Generalized perturbation theory of classical fluids in the critical region

Perturbation theories of classical fluids are extended in the critical region by decomposition of the attractive part of the pair interaction. The generalized theory is in agreement with scaling, has the correct dependence on dimensionality and on the range of the forces and offers a method to compute critical indices.     

Time-dependent perturbation theory with a classical limit

We construct a quantum mechanical perturbation theory which uses the multiple time scale technique. Working with the time translation operator, we use a variant on the method of Bender and Bettencourt. Our perturbation theory smoothly crosses over to the classical result as Planck's -->0. It is seen that this technique has a nonperturbative element built into it.     

The evaluation of some integrals that arise in the perturbation theory of liquids

ABSTRACT

The venerable, but still very useful, perturbation theory of liquids of Barker and Henderson involves some integrals that must be obtained numerically. There is no avoiding this for potentials such as the Lennard–Jones potential. However, to obtain the pressure, Barker and Henderson also obtained the necessary density derivatives numerically. This is cumbersome and care must be used to avoid excessive error. In this paper, it is noted that the earlier and less successful perturbation theory of Frisch et al. employed an alternative method, all but forgotten, for the evaluation of similar integrals. Their method still requires numerical integration. However, it allows an investigator to obtain the integrals for the pressure by analytical differentiation. The resultant integrals must still be evaluated numerically, but the extra step and annoyance of numerical differentiation is avoided. It is noted here that this method of using density derivatives can be incorporated into the Barker–Henderson theory.     

Dynamical perturbation for classical fluids: A solvable model

We investigate on a one-dimensional model the perturbation to the timedependent correlations in a classical fluid when a small interaction is added to a hard core. Various formulas have already been proposed for this correction. We verify on this model, for which everything can be calculated explicitly, that the expressions proposed by Frisch and Berne yield strongly divergent time integrals for the diffusion coefficient. On the contrary, when all corrections are accounted for, the correction to the velocity time correlation is shown to decay like (Int)/t ₂ at large times, yielding a finite first-order correction to the diffusion coefficient. The extension of this calculation to a gas of hard rods in the case of a perturbation with an infinite range is discussed.     

A perturbation theory for the Korteweg-De Vries equation

By writing the perturbed Korteweg-de Vries equation (1) in operator form (2), we derive equations which are a basis for a perturbation method. In particular, in the first approximation, we obtain from them equations describing the evolution of the soliton amplitude and velocity. The present theory may be extended, also, to other nonlinear evolution equations if they are solved, without perturbation, by the inverse-problem method.     

A classical theory of superparamagnetic relaxation

A Fokker-Planck equation for the relaxation of a classical ferromagnetic particle coupled to a classical heat bath is derived from the Nakajima-Zwanzig equation. The equation of motion for the mean magnetization of an ensemble of particles is found to be closed only under special circumstances. In the strong motional narrowing limit the equation of motion reduces to the Bloch equations in the limit MH ? k_BT, i.e. for small particles, and to the Landau-Lifshitz equation in the opposite limit. For the motional narrowing region in toto the particular case of uniaxial anisotropy is analysed, giving an equation of motion which for large particles reduces to a modified Landau-Lifshitz equation with g-shift and a reduced damping constant. This equation cannot be meaningfully identified with the Gilbert equation.Approximate expressions for superparamagnetic relaxation rates by Kramers' method are obtained for the case of (i) triaxial (i.e. orthorhombic) and (ii) cubic (K +ve and ?ve) anisotropy, assuming large energy barriers. The results supplement Brown's expression for uniaxial anisotropy and show a more complicated dependence on the Landau-Lifshitz parameter λ than the linear dependence found by Brown. For small λ the rates tend to constant values compatible with the transition.     

A classical theory of hard squares

A simple phenomenological theory of the hard-square lattice gas is obtained by analyzing a low-order corner transfer matrix variational approximation. The free energy is of Landau type and expressions are obtained for the order parameter and densities. In this approximation, the model exhibits a critical point atz _c =4(3 + 23)/9 with critical exponents given by the classical values: =0_disc,=1/2, =1, =3.     

A finite-dimensional canonical formalism in the classical field theory

A canonical formalism based on the geometrical approach to the calculus of variations is given. The notion of multi-phase space is introduced which enables to define whole the canonical structure (physical quantities, Poisson bracket, canonical fields) without use of functional derivatives. All definitions are of pure geometrical (finite dimensional) character.The observable algebra (physical quantities algebra) obtained here is much smaller then the algebra of all (sufficiently smooth) functionals on the space of states, derived from the standard infinite-dimensional formulation. As it is known, the latter is much too large for purposes of quantization. As the examples prove, our algebra could be an adequate start-point for quantization.For simplifying the language the notion of observable-valued distribution is introduced. Many concrete physical examples are given. E.g. it is shown that some problems connected with gauge in electrodynamics are automatically solved in this approach. The introduced language allows to obtain the Noether theorem in a most natural way.     

A brief introduction to chiral perturbation theory

A brief introduction to the subject of chiral perturbation theory (Xpt) is presented, including a discussion of effective field theory and applications of Xpt in the arena of purely mesonic interactions as well as in the πN sector.     

Defects within the landau theory: A perturbation approach

We discuss the continuum medium theory that enables us to find the defect contribution to phase transition anomalies with the use of only a few phenomenological parameters. The theory is invalid in the immediate vicinity of the phase transition temperature and for high concentration of defects. The possibilities of interpretation of experimental data on the basis of the theory are also discussed.     

A new perturbation theory for transport coefficients

The Green's function method for the calculation of orbital energies developed in the preceding paper is applied to the benzene molecule. It is confirmed that the lowest-order approximation for irreducible diagrams in our theory is equivalent to the usual self-consistent field theory. Higherorder corrections to orbital energies are calculated and theoretical and experimental results are compared.     

Conformal invariance in perturbation theory

Asymptotic conformal invariance is proved to be true at all orders in perturbation theory. The correct Ward Identities for broken conformal invariance are derived: they are the extension of the Callan Symanzik equation from scale to conformal transformations.