Scalar conservation laws with discontinuous flux function: II. On the stability of the viscous profiles

The equation , whereH is Heaviside's step function, appears for example in continuous sedimentation of solid particles in a liquid, in two-phase flow, in traffic-flow analysis and in ion etching. The discontinuity of the flux function atx=0 causes a discontinuity of a solution, which is not uniquely determined by the initial data. By a viscous profile of this discontinuity we mean a stationary solution ofu _t +(F ) _x =u _xx , whereF is a smooth approximation of the discontinuous flux, i.e.,H is smoothed. We present some results on the stability of the viscous profiles, which means that a small disturbance tends to zero uniformly ast. This is done by weighted energy methods, where the weight (depending onf andg) plays a crucial role.     

The metal-semiconductor transition in amorphous Si1−x Cr x films:T 0.19-contribution to the metallic conductivity

A detailed phenomenological re-analysis of previously published conductivity data, (T, x), is presented. It is based on the investigation of differences, (T, x ₁)–(T, x ₂). In this way, the cusp-like low-temperature term is amplified against the other temperature dependent contributions. This term can be described by wherep=0.19±0.03. It is present, if (4.2 K,x) exceeds 260 ^–1 cm^–1, at least up to (4.2 K,x)1350 ^–1 cm^–1 and forT60 K. But it is absent, if (4.2 K,x)180 ^–1 cm^–1. The disappearance of this contribution should be related to the metal-semiconductor transition, taking place atx _c 0.14. On the other hand, the presence of a term proportional toT ^1/2, as predicted by Altshuler and Aronov, seems unlikely.It is argued that the term should be related to the interplay of electron-electron interaction and disorder. The comparison with data from the literature shows that this contribution might also be present in heavily doped crystalline semiconductors.     

Global properties of radial wave functions in Schwarzschild's space-time

Two solutions ₅(x, x _s) and ₆(x, x _s) related to the irregular singular point atx=+ of the radial wave equation in Schwarzschild's space-time are studied as functions of the independent variablex and the parameterx _s. Analytic continuations of ₅ and ₆ are derived and their relation to the flat-space case solutions is established. Explicit expressions for ₃(x, x _s) and ₄(x, x _s) (the solutions about the regular singular point atx=x _s) are given. From these expressions and the analytic continuations of ₅ and ₆ the coefficients relating linearly ₅ and ₆ with _i (i=1, 2, 3, 4) are calculated.     

Uncertainties and an interpretation of nonrelativistic quantum theory

We propose an interpretation of nonrelativistic quantum theory which can be considered a generalized Copenhagen interpretation. The uncertainties (i.e., q and p) in Heisenberg's uncertainty relation q · p/2 can be characterized as (average) errors in an approximate simultaneous measurement if the interpretation proposed here is accepted in nonrelativistic quantum mechanics. Under this interpretation, the (discrete) trajectory of a particle (like Wilson chamber) is significant enough. We propose to analyze this trajectory numerically.     

On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential

In this paper, we first construct multi-lump (nonlinear) bound states of the nonlinear Schrödinger equation for sufficiently small >0, in which sense we call them semiclassical bound states. We assume that 1p< forn=1,2 and 1p<1+4/(n–2) forn3, and thatV is in the class(V) _a in the sense of Kato for somea. For any finite collection {x ₁,...,x _N} of nondegenerate critical points ofV, we construct a solution of the forme ^–iEt/v(x) forE<a, wherev is real and it is a small perturbation of a sum of one-lump solutions concentrated nearx ₁,...,x _N respectively. The concentration gets stronger as 0. And we also prove these solutions are positive, and unstable with respect to perturbations of initial conditions for possibly smaller >0. Indeed, for each such collection of critical points we construct 2 ^N–1 distinct unstable bound states which may have nodes in general, and the above positive bound state is just one of them.     

