values. The scope of this work is to implement perturbation theory for discrete potential systems in modern theories for complex fluids.     

精确的量子化条件和不变量

提出并证明了一维量子系统和三维球对称量子系统的一个精确的量子化条件.在此精确量子化条件中, 除了通常的Nπ项外, 还有一积分项, 称为修正项. 发现该修正项正是在超对称量子力学中所谓的有形状不变势的量子系统的一个不变量，它不依赖于波函数的节点数.对这些系统, 可用基态能级和波函数确定此不变量的值, 从而由精确的量子化条件容易算出全部束缚态的能级. 计算得到能级的正确性又反过来验证了在有形状不变势的量子系统中此修正项确实是不变量.计算的有形状不变势的量子系统, 包括一维的有限方势阱、Morse势及其变形、R 关键词： 量子化条件 超对称量子力学 形状不变势 不变量     

Construction of the Second Constant of Motion for Three-Dimensional Classical Potential Systems

In this paper, we give the construction of thesecond constant of motion for three-dimensional classical potential systems. Correspondingly, the first- and the second-order potential equations are obtained whose solutions can directly provide the integrable systems.     

Analytical formula for the Uehling potential

The closed analytical expression for the Uehling potential is derived. The Uehling potential describes the lowest-order correction on vacuum polarisation in atomic and muon-atomic systems. We also derive the analytical formula for the interaction potential between two electrically charged point particles which includes correction to the vacuum polarisation, but has correct asymptotic behaviour at larger r. Our three-term analytical formula for the Uehling potential opens a new avenue in the study of the vacuum polarisation in light atomic systems.     

Modification of Lytle’s theory: Interpretation of the extended fine structure associated with LIII absorption discontinuity of some ytterbium systems

In Lytle’s theory for the extended fine structure in x-ray absorption spectra, the potential at the boundary of the ‘equivalent sphere’ around the absorbing atom, having volume equal to that of the Wigner-Seitz cell is considered to be infinite. It has been observed that Lytle’s theory is applicable only in the case of metals and metallic systems. In the present paper the extended fine structure associated with the L_III absorption spectra of some systems of ytterbium is interpreted on the basis of Lytle’s model, modified by using a finite potential instead of an infinite one at the boundary of the equivalent sphere. The values of this potential are estimated for eight systems of ytterbium. It has been shown that there exists a correlation between the potentials and covalency of the compounds.     

A derivation of the compressibility equation for liquid metals

The Kirkwood and Buff compressibility equation for multicomponent systems is applied to an assembly of ions and electrons, where the Coulomb potential is replaced by the Yukawa potential. It is then shown that the resulting formula gives the Watabe-Hasegawa-Chihara compressibility equation in the limit of infinitesimal damping constant of the Yukawa potential.     

General Considerations on the Finite-Size Corrections for Coulomb Systems in the Debye--Hückel Regime

We study the statistical mechanics of classical Coulomb systems in a low coupling regime (Debye--Hückel regime) in a confined geometry with Dirichlet boundary conditions for the electric potential. We use a method recently developed by the authors which relates the grand partition function of a Coulomb system in a confined geometry with a certain regularization of the determinant of the Laplacian on that geometry with Dirichlet boundary conditions. We study several examples of fully confining geometry in two and three dimensions and semi-confined geometries where the system is confined only in one or two directions of the space. We also generalize the method to study systems confined in arbitrary geometries with smooth boundary. We find a relation between the expansion for small argument of the heat kernel of the Laplacian and the large-size expansion of the grand potential of the Coulomb system. This allow us to find the finite-size expansion of the grand potential of the system in general. We recover known results for the bulk grand potential (in two and three dimensions) and the surface tension (for two-dimensional systems). We find the surface tension for three-dimensional systems. For two-dimensional systems our general calculation of the finite-size expansion gives a proof of the existence a universal logarithmic finite-size correction predicted some time ago, at least in the low coupling regime. For three-dimensional systems we obtain a prediction for the curvature correction to the grand potential of a confined system.     

Polynomial integrability of the Hamiltonian systems with homogeneous potential of degree − 3

In this paper, we study the polynomial integrability of natural Hamiltonian systems with two degrees of freedom having a homogeneous potential of degree k given either by a polynomial, or by an inverse of a polynomial. For k=−2,−1,…,3,4, their polynomial integrability has been characterized. Here, we have two main results. First, we characterize the polynomial integrability of those Hamiltonian systems with homogeneous potential of degree −3. Second, we extend a relation between the nontrivial eigenvalues of the Hessian of the potential calculated at a Darboux point to a family of Hamiltonian systems with potentials given by an inverse of a homogeneous polynomial. This relation was known for such Hamiltonian systems with homogeneous polynomial potentials. Finally, we present three open problems related with the polynomial integrability of Hamiltonian systems with a rational potential.     

Perturbation theory for fluids of rod-like molecules interacting via the Kihara potential

For systems of molecules interacting via the Kihara core potential a first-order perturbation theory is proposed. As a reference system soft convex bodies are employed with interactions given by the entire repulsive part of the original pair potential (i.e. for surface-to-surface distances smaller than that of the potential minimum). Their equilibrium behaviour is interpreted on the basis of the representative hard convex bodies-parallel convex bodies to the assumed cores with temperature-dependent thickness. The shape of the distribution function was approximated by the Verlet-Weis form. Theoretical expressions were used for the determination of the thermodynamic functions of the Kihara-molecule systems at several reduced temperatures and compared with experimental data for nitrogen.