Variational principles and governing equations in nano-dielectrics with the flexoelectric effect

The flexoelectric effect is very strong and coupled with large strain gradients for nanoscale dielectrics. At the nanoscale, the electrostatic force cannot be ignored. In this paper, we have established the electric enthalpy variational principle for nanosized dielectrics with the strain gradient and the polarization gradient effect, as well as the effect of the electrostatic force. The complete governing equations, which include the effect of the electrostatic force, are derived from this variational principle, and based on the principle the generalized electrostatic stress is obtained, the generalized electrostatic stress contains the Maxwell stress corresponding to the polarization and strain, and stress related to the polarization gradient and strain gradient. This work provides the basis for the analysis and computations for the electromechanical problems in nanosized dielectric materials.     

Variational principles of magnetostatics

Variational principles of magnetostatics are formulated, which permit evaluation of the upper and lover bounds on the energy of a magnetic field produced by a known spatial current density distribution with the help of rather broad classes of trial functions. The potentialities for the use of these variational principles as applied to practical problems is demonstrated with an example of the well-known problem of the calculation of the inductance of a straight solenoid of finite length. In particular, it is shown that the formula appearing in all the textbooks for this inductance, in fact, defines the upper bound for the true value of the inductance.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 70–75, July, 1987.     

Variational principles in the theory of gravitoinertial fields

Variational methods and some features of their application in the theory of gravitoinertial fields are considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 110–114, November, 1974.     

Variational principles and self-organization in two-fluid plasmas

Self-organization of an ordered structure occurs in a plasma under rather restrictive conditions. A new framework for a variational principle invokes a coercive form that results in a criterion for self-organizing relaxation of a two-fluid plasma. The constraints (constants of motion of the ideal model) are adjusted, through a weakly dissipative process, so that the relaxed state, under well-defined conditions, is a stable equilibrium independent of the direct effects of dissipation.     

Variational principles for the calculation of the response function

Several variational principles for inclusive processes are presented and illustrated by simple examples. By choosing appropriate trial functions, the doorway-state, moment- and cumulant-expansion of the response functions are derived from them.     

Variational principles and spatially-homogeneous universes,including rotation

The validity of imposing spatial homogeneity on the variations in the usual action principle for Einstein's equations is studied. It is proved that with this procedure the standard and ADM Lagrangians give correct Einstein equations if and only if the space belongs to Class A of Ellis and MacCallum [1], i.e., the structure constants of the simply transitive group satisfy C _fg ^f =0. The possibility of overcoming this difficulty in the Class B spaces is examined.This work was supported in part by the United States Atomic Energy Commission under Contract Number AT 104-37-39 Project Agreement No. 125.     

Variational principles of electrostatics and higher polarizabilities of conductors

Based on variational principles, new concepts of higher polarizability of an isolated conductor are introduced that generalize well-known concepts of capacity and polarizability. It is shown that the lth-order polarizability in a spherical coordinate system is represented by a (2l+1) × (2l+1) matrix. The corresponding (2l+1) potentials can be considered as a system of lth-order multipoles typical of a conductor and more general than systems of spherical multipoles known in electrodynamics. It is exemplified what new results can be obtained using the higher polarizability technique for solving problems in electrostatics. Krasnoyarsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 61–65, October, 1999.     

Variational principles of magnetostatics of superconductors. I

Fundamental and dual variational principles of magnetostatics of superconductors are formulated. It is shown how test fields can be expressed in terms of vector and scalar potentials. General consequences of the variational magnetostatic formulation are discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 97–102, September, 1991.     

Variational principles of magnetostatics of superconductors. II

Simple examples are provided, where demonstrating the implementation of algorithms formulated by the variational principles of the first part of this paper is quite difficult, because even for the simplest body of practical interest —a loop — solving the problem by classical methods of potential theory is rather difficult. For this reason, presenting the corresponding results requires substantially more space, and obtaining these results can be considered as an independent study.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 103–109, September, 1991.     

Variational principles for scattering in the generator coordinate method

The generator coordinate method (GCM) wave function is used as a trial function in a Kohn type variational principle for scattering phase shifts. It is shown that a GCM trial function is a solution of the variational equations if the Hill-Wheeler integral equation is satisfied subject to an appropriate boundary condition. A new method for introducing the scattering boundary condition is presented. There is a uniqueness theorem for the phase shift.     

On the Casimir effect without cutoff

The formulation of the Casimir effect without cutoffs is discussed. Our derivation emphasizes the decisive role of the free-space electromagnetic energy density. The zero point energy arises as an energy per unit volume, i.e., as local (in x space) energy density. It is given by the vaccum expectation value of the free-space Hamiltonian density in the Fock representation corresponding to the non-trivial geometry. The two Fock representations corresponding to the system with and without plates are proved to be inequivalent.     

