On the Non-linear Generalization of the Gyarmati Principle and Theorem

The validity of the Gyarmati theorem has been extended to non-linear types of constitutive equations governing dissipative processes which have recently been suggested. It has been proved that by extending the validity of the theorem, the validity of Gyarmati's variational principle of thermodynamics is guaranted for non-linear theories adequate to constitutive equations in question as well. The theory was applied for stationary temperature distribution in a rigid bar.     

Some variational principles in electroelastic media under finite deformation

It is proposed that the universal thermodynamic energy variational principle is included in the first law of thermodynamics. Some variational principles in the electroelastic media under finite deformation are derived from this universal thermodynamic variational principle. It is suggested that in the general electroelastic analysis the environment should be considered together with the discussed electroelastic medium. For the variational principle of nonlinear electroelastic media the variation of the electric potential is coupled with the virtual displacement, and the variation of the initial volume should be considered. The Maxwell stress in the initial configuration is naturally derived from this variational principle and it is unique in the second order precision. Supported by the National Natural Science Foundation of China (Grant No. 10472069)     

The variational theory of adiabatic motions,and its applications to linear and non-linear oscillators

This paper presents a variational derivation of the adiabatic periodic motion theorems and related time-integral-of-energy results, including the virial theorem, and some of their applications to linear and non-linear oscillators. Specifically, (i) first, the Maupertuis-Euler-Langrange (MEL) action principle is formulated for the most general (scleronomic and holonomic) system; in the derivation the time-dependent system parameters are treated just like additional generalized co-ordinates and subjected to similar variations; (ii) next, combination of MEL's principle with the first law of thermodynamics yields the adiabatic theorem; subsequent specializations of it lead to additional energetic equations; (iii) the theory is then applied to the one d.o.f. linear and non-linear oscillator; the effects of linear friction and of a harmonic external force are also discussed; useful relations for the adiabatically varying system parameters are thus obtained.     

A Variational Principle for Thermodynamical Waves

When the dissipative processes are dominant in the system, the assumption of local equilibrium holds good and the space time evolution of irreversible system can be described by the variational principle of GYARMATI. However when imposed changes in the state variables are fast, the system can not be in a state of local equilibrium and to define the nonequilibrium state of the system it is necessary to extend the formalism of classical irreversible thermodynamics. The wave approach of Onsagerian thermodynamics is one such pursuit and is a direct generalization of the original Onsager-Machlup proposition. An important consequence of this theory is that it leads to transport equations with finite propagation velocities, which are referred to as thermodynamical waves. In this note we endeavour to write the appropriate form of GYARMATI'S variational principle for thermodynamical waves.     

事件空间中力学系统的微分变分原理

研究事件空间中力学系统的微分变分原理.基于D'Alembert原理，建立了事件空间中力学系统的D'Alembert-Lagrange原理、Jourdain原理、Gauss原理和万有D'Alembert原理，给出了这些原理的Euler-Lagrange参数形式、Nielsen参数形式和Appell参数形式，并导出了万有D'Alembert原理的Mangeron-Deleanu参数形式. 关键词： 分析力学 事件空间 微分变分原理     

Thermodynamics of non-differentiable systems

An axiomatisation of classical thermodynamics previously proposed for a somewhat restricted class of systems whose state spaces are differentiable manifolds is extended to systems whose state spaces are arbitrary connected separable topological spaces. It turns out that such systems need not obey Carathéodory's principle, although they do obey a form of Kelvin's principle.     

Irreversible thermodynamics of fluids

Irreversible thermodynamics of fluids is formulated based on a set of postulates. The theory thus constructed generalizes thermostatics and linear irreversible thermodynamics into the realm of nonlinear irreversible processes. In this theory the extended Gibbs relation and the entropy balance equation appear as a pair of mutually consistent equations under the postulates made. An equivalent theory is also formulated by replacing one of the postulates with another that is basically a variational principle. The variational principle yields the evolution equations for fluxes as the Euler equations that extremize the variational functional postulated. The local form of the extremized variational functional is the entropy balance equation for the irreversible processes in the system. Some further consequences of the theory are also considered. For example, nonequilibrium specific heats are shown to be at least quadratic functions of fluxes and reduce to the equilibrium specific heats in the limit of vanishing fluxes. In order to illustrate an example of possible applications, we have considered nonlinear transport processes in fluids. The connections of the present theory with other theories are discussed.     

Evolution of self-organizing systems from the standpoint of mechanics and thermodynamics

The extension of the principle of least action for many-particle systems has shown that the natural evolution of processes in the environment is governed largely by kinetic-to-potential energy conversion. A relation with the variational principles of nonequilibrium thermodynamics is established. It is shown that the birth and aging of structures are controlled by the principle of minimization of the integral Lagrangian, the role of which in mechanics is not less significant than the role of the law of degradation of energy (the second law of thermodynamics) in thermodynamics.     

