Approximate solutions of the Liouville equation. III. Variational principles and projection operators

Aspects of stationary variational principles for the Laplace-transformed Liouville equation are discussed. Projection techniques are used to derive new stationary principles applicable to the space orthogonal to the space spanned by functions occurring in the conservation laws. As a result, any trial function automatically leads to results satisfying the conservation laws. The procedure is also applied to the parity-even and parity-odd distributions which obey equations governed by the square of the Liouville operator. The technique is extended to eliminate the one-body additive contribution to the solution exactly. Finally, the ideas of the moment method, which leads to the continued-fraction representation of autocorrelation functions, are applied to variational principles. We find continued-fraction variational principles such that a zero trial function yields the usual representation. However, a trial function representing noninteracting particles contains the results of the moment method and in addition yields the exact analytic behavior for free particles.Work supported by a grant from the National Science Foundation.     

Boussinesq方程组的广义变分原理

半反推法是何吉欢为了寻求物理问题的变分原理而提出的,可避免由拉氏乘子法引起的临界变分现象. 应用半反推法分别获得了描述水波运动的两类Boussinesq方程组的一族广义变分原理,并验证了它们的正确性. 关键词： 半反推法 广义变分原理 Boussinesq方程组     

The variational iteration method for solitary patterns solutions of gBBM equation

We consider solitary patterns solutions of generalized Benjamin–Bona–Mahony equations (shortly gBBM). The variational iteration method (shortly VIM) is applied for the numerical solution subject to appropriate initial condition. The numerical solutions of our model equation are calculated in the form of convergence power series with easily computable components. The VIM performs extremely well in terms of accuracy, efficiently, simplicity, stability and reliability.     

Approximate solutions of the Liouville equation. IV. The two-body additive approximation

The two-body additive approximation on the time-dependent Liouville distribution, first introduced in part I of this series, is put into the conventional form of a self-contained kinetic equation for the doublet distribution. From this point of view the approximation consists in truncating the BBGKY chain by expressing the triplet distribution as a functional of lower distributions at the same value of the time variable. To accomplish this, it is necessary to study two associated purely spatial integral equations. The doublet kinetic equation can then be written in terms of solutions of these integral equations and comparison with conventional methods of truncating the BBGKY chain can then be made. For the purpose of comparison a method of truncating the chain based on the Kirkwood superposition approximation is introduced and discussed briefly. The momentum structure of the resulting doublet kinetic equation is similar, but the nonlocality in space of our truncation introduces distinct differences in the spatial structure. The inconsistency between conventional truncations and the exact initial conditions used for the calculation of time-dependent correlation functions is pointed out. This inconsistency is not shared by the two-body additive approximation.     

Numerical solution of sine-Gordon equation by variational iteration method

In this Letter, variational iteration method (VIM) is applied to obtain approximate analytical solution of the sine-Gordon equation without any discretization. Comparisons with the exact solutions reveal that VIM is very effective and convenient.     

Stochastic Liouville equation studies of FT-EPR spectra of singlet-triplet mixing photo-chemically generated radical pair system

A stochastic Liouville equation (SLE) was numerically solved to obtain pulsed Fourier-transform (FT) EPR spectra on a radical pair system created in a photo-induced chemical reaction. Numerical calculations were applied to the photo-chemical reaction of deuterated acetone and 2-propanol at low temperatures. In this reaction system, the antiphase structures of the EPR signals, so called spin-correlated radical pair (SCRP) signals of two identical isopropyl ketyl radicals and spin-polarized free isopropyl ketyl radicals were observed by FT-EPR and continuous wave time-resolved (CW TR) EPR techniques. In the present work, FT-EPR spectra of the antiphase structure signals of the radical pair themselves as well as spin-polarized free radical signals were simulated. Additionally, rising behavior of free radical signals polarized by the radical pair mechanism (RPM) was also clarified. Furthermore two-dimensional (2D) FT-EPR nutation spectra were simulated in the both cases with and without the radical pairs by the use of SLE. In these simulations, strong DC components in the nutation frequency dimension, were well reproduced as was obtained in experiments. It was shown that relaxation during the microwave pulse was essential for the appearance of the DC components.     

