Hawking radiation of <Emphasis Type="Italic">E</Emphasis> < <Emphasis Type="Italic">m</Emphasis> massive particles in the tunneling formalism

We use the tunneling formalism to calculate the Hawking radiation of massive particles. For E ≥ m, we recover the traditional result, identical to the massless case. But E < m particles can also tunnel across the horizon in a Hawking process. We study the probability for detecting such E < m particles as a function of the distance from the horizon and the energy of the particle in the tunneling formalism. We derive a general formula and obtain simple approximations in the near-horizon limit and in the limit of large radii.     

Finite-Size Left-Passage Probability in Percolation

We obtain an exact finite-size expression for the probability that a percolation hull will touch the boundary, on a strip of finite width. In terms of clusters, this corresponds to the one-arm probability. Our calculation is based on the q-deformed Knizhnik–Zamolodchikov approach, and the results are expressed in terms of symplectic characters. In the large size limit, we recover the scaling behaviour predicted by Schramm’s left-passage formula. We also derive a general relation between the left-passage probability in the Fortuin–Kasteleyn cluster model and the magnetisation profile in the open XXZ chain with diagonal, complex boundary terms.     

Asymptotics in ASEP with Step Initial Condition

In previous work the authors considered the asymmetric simple exclusion process on the integer lattice in the case of step initial condition, particles beginning at the positive integers. There it was shown that the probability distribution for the position of an individual particle is given by an integral whose integrand involves a Fredholm determinant. Here we use this formula to obtain three asymptotic results for the positions of these particles. In one an apparently new distribution function arises and in another the distribution function F ₂ arises. The latter extends a result of Johansson on TASEP to ASEP, and hence proves KPZ universality for ASEP with step initial condition.     

On the theory of scattering by complex potentials

The scattering problem for a non-relativistic spinless particle under the influence of a complex effective potential, which is spherically symmetric and tends to zero faster than 1/r at infinity, is considered. Certain general relations, which illuminate the influence of the imaginary part of the potential on the scattering process, are derived with the use of the expression for the probability current density. The rigorous phase-integral method developed by N. Fröman and P. O. Fröman is used for obtaining an exact, general formula for the scattering matrix, or, equivalently, for the phase shift. The formula is expressed in terms of phase-integral approximations of an arbitrary order and certain quantities defined by convergent series. Estimating the latter quantities and omitting small corrections, an approximate formula is derived for the phase shift, valid for the case that only one complex turning point contributes essentially to the phase shift. Criteria for classifying a scattering problem as such a one-turning-point problem are given. The treatment is made general enough to also cover situations of interest in Regge-pole or complex angular momentum theory.     

Surface tension of a molecular fluid

We derive a general expression for surface tension in a dense molecular fluid with an arbitrary many body interaction. This formula reduces to that of Gray and Gubbins under the assumption of pairwise additivity. We then show how this general expression is transformed into a sum rule expression and also into an expression involving the direct correlation function. These last two expressions are analogous to the simple fluid surface tension formulas of Jhon, Desai and Dahler and Yvon-Triezenberg-Zwanzig respectively. We also discuss the case of a fluid in d dimensions.     

Noncolliding Brownian Motion and Determinantal Processes

A system of one-dimensional Brownian motions (BMs) conditioned never to collide with each other is realized as (i) Dyson’s BM model, which is a process of eigenvalues of hermitian matrix-valued diffusion process in the Gaussian unitary ensemble (GUE), and as (ii) the h-transform of absorbing BM in a Weyl chamber, where the harmonic function h is the product of differences of variables (the Vandermonde determinant). The Karlin–McGregor formula gives determinantal expression to the transition probability density of absorbing BM. We show from the Karlin–McGregor formula, if the initial state is in the eigenvalue distribution of GUE, the noncolliding BM is a determinantal process, in the sense that any multitime correlation function is given by a determinant specified by a matrix-kernel. By taking appropriate scaling limits, spatially homogeneous and inhomogeneous infinite determinantal processes are derived. We note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix-kernels are expressed using spectral projections of appropriate effective Hamiltonians. On the common structure of matrix-kernels, continuity of processes in time is proved and general property of the determinantal processes is discussed.     

Schwinger Formula Revisited

We apply the formal W.K.B. method in the complex plane to the quantum field theory to obtain the Schwinger formula for spin and spinless particles; i.e., we obtain the probability that the vacuum state remains unchanged in presence of a constant electric field. Finally, from Schwinger formula we calculate the probability that n pairs are produced.     

