Ward's identity in critical dynamics

We consider the relaxation of an order-parameter fluctuation of wave numberk in a system undergoing a second-order phase transition. In general, close to the critical point, wherek ^{–1 –1} (the correlation length) the relaxation rate has a linear dependence on/k of the form (k, ) = (k, 0)x(1–a/k). In analogy with the use of Ward's identity in elementary particle physics, we show that the numerical coefficienta is readily calculated by means of a mass insertion. We demonstrate, furthermore, that this initial linear drop is the main feature of the full/k dependence of the scaling functionR ^–x (k,), wherex is the dynamic critical exponent andR=(k²+ ²)^1/2 is the distance variable.     

Symmetry vector fields and similarity solutions of a nonlinear field equation describing the relaxation to a maxwell distribution

In the study of the formulation of Maxwellian tails the nonlinear partial differential equation ² u/x +u/x+u ²=0 arises. We determine the Lie point symmetry vector fields and calculate the similarity ansätze. Then we discuss the resulting ordinary differential equations. Finally, the existence of Lie Bäcklund vector fields is studied and a Painlevé analysis is performed.     

On the geometric structure underlying the eikonal equation

Given the eikonal equation _i=1 ³ (/x ⁱ ) ² =n ², we investigate the geometric structure that underlies the law of propagation of the wavefronts (x ¹,x ²,x ³) —ct=0. It turns out that Huygens' principle for the propagation of wavefronts is given in terms of a contact structure. Wavefronts are carried into wavefronts by contact transformations. As regards the wave-particle duality principle that arises in quantum mechanics, there is a natural geometric structure, a symplectic manifold (M ²ⁿ , ), which unifies Fermat's principle and the eikonal equation (Huygens' principle).On leave of absence from Institut für Angewandte Mathematik, Fachbereich Mathematik der Universität Mainz, Mainz, German Federal Republic.     

Hyperbolic initial-boundary-value problem for magnetic flux compression by plane liners

Based on the (relativistic) Maxwell equations with displacement current E/t, the initial-boundary-value problem for the compression of an initially homogeneous magnetic fieldB={0,B(x,t),0} between a fixed liner atx=0 and a detonation-driven liner atx=s(t) is solved analytically. By homogenizing the boundary conditions at the moving boundary, the transient electromagnetic fields are shown to be a superposition of quasistatic elliptic (E/t=0) and hyperbolic (E/t0) wave solutions. The wave equation is solved by a Fourier expansion in time-dependent eigenfunctionsf _n =f _n [nx/s(t)] for the variable region 0xs(t), where the Fourier amplitudes _n (t) are determined by coupled differential equations of second order. It is concluded that the conventional elliptic flux compression theories (E/t=0) hold approximately for nonrelativistic liner speeds , whereas the hyperbolic theory (E/t0) is valid for arbitrary liner speeds .     

On the Surface Pressure for the Edwards-Anderson Model

For the Edwards-Anderson model we introduce an integral representation for the surface pressure (per unit surface) in terms of a quenched moment of the bond-overlap on the surface. We consider free , periodic and antiperiodic ^* boundary conditions (by symmetry ⁽⁾=^(*)), and prove the bounds We show moreover that at high temperatures ⁽⁾ is close to ^2/4 and ⁽⁾ is close to ^2/4 uniformly in the volume .     

The hard-sphere fluid: New exact results with applications

A theorem for convolution integrals is proved and then applied to extend the second zero-separation theorem to the bridge functionb(r) and direct-correlation tail functionsd(r). This theorem allows us to exactly relateb(r)/r andd(r)/ratr=0 for the hard-sphere fluid to the contact value of the radial distribution functiong(r) atr= ⁺. From this we obtain immediately the exact values of b(r)/r and d(r)/r atr=0 through second order in number density . Using our results to compare the exact and Percus-Yevick (PY) bridge function, we find that they differ significantly. After obtaining the bridge function and tail function and their derivatives atr=0 andr= through, we suggest new approximations forb(0) andd(0) as well as an analytical integral-equation theory to improve the PY approximation in the pure hard-sphere fluid. The major deficiency of that approximation has been its poor assessment of the cavity function inside the hard-core region. Our theory remedies this defect in a way that yields ay(r) that is self-consistent with respct to the virial and compressibility relations and also the two zero-separation relations involvingy(r) and its spatial derivative atr=0.     