The stochastic Liouville equation and Padé approximants

The applicability of Padé approximant techniques to solving the stochastic Liouville equation is discussed. The special case of an axially symmetric spin system undergoing isotropic Brownian motion is studied. Two types of expansions are explored which yield efficient algorithms for spectral simulations.     

Hamilton’s principle and associated variational principles in polar thermopiezoelectricity

We express Hamilton’s principle for a regular region of thermopiezoelectric polar materials. First, we obtain a four-field variational principle which leads, as its Euler-Lagrange equations, to the divergence equations and the associated natural boundary conditions only. Next, we adjoin the rest of the fundamental equations into the variational principle through an involutory transformation. Thus, we formulate a differential type of unified variational principles operating on all the field variables. The unified variational principle is extended for the region with a fixed internal surface of discontinuity and for a curvilinear laminated region as well. The principles derived in invariant form are expressible in a system of particular coordinate system most appropriate to the geometry of the regions. They are indicated to recover some of earlier principles as special cases.     

Liouville Integrability of Conservative Peakons for a Modified CH equation

The modified Camassa-Holm (also called FORQ) equation is one of numerous cousins of the Camassa-Holm equation possessing non-smoth solitons (peakons) as special solutions. The peakon sector of solutions is not uniquely defined: in one peakon sector (dissipative^a) the Sobolev H¹ norm is not preserved, in the other sector (conservative), introduced in [2], the time evolution of peakons leaves the H¹ norm invariant. In this Letter, it is shown that the conservative peakon equations of the modified Camassa-Holm can be given an appropriate Poisson structure relative to which the equations are Hamiltonian and, in fact, Liouville integrable. The latter is proved directly by exploiting the inverse spectral techniques, especially asymptotic analysis of solutions, developed elsewhere [3].     

Kinetic equation for a weakly interacting classical electron gas

A kinetic equation is derived for the two-time phase space correlation function in a dilute classical electron gas in equilibrium. The derivation is based on a density expansion of the correlation function and the resummation of the most divergent terms in each order in the density. It is formally analogous to the ring summation used in the kinetic theory of neutral fluids. The kinetic equation obtained is consistent to first order in the plasma parameter and is the generalization of the linearized Balescu-Guersey-Lenard operator to describe spatially inhomogeneous equilibrium fluctuations. The importance of consistently treating static correlations when deriving a kinetic equation for an electron gas is stressed. A systematic derivation as described here is needed for a further generalization to a kinetic equation that includes mode-coupling effects. This will be presented in a future paper.     

Stochastic equations of the Langevin type under a weakly dependent perturbation

Asymptotic expansions for the probability density of the solution of a stochastic differential equation under a weakly dependent perturbation are proposed. In particular, linear partial differential equations for the first two terms of the correlation time expansion are derived. It is shown that in these expansions the boundary layer part appears and non-Gaussianity of the perturbation is important for the Fokker-Planck approximation correction.     

Equations of motion from field equations and a gauge-invariant variational principle for the motion of charged particles

New, gauge-independent, second-order Lagrangian for the motion of classical, charged test particles is proposed. It differs from the standard, gauge-dependent, first-order Lagrangian by boundary terms only. A new method of deriving equations of motion from field equations is developed. When applied to classical electrodynamics, this method enables us to obtain unambiguously the above, second-order Lagrangian from the general energy-momentum conservation principle.     

Stationary solutions of Liouville equations for non-Hamiltonian systems

We consider the class of non-Hamiltonian and dissipative statistical systems with distributions that are determined by the Hamiltonian. The distributions are derived analytically as stationary solutions of the Liouville equation for non-Hamiltonian systems. The class of non-Hamiltonian systems can be described by a non-holonomic (non-integrable) constraint: the velocity of the elementary phase volume change is directly proportional to the power of non-potential forces. The coefficient of this proportionality is determined by Hamiltonian. The constant temperature systems, canonical-dissipative systems, and Fermi-Bose classical systems are the special cases of this class of non-Hamiltonian systems.     

