Information Loss in Coarse-Graining of Stochastic Particle Dynamics

Recently a new class of approximating coarse-grained stochastic processes and associated Monte Carlo algorithms were derived directly from microscopic stochastic lattice models for the adsorption/desorption and diffusion of interacting particles^(12,13,15). The resulting hierarchy of stochastic processes is ordered by the level of coarsening in the space/time dimensions and describes mesoscopic scales while retaining a significant amount of microscopic detail on intermolecular forces and particle fluctuations. Here we rigorously compute in terms of specific relative entropy the information loss between non-equilibrium exact and approximating coarse-grained adsorption/desorption lattice dynamics. Our result is an error estimate analogous to rigorous error estimates for finite element/finite difference approximations of Partial Differential Equations. We prove this error to be small as long as the level of coarsening is small compared to the range of interaction of the microscopic model. This result gives a first mathematical reasoning for the parameter regimes for which approximating coarse-grained Monte Carlo algorithms are expected to give errors within a given tolerance. MSC (2000) subject classifications: 82C80; 60J22; 94A17     

Translationally invariant kink solutions of discrete ϕ<Superscript>4</Superscript> models

The properties of translationally invariant kinks in two discrete ϕ⁴ models are compared with those of the kinks in a classical discrete model. The translationally invariant kink solutions can be found randomly with respect to the lattice sites, i.e., their Peierls–Nabarro potential is exactly equal to zero. It is shown that these solutions have a Goldstone mode, that is, they can move along the lattice at vanishingly small velocities. Thus, the translationally invariant kink is not trapped by the lattice and can be accelerated by an arbitrary small external field and, having an increased mobility, can transfer a range of physical quantities: matter, energy, momentum, etc.     

完整约束力学系统保Lie对称性差分格式

提出一种保完整约束力学系统Lie点对称性的差分格式.其具体做法是:首先求出完整约束力学系统的Lie点对称群,并将其延拓到离散的差分格点上;其次利用其特征方程找出独立的离散不变量;再应用独立的离散不变量来构造不变量的差分格式,且此差分格式在连续极限下应给出原系统的微分方程;最后给出一个例子,作为本文结果的说明.     

Two new discrete integrable systems

In this paper,we focus on the construction of new(1+1)-dimensional discrete integrable systems according to a subalgebra of loop algebra A 1.By designing two new(1+1)-dimensional discrete spectral problems,two new discrete integrable systems are obtained,namely,a 2-field lattice hierarchy and a 3-field lattice hierarchy.When deriving the two new discrete integrable systems,we find the generalized relativistic Toda lattice hierarchy and the generalized modified Toda lattice hierarchy.Moreover,we also obtain the Hamiltonian structures of the two lattice hierarchies by means of the discrete trace identity.     

Approximate sine-Gordon solitons

We look at the recently proposed scheme of approximating a sine-Gordon soliton by an expression derived from two dimensional instantons. We point out that the scheme of Sutcliffe in which he uses two dimensional instantons can be generalised to higher dimensions and that these generalisations produce even better approximations than the original approximation. We also comment on generalisations to other models.     

A Hopping Mechanism for Cargo Transport by Molecular Motors on Crowded Microtubules

We present an algorithm, based on the iteration of conformal maps, that produces independent samples of self-avoiding paths in the plane. It is a discrete process approximating radial Schramm-Loewner evolution growing to infinity. We focus on the problem of reproducing the parametrization corresponding to that of lattice models, namely self-avoiding walks on the lattice, and we propose a strategy that gives rise to discrete paths where consecutive points lie an approximately constant distance apart from each other. This new method allows us to tackle two non-trivial features of self-avoiding walks that critically depend on the parametrization: the asphericity of a portion of chain and the correction-to-scaling exponent.     

Exact Sampling of Self-avoiding Paths via Discrete Schramm-Loewner Evolution

We present an algorithm, based on the iteration of conformal maps, that produces independent samples of self-avoiding paths in the plane. It is a discrete process approximating radial Schramm-Loewner evolution growing to infinity. We focus on the problem of reproducing the parametrization corresponding to that of lattice models, namely self-avoiding walks on the lattice, and we propose a strategy that gives rise to discrete paths where consecutive points lie an approximately constant distance apart from each other. This new method allows us to tackle two non-trivial features of self-avoiding walks that critically depend on the parametrization: the asphericity of a portion of chain and the correction-to-scaling exponent.     

