Exact and approximate solutions for the electron nonsinglet structure function in QED

Two exact, valid up to infinite perturbative order, numerical solutions of the Lipatov equation for the nonsinglet electron structure function in the QED are presented. One of them is of the Monte Carlo type and another is based on the numerical inversion of the Mellin transform. They agree numerically to a very high precision (better than 0.05%). Within the leading logarithmic approximation, the exact solution is compared with the perturbative second and third order exponentiated solutions. It is shown that the perturbative second order solution inspired by the Yennie-Frautschi-Suura (exclusive) exponentiation is much closer to the exact solution than the other ones. New compact analytical formula for the third order exponentiated solution is given. It is shown to be in perfect numerical agreement with the infinite order solution of the Monte Carlo and Mellin type.     

Exact Solutions for Two-Channel Cavity QED Model by Unitary Transformation Method

A fully quantum mechanical picture for an atomic system interacting with two modes in a lossless cavity is presented. It is assumed that one of the modes interacts with the atom through three-photon process and the other one through a single-photon process. The effect of intensity dependent coupling between the atom and the two field modes has also been analyzed. The Unitary Transformation Method presented here not only solves the time dependent problem but also provides the eigensolutions of the interacting Hamiltonian at the same time.     

Exact Solution of the Milburn Equation for the Coupled-Channel Cavity QED Model

We give the exact solution of Milburn equation for a coupled-channel cavity QED model which includes the Stark term and the frequency detuning, and study the influence of the intrinsic decoherence on the atomic inversion of the system.     

Exact results for two-dimensional coarsening

We consider the statistics of the areas enclosed by domain boundaries (‘hulls’) during the curvature-driven coarsening dynamics of a two-dimensional nonconserved scalar field from a disordered initial state. We show that the number of hulls per unit area, n _h (A, t)dA, with enclosed area in the range (A,A + dA), is described, for large time t, by the scaling form n _h (A, t) = 2c _h /(A + λ _h t)², demonstrating the validity of dynamical scaling in this system. Here $ c_h = {1 \mathord{\left/ {\vphantom {1 8}} \right. \kern-0em} 8}\pi \sqrt 3 $ is a universal constant associated with the enclosed area distribution of percolation hulls at the percolation threshold, and λ _h is a material parameter. The distribution of domain areas, n _d (A, t), is apparently very similar to that of hull areas up to very large values of A/λ _h t. Identical forms are obtained for coarsening from a critical initial state, but with c _h replaced by c _h /2. The similarity of the two distributions (of areas enclosed by hulls, and of domain areas) is accounted for by the smallness of c _h . By applying a ‘mean-field’ type of approximation we obtain the form n _d (A, t) ? 2c _d [λ _d (t+t ₀)] ^τ?2/[A+λ _d (t+t ₀)] ^τ , where t ₀ is a microscopic timescale and τ = 187/91 ? 2.055, for a disordered initial state, and a similar result for a critical initial state but with c _d → c _d /2 and τ → τ _c = 379/187 ? 2.027. We also find that c _d = c _h + O(c _h ² ) and λ _d = λ _h (1 + O(c _h )). These predictions are checked by extensive numerical simulations and found to be in good agreement with the data.     

Square-well diatomics Exact low density results

Exact semi-analytical expressions are obtained for the zero density site-site distribution function and the second virial coefficient for homonuclear square-well diatomics. The diatomic molecules considered here are composed of two fused square-well spheres with hard core diameter σ, well diameter λσ, and dimer bond length L, such that 0<Lˇ-σ and 1ˇ-λˇ-1+L/σ. Very accurate, although approximate, analytical expressions are also given for these functions.     

Exact results for correlations in a two-component log-gas

A study of the one-dimensional lattice gas of positive and negative charges interacting via the logarithmic potential is continued. The two-particle distribution functions are evaluated exactly at the couplings=2 and 4. It is proved that the=4 isotherm exhibits an insulator-conductor phase transition at the reduced density 1/2, and the scaling behavior of the correlations near this critical point is given. Similarities of the conjectured phase diagram with that of a one-dimensional one-component log-gas in a periodic potential are noted.     

