Ensemble and trajectory statistics in a nonlinear partial differential equation

We have examined the influence of parametric noise on the solution behavioru(t, x) of a nonlinear initial value() problem arising in cell kinetics. In terms of ensemble statistics, the eventual limiting solution mean and variance are well-characterized functions of the noise statistics, and and depend on . When noise is continuously present along the trajectory, and are independent of the noise statistics and . However, in their evolution toward and , both _u (t, x) and _u ² (t, x) depend on the noise and.     

Finite Range of Large Perturbations in Hamiltonian Dynamics

The dynamics defined by the Hamiltonian , where the _m are fixed random phases, is investigated for large values of A, and for . For a given P ^* and for , this Hamiltonian is transformed through a rigorous perturbative treatment into a Hamiltonian where the sum of all the nonresonant terms, having a Q dependence of the kind cos(kQ – nt + _m) with \Delta \upsilon$$ " align="middle" border="0"> , is a random variable whose r.m.s. with respect to the _m is exponentially small in the parameter . Using this result, a rationale is provided showing that the statistical properties of the dynamics defined by H, and of the reduced dynamics including at each time t only the terms in H such that , can be made arbitrarily close by increasing . For practical purposes close to 5 is enough, as confirmed numerically. The reduced dynamics being nondeterministic, it is thus analytically shown, without using the random-phase approximation, that the statistical properties of a chaotic Hamiltonian dynamics can be made arbitrarily close to that of a stochastic dynamics. An appropriate rescaling of momentum and time shows that the statistical properties of the dynamics defined by H can be considered as independent of A, on a finite time interval, for A large. The way these results could generalize to a wider class of Hamiltonians is indicated.     

Universal Amplitude Ratios in the Critical Two-Dimensional Ising Model on a Torus

Using results from conformal field theory, we compute several universal amplitude ratios for the two-dimensional Ising model at criticality on a symmetric torus. These include the correlation-length ratio x =lim _L (L)/L and the first four magnetization moment ratios V _2n = ²ⁿ / ^{2 n} . As a corollary we get the first four renormalized 2n-point coupling constants for the massless theory on a symmetric torus, G*_2n . We confirm these predictions by a high-precision Monte Carlo simulation.     

Transport Properties of Kicked and Quasiperiodic Hamiltonians

We study transport properties of Schrödinger operators depending on one or more parameters. Examples include the kicked rotor and operators with quasi-periodic potentials. We show that the mean growth exponent of the kinetic energy in the kicked rotor and of the mean square displacement in quasiperiodic potentials is generically equal to 2: this means that the motion remains ballistic, at least in a weak sense, even away from the resonances of the models. Stronger results are obtained for a class of tight-binding Hamiltonians with an electric field E(t) = E ₀+ E ₁cos t. For with 3/2)$$ " align="middle" border="0"> we show that the mean square displacement satisfies for suitable choices of , E ₀, and E ₁. We relate this behavior to the spectral properties of the Floquet operator of the problem.     

Multifractal Spectra of Fragmentation Processes

Let (S(t),t0) be a homogeneous fragmentation of ]0,1[ with no loss of mass. For x]0,1[, we say that the fragmentation speed of x is v if and only if, as time passes, the size of the fragment that contains x decays exponentially with rate v. We show that there is v _typ>0 such that almost every point x]0,1[ has speed v _typ. Nonetheless, for v in a certain range, the random set _v of points of speed v, is dense in ]0,1[, and we compute explicitly the spectrum vDim( _v ) where Dim is the Hausdorff dimension.     

Preferred Invariant Symplectic Connections on Compact Coadjoint Orbits

Let be the Haag--Kastler net generated by the (2) chiral current algebra at level 1. We classify the SL(2, )-covariant subsystems by showing that they are all fixed points nets ^H for some subgroup H of the gauge automorphisms group SO(3) of . Then, using the fact that the net ₁ generated by the (1) chiral current can be regarded as a subsystem of , we classify the subsystems of ₁. In this case, there are two distinct proper subsystems: the one generated by the energy-momentum tensor and the gauge invariant subsystem .     

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.     

Bound States in a Locally Deformed Waveguide: The Critical Case

We consider the Dirichlet Laplacian for astrip in with one straight boundary and a width , where $f$ is a smooth function of acompact support with a length 2b. We show that in the criticalcase, , the operator has nobound statesfor small .On the otherhand, a weakly bound state existsprovided . In thatcase, there are positive c ₁,c ₂ suchthat the corresponding eigenvalue satisfies for all sufficiently small.     

Multiplicative Cellular Automata on Nilpotent Groups: Structure, Entropy, and Asymptotics

If , and is a finite (nonabelian) group, then is a compact group; a multiplicative cellular automaton (MCA) is a continuous transformation which commutes with all shift maps, and where nearby coordinates are combined using the multiplication operation of . We characterize when MCA are group endomorphisms of , and show that MCA on inherit a natural structure theory from the structure of . We apply this structure theory to compute the measurable entropy of MCA, and to study convergence of initial measures to Haar measure.     

Exactly Solvable su(N) Mixed Spin Ladders

It is shown that solvable mixed spin ladder models can be constructed from su(N) permutators. Heisenberg rung interactions appear as chemical potential terms in the Bethe Ansatz solution. Explicit examples given are a mixed spin- spin-1 ladder, a mixed spin- spin- ladder and a spin-1 ladder with biquadratic interactions.     

Functional Relations in Stokes Multipliers—Fun with x6+αx2 Potential

We consider eigenvalue problems in quantum mechanics in one dimension. Hamiltonians contain a class of double well potential terms, x +x , for example. The space coordinate is continued to a complex plane and the connection problem of fundamental system of solutions is considered. A hidden U ( (2 1)) structure arises in fusion relations of Stokes multipliers. With this observation, we derive coupled nonlinear integral equations which characterize the spectral properties of both ± potentials simultaneously.     