Thermal Conductivity for a Momentum Conservative Model

We introduce a model whose thermal conductivity diverges in dimension 1 and 2, while it remains finite in dimension 3. We consider a system of oscillators perturbed by a stochastic dynamics conserving momentum and energy. We compute thermal conductivity via Green-Kubo formula. In the harmonic case we compute the current-current time correlation function, that decay like t ^−d/2 in the unpinned case and like t ^−d/2–1 if an on-site harmonic potential is present. This implies a finite conductivity in d ≥ 3 or in pinned cases, and we compute it explicitly. For general anharmonic strictly convex interactions we prove some upper bounds for the conductivity that behave qualitatively as in the harmonic cases.     

Effect of the Noise Correlation Time on the Stationary Current of Brownian Particle in a Spatially Symmetric Periodic Potential

We investigate the noise-induced transport of Brownian particle in a deterministic spatial symmetrical periodic potential driven by colored cross correlation between a multiplicative white noise and an additive white noise. We derive the general formula of the stationary current. Based on numerical computation, we found that directed motion of the Brownian particles can be induced by the correlation time τ of cross correlation between the multiplicative noise and the additive noise and the current reversal and the direction of the current is controlled by the τ.     

Bethe Ansatz Solution of Discrete Time Stochastic Processes with Fully Parallel Update

We present the Bethe ansatz solution for the discrete time zero range and asymmetric exclusion processes with fully parallel dynamics. The model depends on two parameters: p, the probability of single particle hopping, and q, the deformation parameter, which in the general case, |q| < 1, is responsible for long range interaction between particles. The particular case q = 0 corresponds to the Nagel-Schreckenberg traffic model with v _max = 1. As a result, we obtain the largest eigenvalue of the equation for the generating function of the distance travelled by particles. For the case q = 0 the result is obtained for arbitrary size of the lattice and number of particles. In the general case we study the model in the scaling limit and obtain the universal form specific for the Kardar-Parisi-Zhang universality class. We describe the phase transition occurring in the limit p→ 1 when q < 0.     

Riemann-Hilbert approach and N-soliton formula for a higher-order Chen-Lee-Liu equation

We consider a higher-order Chen-Lee-Liu (CLL) equation with third order dispersion and quintic nonlinearity terms. In the framework of the Riemann-Hilbert method, we obtain the compact N-soliton formula expressed by determinants. Based on the determinant solution, some properties for single soliton and asymptotic analysis of N-soliton solution are explored. The simple elastic interaction of N solitons is confirmed.     

Effects on generalized growth models driven by a non-Poissonian dichotomic noise

In this paper we consider a general growth model with stochastic growth rate modelled via a symmetric non-poissonian dichotomic noise. We find an exact analytical solution for its probability distribution. We consider the, as yet, unexplored case where the deterministic growth rate is perturbed by a dichotomic noise characterized by a waiting time distribution in the two state that is a power law with power 1 < μ < 2. We apply the results to two well-known growth models; Malthus-Verhulst and Gompertz.     

Die Sprungtemperatur stark koppelnder Supraleiter

We investigate the general structure of the equation for the critical temperature of superconductors, introduced in an earlier work. An expression for the Coulomb pseudopotential allowing a better calculation is found without any cut-off procedure. Using an Einstein type phonon spectrum we give an explicit formula forT _c , which in case of strong coupling superconductors differs from McMillan's expression. The agreement with experimental values is good.     

Integral Formulas for the Asymmetric Simple Exclusion Process

In this paper we obtain general integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice with nearest neighbor hopping rates p to the right and q = 1−p to the left. For the most part we consider an N-particle system but for certain of these formulas we can take the limit. First we obtain, for the N-particle system, a formula for the probability of a configuration at time t, given the initial configuration. For this we use Bethe Ansatz ideas to solve the master equation, extending a result of Schütz for the case N = 2. The main results of the paper, derived from this, are integral formulas for the probability, for given initial configuration, that the m ^th left-most particle is at x at time t. In one of these formulas we can take the limit, and it gives the probability for an infinite system where the initial configuration is bounded on one side. For the special case of the totally asymmetric simple exclusion process (TASEP) our formulas reduce to the known ones.     

