Truncated Hamiltonian versus surface perturbation in nonlinear wave theories

Two fundamentally different theories on nonlinear surface waves are scrutinized. The first is a Stokes-like perturbation about a mean equilibrium level. The second is a perturbation about the surface itself by truncation of the Hamiltonian which relates two canonical variables: the velocity potential and the surface elevation. Both perturbation approaches are carefully generalized to include effects from both surface tension and fluid depth. It is found that either the velocity potential or the elevation can be equivalent in the approaches considered but not both variables simultaneously. The resonant condition determines which canonical variable will be equivalent. A clarification is also provided regarding past comparisons between these theories in terms of wave action equations. This study resolves a long-standing controversy as to whether these two approaches are equivalent.     

Molecular extended thermodynamics of rarefied polyatomic gases and wave velocities for increasing number of moments

Molecular extended thermodynamics of rarefied polyatomic gases is characterized by two hierarchies of equations for moments of a suitable distribution function in which the internal degrees of freedom of a molecule is taken into account. On the basis of physical relevance the truncation orders of the two hierarchies are proven to be not independent on each other, and the closure procedures based on the maximum entropy principle (MEP) and on the entropy principle (EP) are proven to be equivalent.     

Brownian Approximation and Monte Carlo Simulation of the Non-Cutoff Kac Equation

The non-cutoff Boltzmann equation can be simulated using the DSMC method, by a truncation of the collision term. However, even for computing stationary solutions this may be very time consuming, in particular in situations far from equilibrium. By adding an appropriate diffusion, to the DSMC-method, the rate of convergence when the truncation is removed, may be greatly improved. We illustrate the technique on a toy model, the Kac equation, as well as on the full Boltzmann equation in a special case.     

On the classes of integral equations nonderivable by graph theory means: An improved molecular theory for classical liquids in equilibrium

Starting from the definition of molecular distribution functions a new hierarchy of integral equations for the equilibrium classical liquids is derived. An outline is given of why this hierarchy cannot be derived by graph theory means. In the theory, the sum of elementary diagrams is given explicitly as a functional of the higher order correlation functions. Analysing this functional the most general truncation scheme within the framework of the first order theories is presented. The pairwise additivity of the intermolecular potential is not assumed.     

A master equation description of coupled systems. II

Dynamical correlations between two coupled systems are described by means of an infinite hierarchy of equations. The condition for a consistent truncation of the hierarchy is given. It allows to derive a closed equation describing the dynamics of one system. The structure of the equations for the stationary and non-stationary case is discussed and a comparison with other approaches is made.     

Diffusion in a bistable potential

The problem of diffusion of a particle in a bistable potential is studied on the basis of the one-dimensional Smoluchowski equation for the space- and time-dependent probability distribution. The potential is modeled as two parabolic wells separated by a parabolic barrier. For the model potential the Smoluchowski equation is solved exactly by a Laplace transform with respect to time for the initial condition that at time zero the probability distribution is given by a thermal equilibrium distribution in one of the wells. In the limit of a high barrier the rate of transition to the other well is given by an asymptotic result due to Kramers. For a potential barrier of moderate height there are significant corrections to the asymptotic result.     

The Vlasov Equilibrium in the Electrostatically Plugged Tandem Mirror Machine MM-4U

The Vlasov equilibrium in the electrostatically plugged tandem mirror machine with double cusps MM-4U is discussed. The plasma potential, density, and non-neutrality profiles are obtained by solving the Poisson-Maxwell equations on the basis of a solution of the Vlasov equation in equilibrium. The truncation conditions of the energy of plasma particles are included. The results show that the potential reaches a maximum and the density gets a minimum at the throat of the machine, i.e., at z = ± 0.36 m. This consorts with the experimental measurements. The non-neutrality profiles show that there are four ion wells in the machine, two at the throats and another two at the centers of cusp.     

Evolution on a Smooth Landscape: The Role of Bias

The role of mutational bias in evolution on a smooth landscape is investigated. We consider both a finite-length genome where the bias increases linearly with the fitness, and an infinite genome with a fixed bias. We present simulations of finite populations in a waiting time model, showing both the nonequilibrium dynamics and the equilibrium fitness distributions that are reached. We compute the equilibrium analytically in several cases, using approximate direct solution of the master equations and truncated hierarchies.     