Exact results for curvature-driven coarsening in two dimensions

We consider the statistics of the areas enclosed by domain boundaries ("hulls") during the curvature-driven coarsening dynamics of a two-dimensional nonconserved scalar field from a disordered initial state. We show that the number of hulls per unit area that enclose an area greater than A has, for large time t, the scaling form Nh(A,t)=2c/(A+lambdat), demonstrating the validity of dynamical scaling in this system, where c=1/8pisquare root 3 is a universal constant. Domain areas (regions of aligned spins) have a similar distribution up to very large values of A/lambdat. Identical forms are obtained for coarsening from a critical initial state, but with c replaced by c/2.     

Exact results for a dilute Potts model

Exact results are obtained for the annealed, dilute,q-component Potts model on the decorated square lattice. The phase diagram is found to consist of a high-temperature region, a low-temperature region, and a two-phase region in between which arises only forq>4: exact expressions for the phase boundary and the critical probability are derived. At the critical point the specific heat is generally finite and has a cusp; the slope of the cusp is finite forq=4 and infinite (vertical) forq=2 and 3.Work supported in part by NSF Grant No. DMR 78-18808.     

Exact results on the random Potts model

We present a number of exact results on the random-bond,q-state Potts model. The quenched model on any finite planar graph or lattice is shown to obey a duality relation for general type of bond-randomness. In the annealed case, the solution of the model reduces to that of the regular (nonrandom) Potts model on the corresponding lattice. Explicit knowledge of the critical parameters of theq-state Potts model in two dimensions allows us to evaluate exactly the phase diagram of the annealed model on the square, triangular and honeycomb lattices. We discuss the behavior near the (random) critical point and comment on the relationship between the quenched and annealed systems. The exact phase diagram of the annealed system is obtained for the bond-diluted model and the spin-glass model with and without dilutions.Work supported in part by NSF grant No. DMR-78-18808     

Directed compact lattice animals: Exact results

The partially directed compact lattice animal model on the square lattice is solved exactly for the cluster number and average cluster radius along the directed axis in terms of the appropriate generating functions. For the critical exponents we find=0 and =1. Caliper size distribution along the directed axis is also calculated analytically. It is used to confirm =1 and to study some finite-size scaling properties for this model. For the perpendicular cluster radius distribution, a combination of analytic arguments and computer results leads to a conjecture on the exact form of the appropriate generating function and to the result =1/2. Some calculations are reported for the triangular lattice and for hypercubic lattices ind>2.     

Exact results for Wilson loops in orbifold ABJM theory

We investigate the exact results for circular 1/4 and 1/2 BPS Wilson loops in the d = 3 N= 4 super Chern-Simons-matter theory that could be obtained by orbifolding Aharony-Bergman-Jafferis-Maldacena(ABJM)theory.The partition function of the N=4 orbifold ABJM theory has been computed previously in the literature.In this paper,we re-derive it using a slightly different method.We calculate the vacuum expectation values of the circular 1/4 BPS Wilson loops in fundamental representation and of circular 1/2 BPS Wilson loops in arbitrary representations.We use both the saddle point approach and Fermi gas approach.The results for Wilson loops are in accord with the available gravity results.     

Exact results for anomalous transport in one-dimensional hamiltonian systems

Anomalous transport in one-dimensional translation invariant hamiltonian systems with short range interactions is shown to belong in general to the Kardar-Parisi-Zhang universality class. Exact asymptotic forms for density-density and current-current time correlation functions and their Fourier transforms are given in terms of the Pr?hofer-Spohn scaling functions, obtained from their exact solution for the polynuclear growth model. The exponents of corrections to scaling are found as well, but not so the coefficients. Mode coupling theories developed previously are found to be adequate for weakly nonlinear chains but in need of corrections for strongly anharmonic interparticle potentials. A simple condition is given under which Kardar-Parisi-Zhang behavior does not apply, sound attenuation is only logarithmically superdiffusive, and heat conduction is more strongly superdiffusive than under Kardar-Parisi-Zhang behavior.