Second-Harmonic-generation studies of inclined thin films

The nonlinear surface susceptibility _S ⁽²⁾ is generally referred to the system of Cartesian coordinates of thesample. However, since actual measurements take place with respect to thelaboratory system of coordinates, appropriate transformation of _S ⁽²⁾ is necessary to deduce the underlying susceptibility-components. We demonstrate this by analyzing the second-harmonic signal generated by an inclined sample upon rotation around thes-axis of the laboratory system of coordinates. The sample consists of two Langmuir-Blodgett-type monolayers of rotational symmetry around the sample normal, the two layers being separated by a plane-parallel flat of 1 mm thickness. The occurrence of a symmetry-forbiddenI _SS ² -signal is discussed in detail.     

Parametrizing Equivalence Classes of Invariant Star Products

Let (M,,) be a symplectic manifold endowed with a symplectic connection . Let Symp(M,) be the group of symplectic transformations of (M,) and Aff(M,) be the group of affine transformations of the affine manifold (M, ). In this Letter, we show that, for any subgroup G of Symp(M,) Aff(M,), the set of G-equivalence classes of G-invariant star products on (M,) is canonically parametrized by the set of sequences of elements belonging to the second de Rham cohomology space of the G-invariant de Rham complex on M.     

On reduction of a two-dimensional generalized Toda lattice to an ordinary differential equations system

Reduction of a two-dimensional generalized Toda lattice to an ordinary differential equations system, which defines the functional dependence of Toda functions on the corresponding separable solutions, is given. It is shown, in particular, that between all the equations of the form ²/x t=() only Liouville (L), sine-Gordon (SG) and Bullough-Dodd (BD) equations, which are associated with simple Lie algebras of a finite growth of rank 1, lead to the third Painlevé equation (P3). The last circumstance allows one, probably, to assume the existence of a deep relation between the complete integrability condition for the corresponding class of dynamical systems and the criterion of the absence of movable critical points in the solutions of an ordinary differential equation system of the second order. If this is so, it means that Painleé's equations and transcendents can be generalized for multi-component cases.Tbilisi State University     

Observed difference between the probabilities of recoilless resonance absorptionf′ and reemissionf″ of gamma-quanta in single crystals of Na2[Fe(CN)5NO]·2 H2O

The recoilless absorption probability factor,f, and recoilless reemission,f, both measured on Na₂[Fe(CN)₅NO]·2 H₂O single crystals using the black filter technique, were found to be different. Unexpectedly, the results found weref>f. In the calculation off, selfabsorption in the scatterer, non-ideality of the black filter and the influence of non-resonant scattering processes have all been taken into account. By varying the scattering geometry for the incoming and outgoing -beam relative to the crystallographic axes only a change in the reemitted valuesf _a, f_b, f_c could be detected because of the long lifetime of the excited nucleus (10^–7 s) relative to the lattice vibration frequencies (10¹² Hz).     

r-Functions with quasi-dynamical spectral parameter

For certain 1 + 1-dimensional classical field theories, whose equations of motion can naturally be written in Lax form by introducing a quasi-dynamical spectral parameter (using sdiff2, the Lie algebra of symplectic diffeomorphisms of a two-dimensional manifold), the previously derived Poisson-commutativity of an infinite set of conserved charges also follows from the existence of an -function, whose functional form is given, and checked explicitly (as well as its Yang-Baxter equation) for an interaction that is exponential (respectively, inverse square) in the spatial derivative of the field.     

Differential equations in the spectral parameter

We determine all the potentialsV(x) for the Schrödinger equation (– _x ² +V(x))=k² such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formB(k, _k )ø=(x)ø. For each suchV(x) we determine the algebra of all possible operatorsB and the corresponding functions (x)This research was partially supported by NSF grant DMS 84-03232 and ONR contract NOOO14-84-C-0159     

Sharpness of the phase transition in percolation models

The equality of two critical points — the percolation thresholdp _H and the pointp _T where the cluster size distribution ceases to decay exponentially — is proven for all translation invariant independent percolation models on homogeneousd-dimensional lattices (d1). The analysis is based on a pair of new nonlinear partial differential inequalities for an order parameterM(,h), which forh=0 reduces to the percolation densityP — at the bond densityp=1–e ^– in the single parameter case. These are: (1)MhM/h+M ²+MM/, and (2) M/|J|MM/h. Inequality (1) is intriguing in that its derivation provides yet another hint of a ³ structure in percolation models. Moreover, through the elimination of one of its derivatives, (1) yields a pair of ordinary differential inequalities which provide information on the critical exponents and . One of these resembles an Ising model inequality of Fröhlich and Sokal and yields the mean field bound 2, and the other implies the result of Chayes and Chayes that . An inequality identical to (2) is known for Ising models, where it provides the basis for Newman's universal relation and for certain extrapolation principles, which are now made applicable also to independent percolation. These results apply to both finite and long range models, with or without orientation, and extend to periodic and weakly inhomogeneous systems.Research supported in part by the NSF Grant PHY-8605164Also in the Physics Department     

Path integrals on half-line

In the framework of path integrals we present a solution to the Schrödinger equation for a free particle confined to the half-linex > 0. A solution in question corresponds to the boundary condition (/x) (0,t)= (0,t) where is a real constant.     