Black holes and information theory

During the past three decades investigators have unveiled a number of deep connections between physical information and black holes whose consequences for ordinary systems go beyond what has been deduced purely from the axioms of information theory. After a self-contained introduction to black hole thermodynamics, we review from its vantage point topics such as the information conundrum that emerges from the ability of incipient black holes to radiate, the various entropy bounds for non-black hole systems (holographic bound, universal entropy bound, etc.) which are most easily derived from black hole thermodynamics, Bousso's covariant entropy bound, the holographic principle of particle physics, and the subject of channel capacity of quantum communication channels.     

On the universality of thermodynamics' Legendre transform structure

It is shown that the Legendre transform structure of thermodynamics does not depend upon the functional form of the entropy. It is a just a necessary consequence of Jaynes' maximum entropy principle. Moreover, in the important special case of the canonical ensemble, the Legendre transform structure of thermodynamics is shown to be independent of the functional form of the mean energy constraint as well.     

The variational symmetries and conservation laws in classical theory of Heisenberg (anti)ferromagnet

The non-linear partial differential equations describing the spin dynamics of Heisenberg ferro- and antiferromagnet are studied by Lie transformation group method. The generators of the admitted variational Lie symmetry groups are derived and conservation laws for the conserved currents are found via Noether's theorem.     

Complementary variational principles and their application to rheo-linear and rheo-non-linear vibrations

A complementary (to Hamilton's) variational principle, and its subsequent energetic specialization to periodic systems are developed, and, subsequently, applied to the approximate determination of the stability/instability boundaries, first of the linear Mathieu equation, and then of two common types of non-linear Mathieu-like equations. Extensions of the methodology are also indicated.     

Nonlinear distribution functions for the Vlasov-Poisson system

The nonlinear distribution function introduced by Allis has been used to investigate the stability of the solution of Vlasov-Poisson’s equations. The ‘average’ Lagrangian is calculated on the basis of this distribution function, and the ‘average’ variational principle of Witham is applied to discuss modulational stability. It is found that the distribution function of Allis exactly gives rise to the Lighthill’s stability condition of non-linear waves.     

Variational properties of irreversible processes

The thermodynamic integral principle, equivalent to the Onsager theory of irreversible thermodynamics, is analyzed in detail for a purely dissipative system. Different reformulations of the principle are also given together with the derivation of the corresponding Euler—Lagrange equations. One of them, the dual field formulation, is of special interest: It is an exact variational principle in terms of the intensive parameters and their dual fields introduced in place of the thermodynamic current densities. Finally, the possibility of deducing variational statements in terms of volume and surface dissipation functionals is discussed.     

On the static and kinetic methods of conservative system stability,and Rayleigh's principle: A reexamination

The relation between the static and kinetic variational methods of the stability of equilibrium analysis of conservative systems and the corresponding static and kinetic Rayleigh's principles is reexamined. Specifically made explicit are the connections between (i) the virtual work principle (for the adjacent equilibrium configuration) and Rayleigh's principle of extremum critical loads and buckled modes, and (ii) Hamilton's principle (for the adjacent non-equilibrium configuration) and Rayleigh's principle of extremum frequencies and mode shapes, through simple familiar examples. These connections are found by considering, in addition to the familiar mode amplitude variations, variations of the load which in turn produce variations in the space (i) and time (ii) domain lengths, respectively; one is thus led to a variable endpoints variational problem (instead of the customary fixed endpoints one) which, by invoking the energetics of these adjacent configurations, is simplified and finally brought to the standard Rayleigh's principle form.     

近海波-流相互作用的缓坡方程理论体系

在对近海一大类水波模型——缓坡方程的总体考察、分析和判断基础上,引进一个表征近海普遍波-流相互作用机制的算子,进而借助普适的Hamilton水波变分原理,建立了一个关联于时间的近海波-流相互作用缓坡方程理论体系,从形式到内容上达到了一种高度协调和统一.     

Brillouin-Wigner and Feenberg perturbation methods in many-body theory

Brillouin-Wigner (BW) perturbation formulae can be rearranged into a form first proposed by Feenberg. Feenberg's perturbation formulae also follow from a variational principle. They are successfully tested at two typical problems, for which ordinary perturbation techniques completely fail. The first is to find conditions for the bosonization of the Tomonaga model perturbatively. The second is to clarify whether non-Fermi liquid behavior of the momentum distribution function of the Luttinger model can be achieved perturbatively.