Progress in lattice gauge theory

Two topics of lattice gauge theory are reviewed. They include string tension and β-function calculations by strong coupling Hamiltonian methods for SU(3) gauge fields in 3 + 1 dimensions, and a 1/N-expansion for discrete gauge and spin systems in all dimensions. The SU(3) calculations give solid evidence for the coexistence of quark confinement and asymptotic freedom in the renormalized continuum limit of the lattice theory. The crossover between weak and strong coupling behavior in the theory is seen to be a weak coupling but non-perturbative effect. Quantitative relationships between perturbative and non-perturbative renormalization schemes are obtained for the O(N) nonlinear sigma models in 1 + 1 dimensions as well as the range theory in 3 + 1 dimensions. Analysis of the strong coupling expansion of the β-function for gauge fields suggests that it has cuts in the complex 1/g²-plane. A toy model of such a cut structure which naturally explains the abruptness of the theory's crossover from weak to strong coupling is presented. The relation of these cuts to other approaches to gauge field dynamics is discussed briefly.The dynamics underlying first order phase transitions in a wide class of lattice gauge theories is exposed by considering a class of models-P(N) gauge theories - which are soluble in the N → ∞ limit and have non-trivial phase diagrams. The first order character of the phase transitions in Potts spin systems for N #62; 4 in 1 + 1 dimensions is explained in simple terms which generalizes to P(N) gauge systems in higher dimensions. The phase diagram of Ising lattice gauge theory coupled to matter fields is obtained in a $\text{1}\text{N}$ expansion. A one-plaquette model (1 time-0 space dimensions) with a first-order phase transitions in the N → ∞ limit is discussed.     

Percolation of the Loss of Tension in an Infinite Triangular Lattice

We introduce a new class of bootstrap percolation models where the local rules are of a geometric nature as opposed to simple counts of standard bootstrap percolation. Our geometric bootstrap percolation comes from rigidity theory and convex geometry. We outline two percolation models: a Poisson model and a lattice model. Our Poisson model describes how defects--holes is one of the possible interpretations of these defects--imposed on a tensed membrane result in a redistribution or loss of tension in this membrane; the lattice model is motivated by applications of Hooke spring networks to problems in material sciences. An analysis of the Poisson model is given by Menshikov et al. ⁽⁴⁾ In the discrete set-up we consider regular and generic triangular lattices on the plane where each bond is removed with probability 1–p. The problem of the existence of tension on such lattice is solved by reducing it to a bootstrap percolation model where the set of local rules follows from the geometry of stresses. We show that both regular and perturbed lattices cannot support tension for any p<1. Moreover, the complete relaxation of tension--as defined in Section 4--occurs in a finite time almost surely. Furthermore, we underline striking similarities in the properties of the Poisson and lattice models.     

A numerical method to compute exactly the partition function with application toZ(n) theories in two dimensions

I present a new method to exactly compute the partition function of a class of discrete models in arbitrary dimensions. The time for the computation for ann-state model on anL ^d lattice scales like . I show examples of the use of this method by computing the partition function of the 2D Ising and 3-state Potts models for maximum lattice sizes 10×10 and 8×8, respectively. The critical exponentsv and and the critical temperature one obtains from these are very near the exactly known values. The distribution of zeros of the partition function of the Potts model leads to the conjecture that the ratio of the amplitudes of the specific heat below and above the critical temperature is unity.     

Heavy ion radiation damage annealing models-a new interpretation

The annealing behaviour of heavy ion radiation damage trails in Lexan Polycarbonate is studied on the basis of two different models proposed by Modgil and Virk¹¹ and Price et al. ¹⁵ The shortcomings of these two approaches are brought into focus and a new formulation proposed to remove some inherent drawbacks.     