Influence of gravity on surface melting

A chemical network system for use in the study of reaction cascades is described by the nonlinear rate equation , in which (x) is derived from the –G/x of Taylor's expansion of the order parameterx of the thermodynamic potential, Gibbs' functionG(T,P,x), at about the critical pointC(T _C ,P _C ) of the control variables (parameters)T andP. The behavior of the system around the stable pointx=0 is analyzed only with the sign ofk ₁(T, P), becausek ₂(T, P) is always positive. The system is not closed and is affected by physical and chemical changes in the neighbor systems. WhenT andP fluctuate ( ) through the changes andk ₁ passes zero, the system in the steady state becomes instable andk ₁ jumps at the reaction threshold. Then, the products are formed at the number of moleculesx=(|k ₁|/k ₂)^1/2. To describe such a transition,k ₁ giving theG curvature atx=0 is represented phenomenologically by an approximate function, tanh(G/RT), for a metastable state with a relatively smallk ₁(>0). The reaction takes place in a cascade in accordance with a cubic state equation obtained from tanh(G/RT), which describes a jump of the reaction energy G at the reaction threshold.     

On the Fokker-Planck equation with time-dependent drift and diffusion coefficients and its exponential solutions

The static critical behavior of a mixed hydrogen-bonded ferro-antiferroelectric system is investigated with use of a pseudo-spin Ising model in a transverse field, which describes the tunneling of protons. Assuming an infinite-ranged random interbond coupling, then0 replica method is applied to evaluate the averaged free energy, from which the replica-symmetric solution of the Sherrington-Kirkpatrick type is obtained. It is shown that a proton pseudo-spin glass (PG) phase exists below a freezing temperatureT _f , whereT _f and the order parameters for the PG and ferroelectric phase are parametrized by the tunneling frequency . For _c, where _c = and is the random distribution width, no ordering is possible even forT0. Numerical solutions for the PG order parameter and the dielectric susceptibility atT0 are obtained, and the phase diagram for a simple model of a mixed system atT0 is presented.Dedicated to Prof. Dr. H.E. Müser on the occasion of his 60th birthday     

Exact Large Deviation Functional of a Stationary Open Driven Diffusive System: The Asymmetric Exclusion Process

We consider the asymmetric exclusion process (ASEP) in one dimension on sites i=1,...,N, in contact at sites i=1 and i=N with infinite particle reservoirs at densities _a and _b . As _a and _b are varied, the typical macroscopic steady state density profile ¯(x), x[a,b], obtained in the limit N=L(b–a), exhibits shocks and phase transitions. Here we derive an exact asymptotic expression for the probability of observing an arbitrary macroscopic profile , so that is the large deviation functional, a quantity similar to the free energy of equilibrium systems. We find, as in the symmetric, purely diffusive case q=1 (treated in an earlier work), that is in general a non-local functional of (x). Unlike the symmetric case, however, the asymmetric case exhibits ranges of the parameters for which is not convex and others for which has discontinuities in its second derivatives at (x)=¯(x). In the latter ranges the fluctuations of order in the density profile near ¯(x) are then non-Gaussian and cannot be calculated from the large deviation function.     

Moduli spaces of curves and representation theory

We establish a canonical isomorphism between the second cohomology of the Lie algebra of regular differential operators on ^x of degree 1, and the second singular cohomology of the moduli space of quintuples (C, p, z, L, []), whereC is a smooth genusg Riemann surface,p a point onC, z a local parameter atp, L a degreeg–1 line bundle onC, and [] a class of local trivializations ofL atp which differ by a non-zero factor. The construction uses an interplay between various infinite-dimensional manifolds based on the topological spaceH of germs of holomorphic functions in a neighborhood of 0 in ^x and related topological spaces. The basic tool is a canonical map from to the infinite-dimensional Grassmannian of subspaces ofH, which is the orbit of the subspaceH _– of holomorphic functions on ^x vanishing at , under the group AutH. As an application, we give a Lie-algebraic proof of the Mumford formula: _n =(6n ²–6n+1)₁, where _n is the determinant line bundle of the vector bundle on the moduli space of curves of genusg, whose fiber overC is the space of differentials of degreen onC.     