Topology of solutions to the Liouville equation

Suggestions concerning the generalization of geometric quantization to the case of non-linear field theories are given. Preliminary results for the Liouville field theory are presented.     

Correlation functions for diatomic symmetric molecules from the RISM equation

Using the solution of the RISM equation for diatomic symmetric molecules outlined in a previous paper, the site-site radial distribution function (RDF) is calculated and compared with the Monte Carlo results and the numerical RDF of Lowden and Chandler. The RDF calculated here and the numerical RDF of Lowden and Chandler agree well at intermediate and high densities. At low density, however, both have systematic errors. The agreement between the RDF calculated here and the Monte Carlo results suggests that a simplified formulation of the RISM solution may serve well as a reference system in a perturbation theory for diatomic fluids.     

含弛豫项的广义光学Bloch方程的近似解

本文用叠代法求得了含弛豫项的广义光学Bloch方程的近似解。与计算机给出的数值积分解的比较表明,一阶叠代解具有足够好的精度。由此得出了上能级占有几率随时间变化的解析表达式及多光子吸收、Bloch-Siegert频移等有用结果。     

Construction of solitary solutions for nonlinear dispersive equations by variational iteration method

Variational iteration method is implemented to construct solitary solutions for nonlinear dispersive equations. In this scheme the solution takes the form of a convergent series with easily computable components. The chosen initial solution or trial function plays a major role in changing the physical structure of the solution. Many models are approached and the obtained results reveal that the method is very effective and convenient for constructing solitary solutions.     

Approximate solutions of SchrSdinger equation for Eckart potential with centrifugal term

The approximate analytical solutions of the Schrodinger equation for the Eckart potential are presented for the arbitrary angular momentum by using a new approximation of the centrifugal term. The energy eigenvalues and the corresponding wavefunctions are obtained for different values of screening parameter. The numerical examples are presented and the results are in good agreement with the values in the literature. Three special cases, i.e., s-wave, ξ= λ=1, and β=0, are investigated.     

Approximate analytic solutions for a generalized Hirota—Satsuma coupled KdV equation and a coupled mKdV equation

<正>This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries(KdV) equation and a coupled modified Korteweg-de Vries(mKdV) equation. This method provides a sequence of functions which converges to the exact solution of the problem and is based on the use of the Lagrange multiplier for the identification of optimal values of parameters in a functional.Some examples are given to demonstrate the reliability and convenience of the method and comparisons are made with the exact solutions.     

New hierarchy for the Liouville equation,irreversibility and Fokker‐Planck‐like structures

The issue of irreversibility is revisited for a closed system formed by N classical non‐relativistic particles inside a volume Ω, interacting through two‐body potentials, for large N and Ω. The classical phase‐space distribution function f, multiplied by suitable Hermite polynomials and integrated over all momenta, yields new moments. The Liouville equation and the initial distribution f_in imply a new non‐equilibrium linear infinite hierarchy for the moments. That hierarchy differs from the BBGKY one for distribution functions and displays some suggestive Fokker‐Planck‐like structures. A physically motivated ansatz for f_in (which introduces statistical assumptions), used by previous authors, is chosen. All moments of order n ≥ n₀ are expressed in terms of those of order n₀ — 1 and of f_in. The properties of the Fokker‐Planck‐like structures (hermiticity, non‐negative eigenvalues) allow for implementing a natural long‐time approximation in the hierarchy, so as to introduce relaxation to equilibrium and irreversibility, consistently with the hydrodynamical balance equations. Further (more restrictive) assumptions and approximations lead to new irreversible models, generalizing non‐trivially the Fokker‐Planck equation. They are described through a truncated hierarchy of linear equations for moments of order n ≤ n₀ — 1 (n₀ being finite). The connections with Brownian particle dynamics and Fluid Dynamics are analyzed, for consistency.