Uniqueness of Gibbs State for Non-Ideal Gas in ℝd: The Case of Multibody Interaction

We study the question of existence and uniqueness of non-ideal gas in ^d with multi-body interactions among its particles. For each k-tuple of the gas particles, 2km ₀<, their interaction is represented by a potential function _k of a finite range. We introduce a stabilizing potential function , such that (x ₁,..., ) grows sufficiently fast, when diam{x ₁,..., } shrinks to 0. Our results hold under the assumption that at least one of the potential functions is stabilizing, which causes a sufficiently strong repulsive force. We prove that (i) for any temperature there exists at least one Gibbs field, and (ii) there exists exactly one Gibbs field at sufficiently high temperature, such that for any >0, C(V ₀)< for all volumes V smaller than a certain fixed finite volume V ₀. The proofs use the criterion of the uniqueness of Gibbs field in non-compact case developed in ref. 4, and the technique employed in ref. 1 for studying a gas with pair interaction.     

Precise calculation of the dynamical exponent of two-dimensional percolation

We report numerical data obtained on the special-purpose computer PERCOLA for the exponent of the electrical conductivity of 2D percolation. The extrapolation yields and a correction to the scaling exponent=1.2±0.2.     

An analyticity bound for two-dimensional Ising model at low temperature

We study the coexistence phase in the two-dimensional Ising model. Optimizing the cluster expansion technique, we are able to prove the phase separation phenomenon, with the Onsager value for the surface tension, in a range , where estimates from above the critical within 19% and essentially coincides with the entropic bound.     

Landau Hamiltonians with Unbounded Random Potentials

We prove the almost sure existence of a pure point spectrum for the two-dimensional Landau Hamiltonian with an unbounded Anderson-like random potential, provided that the magnetic field is sufficiently large. For these models, the probability distribution of the coupling constant is assumed to be absolutely continuous. The corresponding densityg has support equal to , and satisfies , for some > 0. This includes the case of Gaussian distributions. We show that the almost sure spectrum is , provided the magnetic field B0. We prove that for each positive integer n, there exists a field strength B _n , such that for all B>B _n , the almost sure spectrum is pure point at all energies except in intervals of width about each lower Landau level , for m < n. We also prove that for any B0, the integrated density of states is Lipschitz continuous away from the Landau energiesE _n (B). This follows from a new Wegner estimate for the finite-area magnetic Hamiltonians with random potentials.     

On a Shock Front in Burgers Turbulence

We investigate how chaos propagates in the solution of Burgers equation _t u+u _x u=0 with initial condition u(,0) distributed as a white noise on and 0 on . We describe the evolution of the shock front that travels to the left. Asymptotics are given for both large and small time t.     

Integral Presentations for the Universal R-Matrix

We present an integral formula for the universal R-matrix of quantum affine algebra U _q with Drinfeld comultiplication. We show that the properties of the universal R-matrix follow from the factorization properties of the cycles in proper configuration spaces. For general g we conjecture that such cycles exist and unique. For U _q we describe precisely the cycles and present a new simple expression for the universal R-matrix as a result of calculation of corresponding integrals.     

Local Primitive Causality and the Common Cause Principle in Quantum Field Theory

If (V) is a net of local von Neumann algebras satisfying standard axioms of algebraic relativistic quantum field theory and V ₁ and V ₂ are spacelike separated spacetime regions, then the system ( (V ₁ ), (V ₂ ), ) is said to satisfy the Weak Reichenbach's Common Cause Principle iff for every pair of projections A (V ₁ ), B (V ₂ ) correlated in the normal state there exists a projection C belonging to a von Neumann algebra associated with a spacetime region V contained in the union of the backward light cones of V ₁ and V ₂ and disjoint from both V ₁ and V ₂ , a projection having the properties of a Reichenbachian common cause of the correlation between A and B. It is shown that if the net has the local primitive causality property then every local system ( (V ₁ ), (V ₂ ), ) with a locally normal and locally faithful state and suitable bounded V ₁ and V ₂ satisfies the Weak Reichenbach's Common Cause Principle.     

BV Structure of the Cohomology of Nilpotent Subalgebras and the Geometry of(W) Strings

Given a simple, simply laced, complex Lie algebra corresponding to the Lie group G, let be thesubalgebra generated by the positive roots. In this Letter we construct aBV algebra whose underlying graded commutative algebra is given by the cohomology, with respect to , of the algebra of regular functions on G with values in . We conjecture that describes the algebra of allphysical (i.e., BRST invariant) operators of the noncritical string. The conjecture is verified in the two explicitly known cases, ₂ (the Virasoro string) and ₃ (the string).     

Finitary Spacetime Sheaves of Quantum Causal Sets: Curving Quantum Causality

A locally finite, causal, and quantal substitute for a locally Minkowskian principal fiber bundle of modules of Cartan differential forms over a bounded region X of a curved C -smooth spacetime manifold M with structure group G that of orthochronous Lorentz transformations L ⁺ := SO(1,3), is presented. is usually regarded as the kinematical structure of classical Lorentzian gravity when the latter is viewed as a Yang-Mills type of gauge theory of a sl(2, {})-valued connection 1-form . The mathematical structure employed to model this replacement of is a principal finitary spacetime sheaf of quantum causal sets with structure group G _n, which is a finitary version of the continuous group G of local symmetries of General Relativity, and a finitary Lie algebra g _n-valued connection 1-form on it, which is a section of its subsheaf . is physically interpreted as the dynamical field of a locally finite quantum causality, whereas its associated curvature as some sort of finitary and causal Lorentzian quantum gravity.