Scaling of the surface tension of phase-separated polymer solutions

The tension of the interface between the equilibrium phases of a phase-separated polymer solution is obtained in the simplest mean-field approximation from the functional equation for the composition profile of the interface. For temperaturesT near the critical solution temperatureT _c, i.e., for Flory parameter near _c, and for high degrees of polymerizationN, the profile and tension scale with=N ^1/2(– _c), just as do the compositions of the coexisting phases in mean-field approximation. The surface tension in the asymptotic limitN, _c at fixedx, is found to be given bya ²/kT _c (2c'/c)^1/2 N ^-5/4(x), wherea is the lattice spacing of an underlying lattice (or, roughly, the length of a monomer),c andc are the vertical and total coordination numbers of the lattice, and(x) is a scaling function, known for allx, with the asymptotic behavior asx0 and asx. The latter implies that becomes independent ofN asN at fixedT nearT _c; the former implies that becomes proportional toN ^–1/2(1–T/T _c)^3/2 asTT _c at fixedN1, as found previously.     

A Note on Anisotropic Percolation

We consider an anisotropic independent bond percolation model on , i.e. we suppose that the vertical edges of are open with probability p and closed with probability 1–p, while the horizontal edges of are open with probability p and closed with probability 1– p, with 0 < p, < 1. Let , with x₁ < x₂, and . It is natural to ask how the two point connectivity function P_p,({0 x}) behaves, and whether anisotropy in percolation probabilities implies the strict inequality P_p,({0 x})> P_p,({0 x}). In this note we give affirmative answer at least for some regions of the parameters involved.Mathematics Subject Classifications (2000). 82B20, 82B41, 82B43.     

μ +SR in solids — trends and expectations

Conventional ⁺SR studies are classified according to their relation with Solid State Physics, and an attempt is made to identify current trends in the field of ⁺SR in solids. Desirable trends and development in ⁺SR techniques are also discussed shortly. Rather than risk attempts at predicting the future [1] of ⁺SR in solids, it is perhaps prudent merely to identify trends in the field and let individual readers to extrapolate. We shall start with trends in the main stream conventional ⁺SR and then discuss some desirable technical development.     

Kinetics and mechanism of copper fracture due to deformation in surface-active paths

The general kinetic characteristics of copper fracture in the presence of surface-active bismuth-lead baths during creep and elongation under tension are explained. It is shown that the subcritical stage of crack development controls the process, whereupon the effects of stresses , temperature, strain rate , surface energy at the copper-bath interface _SL, and surface energy at the grain boundaries _b on the rate of crack development l/ are analyzed. The basic conclusions are that: a)l/=(–) ( and being constants here); b) the crack development activating energy ) the reduction of energy _b, achieved by intergranular internal adsorption of 0.5% antimony, lowers the value of about 50 times; d) a 30% increase in surface energy _SL reduces the cracking rate 30 times, according to the relation (where A=6 · 10^–15 cm²); and e) .Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 7–15, May, 1976.     

Non-hermitian quantum mechanics in non-commutative space

A recent investigation of the possibility of having a -symmetric periodic potential in an optical lattice stimulated the urge to generalize non-hermitian quantum mechanics beyond the case of commutative space. We thus study non-hermitian quantum systems in non-commutative space as well as a -symmetric deformation of this space. Specifically, a -symmetric harmonic oscillator together with an iC(x ₁+x ₂) interaction are discussed in this space, and solutions are obtained. We show that in the deformed non-commutative space the Hamiltonian may or may not possess real eigenvalues, depending on the choice of the non-commutative parameters. However, it is shown that in standard non-commutative space, the iC(x ₁+x ₂) interaction generates only real eigenvalues despite the fact that the Hamiltonian is not -symmetric. A complex interacting anisotropic oscillator system also is discussed.     