Non Equilibrium Current Fluctuations in Stochastic Lattice Gases

We study current fluctuations in lattice gases in the macroscopic limit extending the dynamic approach for density fluctuations developed in previous articles. More precisely, we establish a large deviation principle for a space-time fluctuation j of the empirical current with a rate functional I(j). We then estimate the probability of a fluctuation of the average current over a large time interval; this probability can be obtained by solving a variational problem for the functional I. We discuss several possible scenarios, interpreted as dynamical phase transitions, for this variational problem. They actually occur in specific models. We finally discuss the time reversal properties of I and derive a fluctuation relationship akin to the Gallavotti-Cohen theorem for the entropy production.     

Asymptotics for the Fredholm determinant of the sine kernel on a union of intervals

In the bulk scaling limit of the Gaussian Unitary Ensemble of hermitian matrices the probability that an interval of lengths contains no eigenvalues is the Fredholm determinant of the sine kernel over this interval. A formal asymptotic expansion for the determinant ass tends to infinity was obtained by Dyson. In this paper we replace a single interval of lengths bysJ, whereJ is a union ofm intervals and present a proof of the asymptotics up to second order. The logarithmic derivative with respect tos of the determinant equals a constant (expressible in terms of hyperelliptic integrals) timess, plus a bounded oscillatory function ofs (zero ifm=1, periodic ifm=2, and in general expressible in terms of the solution of a Jacobi inversion problem), pluso(1). Also determined are the asymptotics of the trace of the resolvent operator, which is the ratio in the same model of the probability that the set contains exactly one eigenvalue to the probability that it contains none. The proofs use ideas from orthogonal polynomial theory.Research supported by National Science Foundation grant DMS-9216203.     

Spatial structure and stability based on random walks

A general expression for a recursion formula which describes a random walk with coupled modes is given. In this system, the random walker is specified by the jumping probabilities P⁺ and P^– which depend on the modes. The transition probability between the modes is expressed by a jumping probabilityR ^(ij) (orr _ij). With the aid of this recursion formula, spatial structures of the steady state of a coupled random walk are studied. By introducing a Liapunov function and entropy, it is shown that the stability condition for the present system can be expressed as the principle of the extremum entropy production.On leave of absence from Tohoku University, Department of Applied Science, Faculty of Engineering, Sendai, 980 Japan.     

Acoustic breathing mode frequencies in cylinders,cylindrical shells and composite cylinders of general anisotropic crystals: Application to nanowires

We present here a study of the acoustic breathing modes for infinitely long cylinders, cylindrical shells and composite cylinders of general anisotropic crystals. We assume cylindrical anisotropy for the systems studied. We obtain expressions in closed form for their frequencies in the case of cylinders and cylindrical shells, valid for any anisotropic material, thus including up to 21 independent elastic constants. In the case of the lowest breathing mode of a thin cylindrical shell we obtain a simple analytical formula. This can be used to obtain a first estimate of the breathing mode frequency in nanotubes for any material. In the case of core–shell and composite cylinders we obtain the expressions for the secular determinant. We calculate the frequencies of the lowest acoustic breathing mode of Au, CdSe, InAs, GaAs, Ag and Bi nanowires obtained recently by different experimental groups. We also present results for the acoustic breathing modes of Au/Ag and ZnS/SiO₂ core–shell nanowires produced recently.     

Conformal field theories via Hamiltonian reduction

Constraining theSL(3) WZW-model we construct a reduced theory which is invariant with respect to the new chiral algebraW ₃ ² . This symmetry is generated by the stress-energy tensor, two bosonic currents with spins 3/2 and theU(1) current. We conjecture a Kac formula that describes the highly reducible representation for this algebra. We also discuss the quantum Hamiltonian reduction for the general type of constraints that leads to the new extended conformal algebras.Address after September 1990: Lyman Laboratory, Harvard University, Cambridge, MA 02138, USA     

Coherent transport in disordered metals out of equilibrium

We derive a formula for the quantum corrections to the electrical current for a metal out of equilibrium. In the limit of linear current-voltage characteristics our formula reproduces the well known Altshuler-Aronov correction to the conductivity of a disordered metal. The current formula is obtained by a direct diagrammatic approach, and is shown to agree with what is obtained within the Keldysh formulation of the non-linear sigma model. As an application we calculate the current of a mesoscopic wire. We find a current-voltage characteristics that scales with eV/kT, and calculate the different scaling curves for a wire in the hot-electron regime and in the regime of full non-equilibrium. Received 13 June 2001