Monte Carlo studies of two measures of polymer chain size as a function of temperature

Monte Carlo simulations of single polymer chains with both excluded volume and nearest-neighbor interaction energies are discussed. Two measures of chain size are obtained in the simulation, the radius of gyration of the polymer chain and the inverse radius of the polymer chain. Both of these are reported as a function of temperature, or interaction energy, and chain length,N. The possibility of estimating the fractal dimensions of these measures from the Monte Carlo data is discussed in the context of two different interpolation functions for the temperature dependence of the fractal dimensions. The approach to the fractal dimension as a function of chain length,N, is studied. It is suggested that the approach to fractal dimension of the measures of chain size of polymers is slow, perhaps a fractional power itself.     

The weak-coupling phase of lattice spin and gauge models

We analyze the lattice weak-coupling (w.c.) expansion of O(N), CP^N?1 and chiral spin models, and of large-N reduced chiral and gauge models.We find that the w.c. expansion always agrees with mean field results, whenever comparable, for arbitrary space-time dimensions, and that the expansion of the reduced models agrees with that of the original ones. However, w.c. results disagree with one-dimensional large-N and (old and new) exact results. We explain this phenomenon as a failure of the analytic continuation from higher dimensions that defines lattice w.c. perturbation theory for massless models (even if infrared singularities always cancel).We use an improved version of the mean field (m.f.) technique suitable for reduced models. We compute the m.f. approximation of chiral models and use this result to determine the large-d (m.f.) behaviour of reduced gauge models, finding agreement with standard Wilson theory results.We give a new characterization of large-N chiral models in terms of the single-link integral for the adjoint representation of SU(N).     

Local theory of solutions for the complex grassmannian andCP N−1 sigma models

We study the classical solutions of the complex Grassmannian nonlinear sigma models and of theCP ^N?1 model in two euclidean dimensions. Exact solutions of various types, which seem to be complete, are constructed explicitly in an elementary way, namely in terms of holomorphic functions and the Gramm-Schmidt orthonormalization procedure. A new type of discrete symmetry transformations which map one solution into another is presented.     

A new simulation method for the determination of forces in polymer/colloid systems

We introduce a new simulation method, which we call the contact-distribution method, for the determination of the Helmholtz potential for polymer/colloid systems from lattice Monte-Carlo simulations. This method allows one to obtain forces between finite or semi-infinite objects of any arbitrary shape and dimensions in the presence of polymer chains in solution or physisorbed or chemisorbed at interfaces. We illustrate the application of the method using two examples: (i) the interaction between the tip of an atomic force microscope (AFM) and a single, end-grafted polymer chain and (ii) the interaction between an AFM tip and a polymer brush. Numerical results for the first two cases illustrate how the method can be used to confirm and extend scaling laws for forces and Helmholtz potentials, to examine the effects of the shapes and sizes of the objects and to examine conformational transitions in the polymer chains. Received: 15 May 1998 / Revised: 11 June 1998 / Accepted: 12 June 1998     

Nonstandard parafermions and string compactification

Nonstandard parafermions are built and their central charges and dimensions are calculated. We then construct new N=2 superconformal field theories by tensoring the parafermions with a free boson. We study the spectrum and modular transformations of these theories. Superstring and heterotic strings in four dimensions are then obtained by tensoring the new superconformal field theories along with some minimal models. The generations and antigenerations are studied. We give an example of the ¹²(5,7) theory which is shown to have two net generations.     

Two new integrable lattice hierarchies associated with a discrete Schrödinger nonisospectral problem and their infinitely many conservation laws

In this Letter, by means of using discrete zero curvature representation and constructing opportune time evolution problems, two new discrete integrable lattice hierarchies with n-dependent coefficients are proposed, which relate to a new discrete Schrödinger nonisospectral operator equation. The relation of the two new lattice hierarchies with the Volterra hierarchy is discussed. It has been shown that one lattice hierarchy is equivalent to the positive Volterra hierarchy with n-dependent coefficients and another lattice hierarchy with isospectral problem is equivalent to the negative Volterra hierarchy. We demonstrate the existence of infinitely many conservation laws for the two lattice hierarchies and give the corresponding conserved densities and the associated fluxes formulaically. Thus their integrability is confirmed.