Temperature scaling of the integrated dynamical susceptibility in YBa2Cu3O6.5(T c =50 K)

The magnetic neutron scattering cross section in a crystal YBa₂Cu₃O_6.5(T _c =50 K) has been measured for energies between 4.5 meV and 15 meV and temperatures between 10 K and 275 K. From these data we extract the generalized susceptibility integrated around the (, ) position These data are combined with those at 8.3 and 33.1 meV from a previous study. It is found thatI(, T) exhibits the simple temperature scaling form found previously in La_2–x Sr _x CuO₄. Specifically, in YBa₂Cu₃O_6.5,I(, T)=I(, 0)2/tan^–1 (/0.9T). Implications of this behavior for the d.c. and infrared conductivity are discussed.     

On the asymptotic behavior of the solutions of the Caldirola-Kanai equation

We study in this Letter the asymptotic behavior, as t+, of the solutions of the one-dimensional Caldirola-Kanai equation for a large class of potentials satisfying the condition V(x)+ as |x|. We show, first of all, that if I is a closed interval containing no critical points of V, then the probability P _t (t) of finding the particle inside I tends to zero as t+. On the other hand, when I contains critical points of V in its interior, we prove that P _t (t) does not oscillate indefinitely, but tends to a limit as t+. In particular, when the potential has only isolated critical points x ₁, ..., x _N our results imply that the probability density of the particle tends to in the sense of distributions.Supported by Fulbright-MEC grant 85-07391.     

First-order phase transitions in finite-size strips with interfaces

Finite-size rounding of the magnetization discontinuity at the magnetic phase transition atH=0 (T<T _c ) in 2d Ising-type strips of sizeL ×L , with ± boundary conditions alongL inducing an interface of lengthL , is studied by phenomenological considerations and transfer matrix techniques. Scaling expressions are derived forL =O(L ) and also in the infinite strip limitL . Most of the results can be extended to the 3d case.     

Absence of asymptotic-time symmetry breaking in zero-bias spin-boson model with partial relaxation

The symmetric spin-boson model without external field is treated for any type of coupling to the boson bath and any initial bath density matrix. With initially fully aligned spin (_z (0)= =1), the proof is given that a partial relaxation (_z (+) t1<) implies that there is no asymptotic-time (up-and-down) symmetry breaking (i.e. that _z (+)=0). For the problem of a particle (interacting with free bosons) in a symmetric double well without spatial symmetry breaking before the infinite time limit, this means that att + the particle distribution becomes symmetric (irrespective of the full initial asymmetry) unless the particle fully remains (att + ) in Ihe starting well.     

Remarks on spectra of modular operators of von Neumann algebras

It is shown that if is an invariant state of an asymptotically abelianC* algebra , then the spectrum of modular operator for is contained in the spectrum of any other modular operator for the von Neumann algebra .It is also shown that a modular operator can not have an isolated spectrum with a finite multiplicity at 1 unless the associated Hilbert space is of finite dimension. It is further shown that if a modular operator has an isolated spectrum with a finite multiplicity atx 1, then the von Neumann algebra is a direct sum of ₁ and ₂ where ₁ is represented on a finite dimensional Hilbert space and the modular operator for ₂ does not have its spectrum atx.Applications to Connes invariant are given.On leave from Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan.     

Phase transition in an Ising-spinglass-model including next-nearest-neighbor-interaction?

The Transfermatrix-Method [1, 2] is applied to an Ising-spinglass-model on the square lattice with nearest neighbor interactionsJ ₁ positive and next nearest neighbor interactionsJ ₂ negative, when a fraction 1-x of the spins is removed at random. An example for this model is Eu _x Sr_1-x S [3, 4]. The question concerning the existence of a springlass-phase is not yet clarified. Investigation of the correlation function s ₀s_R² (correlation between two spins with distanceR), which is related to the Edwards-Anderson-orderparameter shows in the present work evidence against a spinglass-phase at finiteT. Analogous to the more common ±J-model (competing nearest neighbor interactions at random) a phase transition should only occur atT=0.