Experimental study of ternary fission of U235 with thermal and resonance neutrons

The alpha particle spectra from ternary fission of U²³⁵ have been investigated in the thermal and the resonance neutron energy regions. The resonance neutrons have been obtained by passing the reactor neutron flux through a boron filter. A silicon surface barrier detector telescope technique for alpha particle detection has been used. In the resonance neutron energy region the ratio _f / _f has been found to be 1·02 ± 0·2 times the value of _f / _f in the thermal region. The non-gaussian tail of the LRA energy distribution has been observed.The authors are indebted to Dr. I.Wilhelm, Mrs. M.Pospíilová, Mr. J.vanda for their kindly help in the experimental data handling and to Mr. V.Kvítek for his assistance during the experiment.     

The chiral determinant and the eta invariant

For { _y },y, a one parameter family of invertible Weyl operators of possibly non-zero index acting on spinors over an even dimensional compact manifoldX, we express the phase of the chiral determinant det _– in terms of the invariant of a Dirac operator acting on spinors over ×X.Supported in part by NSF Grant No. PHY-82-15249Supported in part by NSF Grant PHY 8605978 and the Robert A. Welch Foundation     

An upper bound on the free energy for classical systems with Coulomb interactions in a varying external field

The thermodynamic limit is taken using a sequence of regions all the same shape as a given region of volume ||, with a specified distribution of normal field component on . We show that with magnetostatic interactions the limiting free energy density is bounded above by jhen where (,B) is the free energy density for a system of density in a uniform external fieldB and the inf is taken over all divergence-free fieldsB with given normal component on and all densities (x) compatible with particle number constraints of the form where _i is a sub-region of . A physical argument suggests that this upper bound is the true thermodynamic limit, and that it takes account demagnetization effects. Electrostatic interactions can be treated similarly.     

Contribution to the thermodynamics of the isomorphous γ ⇆ α-transition in metallic cerium

Using the method of the theory of thermodynamic stability and the form of the lines of the phase equilibrium of the isomorphous -transition in metallic cerium, the authors investigate the behavior of the basic thermodynamic characteristics of this transition in the two-phase region and in a neighborhood of the critical point. Attention is focused primarily on the little-studied adiabatic quantities (T/C_V, (–P/V)_S, (T/V)_S). It is shown that along the phase tie-lines these quantities do not depend on V and S; in the limit of the critical point all adiabatic quantities have nonzero minima, and all isodynamic quantities (T/Cp, (–P/V)_T, (T/V)_P) approach zero according to the same law. The obtained thermodynamic results are compared with existing experimental data and models which can be interpreted thermodynamically. It is concluded that critical phenomena in cerium correspond to critical behavior of the first type.Dnepropetrovsk State University Dedicated to the 300th Anniversary of the Unification of the Ukraine with Russia.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 66–70, May, 1991.     

Correlation function formulas for some infrared asymptotic-free scalar field lattice models via the block renormalization group

We consider a class of scalar field lattice models with action 1/2()+V(), V small. After n block renormalization group transformations, new formulas are obtained for the finite lattice generating and correlation functions. For some infrared asymptotic-free models in the thermodynamic and n limits, the formulas for correlation functions are especially simple, isolate the correct dominant long-distance behavior, and can be used to control the subdominant contributions.     

A Monomial Basis for the Virasoro Minimal Series M(p,p′) : The Case 1<p′/p<2

Quadratic relations are given explicitly in two cases of chiral conformal field theory, and monomial bases of the representation spaces are constructed by using the Fourier components of the intertwiners. The first case is the (2,1) primary fields for the (p,p)-minimal series M_r,s (1rp–1,1sp–1) for the Virasoro algebra where 1<p/p<2. We restrict ourselves to the case p3, for which the (2,1) primary field exists. The second case is the intertwiners corresponding to the two-dimensional representation for the level k integrable highest weight modules V() (0k) for the affine Lie algebra