Variational principles for constructing few-mode models of the laminar-to-turbulent transition

A method based on variational principles is proposed for constructing few-mode models of distributed nonlinear systems to a higher accuracy of approximation than the Galerkin method for a prespecified basis of trial functions. This method may be useful for studying the initial stages of the laminar-to-turbulent transition, when the increase in the number of trial functions sharply increases the stiffness of the reduced system, so that it is advisable to use optimal few-mode models. The proposed method is used to construct and analyze a modification of the Lorenz model for a Saltzman system. Zh. Tekh. Fiz. 67, 1–5 (May 1997)     

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.     

Variational principles for calculation of transport coefficients of relativistic multi-component systems

Variational expressions that give lower bounds on the transport coefficients of a multi- component relativistic system are derived from the set of coupled Boltzmann equations that are assumed to govern its behavior. No assumptions concerning the types of interactions that may exist between the various particles comprising the system other than that they are point-like are made. In particular we do not restrict ourselves to binary interactions and we allow particle creation and destruction to occur. We also take account of any external fields that may be present.     

Variational principles for multi-walled carbon nanotubes undergoing buckling based on nonlocal elasticity theory

Variational principles are derived for multi-walled carbon nanotubes undergoing buckling using the semi-inverse method. Derivations are based on the continuum modelling of nanotubes taking into account small scale effects via the nonlocal theory of elasticity. Natural and geometric boundary conditions for multi-walled nanotubes are derived which leads to a set of coupled boundary conditions.     

Variational principles,renormalization group,and Kadanoff's universality

Variational principles satisfied by various thermodynamic functionals are used to set up a completely renormalized scheme for the analysis of critical phenomena. Different aspects of Kadanoff's universality can be expressed in a simple fashion in our language. The main result of the paper is a unified derivation of the scaling laws combining the variational principles with renormalization group techniques which are especially simple in this formalism. A suitable choice of the normalization point has led to a new renormalization group transformation. The corresponding differential equation can be solved even in the nonasymptotic region. The discussion of the asymptotic theory and of the approach to it is therefore simpler. The connection with the more traditional approaches is discussed. The calculation of the critical indices is reduced to only two of them which are directly expressed in terms of renormalized quantities. From this point onwards their evaluation proceeds along standard lines. Special emphasis has been given to the illustration of the power and conceptual simplicity of the method.     

Variational principles on principal fiber bundles: A geometry theory of Clebsch potentials and Lin constraints

The geometric theory of Lin constraints and variational principles in terms of Clebsch variables proposed recently by Cendra and Marsden [1987] will be generalized to include those systems defined not only on configuration spaces which are products of Lie groups and vector spaces but on configuration spaces which are principal bundles with structural group G. This generalization includes, for example, fluids with free boundaries, Yang-Mills fields, and it will be very useful, as it will be shown later, to illustrate some aspects of the theory of particles moving in a Yang-Mills field in both its variational and Hamiltonian aspects.     

系统规模对群体行为的效果

影响细胞群体行为的因素是多种多样的, 除了以前研究的细胞通讯方式和环境因素外, 还与现有文献没有或很少研究的细胞数目(即系统规模)有关. 本文研究了系统规模对一类合成多细胞通讯系统的聚类行为的影响. 在该系统中, 单个系统是由压制振动子和基于延迟的松弛振子整合而成的振子, 而振子之间通过群体感应机制相互耦合. 通过分岔分析和数值模拟发现: 细胞数目的增加不仅可以改变平衡态聚类稳定性区间的大小并诱导新的聚类行为, 而且有利于扩大平衡态聚类的吸引域, 表明细胞分化可能与系统规模有密切关系; 细胞数目的增加还可以极大地丰富平衡态聚类和振动聚类的表现形式和共存方式, 为生物体对环境的适应性提供了良好的基础. 我们的结果不仅扩充了耦合系统的动力学行为, 也为理解多细胞现象奠定了基础. 关键词： 系统规模 群体感应 群体行为 耦合振子     

Effect of spatial patterns on population size

We investigate the role of the spatial pattern and temporal dynamics on the population properties of a diffusive Gause–Lotka–Volterra system. The average total population size is insensitive to the temporal dynamics, whereas a significant decrease of the population size can be found as the spatial diffusion effect is increased, implying that the spatial pattern plays an important role. At large diffusion coefficients, a saturation of the spatial pattern variation is observed, which can be understood by a spatial scaling analysis of the system. The existence of multiple attractors can also indicate that spatial patterns play a more important role than temporal dynamics in dominating the population size.     

Variational principles for two kinds of extended Korteweg—de Vries equations

Variational principles are constructed using the semi-inverse method for two kinds of extended Korteweg—de Vries (KdV) equations, which can be regarded as simple models of the nonlinear oceanic internal waves and atmospheric long waves, respectively. The obtained variational principles have also been proved to be correct.