Interpretation of global integrability conditions for stationary Einstein-Maxwell equations

In constructing solutions of the stationary Einstein-Maxwell equations two global integrability conditions arise. That one of these conditions may be interpreted as an equilibrium condition has been shown in an earlier paper [1]. In that paper I also put forward the idea that the second condition also was an equilibrium condition. In this work I suggest a more reasonable interpretation for this second condition.     

Lie Algebras Associated with Group U（n）

Starting from the subgroups of the group U（n）, the corresponding Lie algebras of the Lie algebra Al are presented, from which two well-known simple equivalent matrix Lie algebras are given. It follows that a few expanding Lie algebras are obtained by enlarging matrices. Some of them can be devoted to producing double integrable couplings of the soliton hierarchies of nonlinear evolution equations. Others can be used to generate integrable couplings involving more potential functions. The above Lie algebras are classified into two types. Only one type can generate the integrable couplings, whose Hamiltonian structure could be obtained by use of the quadratic-form identity. In addition, one condition on searching for integrable couplings is improved such that more useful Lie algebras are enlightened to engender. Then two explicit examples are shown to illustrate the applications of the Lie algebras. Finally, with the help of closed cycling operation relations, another way of producing higher-dimensional Lie algebras is given.     

Lie Algebras Associated with Group U(n)

Starting from the subgroups of the group U(n), the corresponding Lie algebras of the Lie algebra A1 are presented, from which two well-known simple equivalent matrix Lie algebras are given. It follows that a few expanding Lie algebras are obtained by enlarging matrices. Some of them can be devoted to producing double integrable couplings of the soliton hierarchies of nonlinear evolution equations. Others can be used to generate integrable couplings involving more potential functions. The above Lie algebras are classified into two types. Only one type can generate the integrable couplings, whose Hamiltonian structure could be obtained by use of the quadratic-form identity. In addition, one condition on searching for integrable couplings is improved such that more useful Lie algebras are enlightened to engender. Then two explicit examples are shown to illustrate the applications of the Lie algebras. Finally, with the help of closed cycling operation relations, another way of producing higher-dimensional Lie algebras is given.     

On the approach to kinetic theory due to Gross

A classical kinetic theory introduced by Gross is explored in further detail. The theory consists of a sequence of approximations to the Liouville distribution function, with each approximation leading to a truncation of the BBGKY hierarchy at successively higher order. We formulate the truncation scheme at general order in terms of a set of time-dependent equilibrium correlation functions. It has the correct symmetries and, as is implied by the work of Gross with the first two approximations, is such that the interparticle potential appears only implicitly via static equilibrium correlation functions. We arrange the theory as a sequence of linear kinetic equations for the phase-space density correlation function, and solve for the collision kernels which result in each order. The collision kernel of the second approximation, which involves only binary dynamics, is shown to be a mean-field generalization of the known low-density kernel. The third approximation gives a similar generalization of the triple-collision kernel. The nth approximation also reproduces the frequency moments of S(kω) through order ω²ⁿ. More generally, the approximations are shown to give a continued-fraction expansion of the collision kernel, with the levels governed by the dynamics of successively larger numbers of particles. This is a renormalized kinetic theory in the sense that the potential is eliminated and clusters of particles are never isolated.     

静电堵漏串级镜MM－4U的约束位形的理论研究

本文分析了双会切静电堵漏串级磁镜ＭＭ－４Ｕ中粒子损失的可能机制，考虑了等离子体粒子能量的切断，从平衡的Ｖｌａｓｏｖ方程的解出发，数值求解了一组静态的达朗倍尔方程，即Ｐｏｉｓｓｏｎ方程和Ｍａｘｗｅｌｌ方程，得到了等离子体电位、密度及非中性度（净余电子数密度）分布。结果表明，堵漏电压及堵漏电极极板的几何位形、磁场位形及磁场大小、电子温度及电子－离子温度比在串级磁镜的约束位形的形成中起着重要的作用。     

Virasoro symmetry of constrained KP hierarchies

The conventional formulation of additional nonisospectral sysmmetries for the full Kadomtsev-Petviashvili (KP) integrable hierarchy is not compatible with the reduction to the important class of constrained KP (cKP) integrable models. This paper solves explicitly the problem of compatibility of the Virasoro part of additional symmetries with the underlying constraints of cKP hierarchies. Our construction involves an appropriate modification of the standard additional-symmetry flows by adding a set of “ghost symmetry” flows. We also discuss the special case of cKP — truncated KP hierarchies, obtained as Darboux-Bäcklund orbits of initial purely differential Lax operators. Our construction establishes the condition for commutativity of the additional-symmetry flows with the discrete Darboux-Bäcklund transformations of cKP hierarchies leading to a new derivation of the string-equation constraint in matrix models.     

