Equilibrium Energy and Entropy of Vortex Filaments in the Context of Tornadogenesis and Tornadic Flows

In this work, we study approximations of supercritical or suction vortices in tornadic flows and their contribution to tornadogenesis and tornado maintenance using self-avoiding walks on a cubic lattice. We extend the previous work on turbulence by A. Chorin and collaborators to approximate the statistical equilibrium quantities of vortex filaments on a cubic lattice when both an energy and a statistical temperature are involved. Our results confirm that supercritical (smooth, “straight”) vortices have the highest average energy and correspond to negative temperatures in this model. The lowest-energy configurations are folded up and “balled up” to a great extent. The results support A. Chorin’s findings that, in the context of supercritical vortices in a tornadic flow, when such high-energy vortices stretch, they need to fold and transfer energy to the surrounding flow, contributing to tornado maintenance or leading to its genesis. The computations are performed using a Markov Chain Monte Carlo approach with a simple sampling algorithm using local transformations that allow the results to be reliable over a wide range of statistical temperatures, unlike the originally used pivot algorithm that only performs well near infinite temperatures. Efficient ways to compute entropy are discussed and show that a system with supercritical vortices will increase entropy by having these vortices fold and transfer their energy to the surrounding flow.

COMPUTER SIMULATION OF A SINGLE POLYMER CHAIN IN DIFFERENT SOLVENTS

In the present paper, the behavior of a single polymer chain under various solvent conditions was modeled by self-avoiding walks (SAW) with nearest neighbors attraction Ae on a simple cubic lattice. Determination of the 0-condition was based on the numerical results of the mean square radius of gyration and end-to-end distance. It was found that at the 0 temperature Delta epsilon /kT equals -0.27. The exponents a in the Mark-Houwink equation with different interaction parameters are consistent with the results of experiments: under 0-condition, alpha= 0.5, and for a good solvent alpha =0.74-0.84, respectively.

CONFORMAL RESTRICTION MEASURES ON LOOPS SURROUNDING AN INTERIOR POINT

A conformal restriction measure is a probability measure which is used to describe the law of a random connected subset in a simply connected domain that satisfies a certain conformal restriction property. Usually there are three kinds of conformal restriction measures: one (called the chordal restriction measure) has two given boundary points of the random set, the second (called the radial restriction measure) has one boundary point and one interior point in the random set, and the third (called the tri-chordal restriction measure) has three boundary points in the random set. In this article, we will define a new probability measure such that the random set associated to it contains one given interior point and does not intersect with the boundary. Furthermore, we will show that this measure can be characterized by one parameter;we will also construct this one-parameter family of measures in two ways and obtain several properties.

Irreversible Markov chain Monte Carlo algorithm for self-avoiding walk

We formulate an irreversible Markov chain Monte Carlo algorithm for the self-avoiding walk （SAW）, which violates the detailed balance condition and satisfies tile balance condition. Its performance improves significantly compared to that of the Berretti-Sokal algorithm, which is a variant of the Metropolis Hastings method. The gained efficiency increases with spatial dimension （D）, from approximately 10 times in 2D to approximately 40 times in 5D. We simulate the SAW on a 5D hypercubic lattice with periodic boundary conditions, for a linear system with a size up to L = 128, and confirm that as for the 5D Ising model, the finite-size scaling of the SAW is governed by renormalized exponents, υ^＊ = 2/d and γ/υ^＊ = d/2. The critical point is determined, which is approximately 8 times more precise than the best available estimate.

Computer simulation of self-avoiding walk terminally attached to wall

The self-avoiding walk (SAW) is an important model greatly different from the normalrandom walk in mathematics. Since a site that has been occupied once cannot be visitedagain by the walker, the SAW is not a Markov chain. Generally speaking, it is difficult togive an accurate analytical expression of the problem dealing with the SAW. It is wellknown that the SAW model is widely used in physics, chemistry and biology. For exam-

Wiring the Brain by Clustered Protocadherin Neural Codes

There are more than a thousand trillion specific synaptic connections in the human brain and over a million new specific connections are formed every second during the early years of life. The assembly of these staggeringly complex neuronal circuits requires specific cell-surface molecular tags to endow each neuron with a unique identity code to discriminate self from non-self. The clustered protocadherin(Pcdh) genes, which encode a tremendous diversity of cell-surface assemblies, are candidates for neuronal identity tags. We describe the adaptive evolution,genomic structure, and regulation of expression of the clustered Pcdhs. We specifically focus on the emerging3-D architectural and biophysical mechanisms that generate an enormous number of diverse cell-surface Pcdhs as neural codes in the brain.

Conformational statistics of polymer chain terminally attached to wall(Ⅱ)——SAW model tail chain

The SAW tail chains were studied.The permitted conformational number and the mean square end-to-end distance as a function of the chain length N for such a model tail chain were obtained by computer simulations,including the exact enumeration and Monte Carlo method.These two basic quantities obeyed the relations deduced from the scaling law.The critical exponents and the lattice indexes were given by fitting the data of the computer experiments.It has been shown that there is a certain extension in the size of the SAW tail chains as well as the NRW tail chains in the direction normal to the wall.The normal component of the mean square end-to-end distance is almost twice as large as the parallel component of the short chain SAW.However,as N→∞,the effect of the wall on the chain conformation becomes a little weak because of the self-avoiding behavior for the model.That is quite different from the case of the NRW tail chain.