Deviations from minimum entropy production at steady states of reacting chemical systems arbitrarily close to equilibrium

Our analysis of reacting systems displaced from equilibrium by a matter flux across the boundaries has shown that the state of minimum entropy production differs from the steady state, even in the near equilibrium regime. When the displacement δ from equilibrium is small, the derivative of the dissipation at the steady state and the dissipation itself are both of order δ². The state of least dissipation is displaced from the steady state by terms of order δ² in the species concentrations. The theorem of minimum entropy production may be derived by first truncating the series expansions for the reaction rates, affinities and dissipation assuming that δ is small, and then differentiating to locate the minimum of the dissipation within the resulting idealized model. For chemical systems with an arbitrarily small but macroscopic displacement from equilibrium, this truncation procedure establishes that the dissipation is comparatively small in a neighborhood of the steady state; but it causes large relative errors in the values of the concentration derivatives and time derivatives of the dissipation within that neighborhood, because the operations of series truncation and differentiation do not commute near the steady state, when δ is small but nonzero. Near the steady state, the concentration derivative of the term of order δ³ in the dissipation is comparable to or larger than the derivative of the δ² term.     

Equilibrium states for a one dimensional lattice gas

The thermodynamic equilibrium state can be defined directly for an infinite system via an equilibrium condition or via the variational principle. Both definitions are used to calculate the equilibrium state for a one dimensional lattice gas with finite range interactions.     

Conditional Expectations Relative to a Product State and the Corresponding Standard Potentials

For a lattice system with a finite number of Fermions and spins on each lattice point, conditional expectations relative to an even product state (such as Fermion Fock vacuum) are introduced and the corresponding standard potential for any given dynamics, or more generally for any given time derivative (at time 0) of strictly local operators, is defined, with the case of the tracial state previously treated as a special case. The standard potentials of a given time derivative relative to different product states are necessarily different but they are shown to give the same set of equilibrium states, where one can compare states satisfying the variational principle (for translation invariant states) or the local thermodynamical stability or the Gibbs condition, all in terms of the standard potential relative to different even product states.     

Wild's solution of the nonlinear Boltzmann equation

The relaxation to equilibrium of a spatially uniform pseudo-Maxwellian gas is considered. A modified Wild expansion is defined for solving the nonlinear Boltzmann equation. The positivity and asymptotic conditions, as well as the conservation rules, are maintained at each truncation order. Some particular examples are evaluated. The comparison with exact solutions illustrates the very fast convergence of this method.     

Potential energy surfaces of ozone in the ground state

Equilibrium parameters of ozone, such as equilibrium geometry structure parameters, force constants and dissociation energy are presented by CBS-Q {\it ab initio} calculations. The calculated equilibrium geometry structure parameters and energy are in agreement with the corresponding experimental values. The potential energy function of ozone with a C离解能;空气;能量表面;地面ozone, potential energy surface, barrier, dissociation energyProject supported by the National Natural Science Foundation of China (Grant Nos~10376021 and 10676025), and the Scientific Research Fund of Sichuan Provincial Education Department, China (Grant No~2006A131).2006-10-08Equilibrium parameters of ozone, such as equilibrium geometry structure parameters, force constants and dissociation energy are presented by CBS-Q ab initio calculations. The calculated equilibrium geometry structure parameters and energy are in agreement with the corresponding experimental values. The potential energy function of ozone with a C2v symmetry in the ground state is described by the simplified Sorbie-Murrell many-body expansion potential function according to the ozone molecule symmetry. The contour of bond stretching vibration potential of an O3 in the ground state, with a bond angle （θ） fixed, and the contour of O3 potential for O rotating around O1-O （R1）, with O1-O bond length taken as the one at equilibrium, are plotted. Moreover, the potentials are analysed.