Survival and extinction in cyclic and neutral three-species systems

We study the ABC model ( A + B↦2B, B + C↦2C, C + A↦2A), and its counterpart: the three-component neutral drift model ( A + B↦2A or 2B, B + C↦2B or 2C, C + A↦2C or 2A.) In the former case, the mean-field approximation exhibits cyclic behaviour with an amplitude determined by the initial condition. When stochastic phenomena are taken into account the amplitude of oscillations will drift and eventually one and then two of the three species will become extinct. The second model remains stationary for all initial conditions in the mean-field approximation, and drifts when stochastic phenomena are considered. We analyzed the distribution of first extinction times of both models by simulations of the master equation, and from the point of view of the Fokker-Planck equation. Survival probability vs. time plots suggest an exponential decay. For the neutral model the extinction rate is inversely proportional to the system size, while the cyclic model exhibits anomalous behaviour for small system sizes. In the large system size limit the extinction times for both models will be the same. This result is compatible with the smallest eigenvalue obtained from the numerical solution of the Fokker-Planck equation. We also studied the behaviour of the probability distribution. The exponential decay is found to be robust against certain changes, such as the three reactions having different rates. Received 14 August 2002 and Received in final form 14 February 2003 / Published online: 1 April 2003 RID="a" ID="a"e-mail: ita@physics.ubc.ca     

Some remarks concerning Nambu mechanics

The quantum probability flux of a particle integrated over time and a distant surface gives the probability for the particle crossing that surface at some time. We prove the free flux-across-surfaces theorem, which was conjectured by Combes, Newton and Shtokhamer (Phys. Rev. D. 11 (1975), 366), and which relates the integrated quantum flux to the usual quantum mechanical formula for the cross-section. The integrated quantum flux is equal to the probability of outward crossings of surfaces by Bohmian trajectories in the scattering regime.     

Dynamic and thermodynamic consequences of adsorbate lateral interactions in surface reaction kinetics

The effects of lateral interactions on the two-dimensional distribution of adspecies on solid surfaces and their consequences for reaction kinetics are demonstrated for the bimolecular reactive systemA+BAB. The discussion concentrates on systems where one reactant,A, is stationary while the other,B, is freely diffusing and instantaneously relaxing. A modified Bethe-Peierls-type lattice gas approximation is formulated in order to account for the rapidly-equilibrating distribution ofB atoms. The approximation takes into account all nearest and next nearest neighbor interactions between the adspecies and the nonuniformity of the lattice available toB implied by the presence of immobileA's on the surface. This model is combined with a Monte Carlo simulation of the reactive events in order to calculate reaction rates, e.g., in temperature-programmed processes. The rates are compared with full Monte Carlo simulations (for all kinetic processes), showing good agreement between the two schemes, except at very high coverages, where very long range correlations in the system which are ignored in the lattice gas approximation must be taken into account.This paper is dedicated to Howard Reiss in celebration of his 66th birthday.     

Loop and Path Spaces and¶Four-Dimensional BF Theories:¶Connections, Holonomies and Observables

We study the differential geometry of principal G-bundles whose base space is the space of free paths (loops) on a manifold M. In particular we consider connections defined in terms of pairs (A,B), where A is a connection for a fixed principal bundle P(M,G) and B is a 2-form on M. The relevant curvatures, parallel transports and holonomies are computed and their expressions in local coordinates are exhibited. When the 2-form B is given by the curvature of A, then the so-called non-abelian Stokes formula follows. For a generic 2-form B, we distinguish the cases when the parallel transport depends on the whole path of paths and when it depends only on the spanned surface. In particular we discuss generalizations of the non-abelian Stokes formula. We study also the invariance properties of the (trace of the) holonomy under suitable transformation groups acting on the pairs (A,B). In this way we are able to define observables for both topological and non-topological quantum field theories of the BF type. In the non-topological case, the surface terms may be relevant for the understanding of the quark-confinement problem. In the topological case the (perturbative) four-dimensional quantum BF-theory is expected to yield invariants of imbedded (or immersed) surfaces in a 4-manifold M. Received: 28 March 1998 / Accepted: 12 September 1998     

Growth and addition in a herding model

A model of herding is introduced which is exceptionally simple, incorporating only two phenomena, growth and addition. At each time step either (i) with probability p the system grows through the introduction of a new agent or (ii) with probability q = 1 - p a free agent already in the system is added at random to a group of size k with rate A_k. Two versions of the model, A _k = k and A _k = 1, are solved and in both versions we find two different types of behaviour. When p > 1/2 all the moments of the distribution of group sizes are linear in time for large time and the group distribution is power-law. When p < 1/2 the system runs out of free agents in a finite time. Received 12 February 2002 Published online 9 July 2002     

On the Evaluation of the Evolution Operator Z Reg( R 2, R 1) in the Diakonov–Petrov Approach to the Wilson Loop

We evaluate the evolution operator Z _Reg(R ₂,R ₁) introduced by Diakonov and Petrov for the definition of the Wilson loop in terms of a path integral over gauge degrees of freedom. We use the procedure suggested by Diakonov and Petrov (Physics Letters B 224 (1989) 131) and show that the evolution operator vanishes. PACS numbers: 11.10.-z; 11.15.-q; 12.38.-t; 12.38.Aw; 12.90.+b.     

Phase transitions of a spin-one alternating magnetic superlattice

We examine the critical behavior of a magnetic superlattice which made up of two magnetic materials, A and B. Using the effective field theory with a probability distribution technique that accounts for the single-site spin correlation, we derive the analytical equation for the Curie temperature of the superlattice which alternates as ABAB...AB. The dependence of the Curie temperature on the interface coupling strength J_ab and the layer number of the finite superlattice was calculated. The effects of the surface modification are also studied. Received 2 March 2001     

Structure, stability and magnetic properties of the Fe3O4(1 1 0) surface: Density functional theory study

Six surface models for the Fe₃O₄(1 1 0) surface were studied using the density functional theory (DFT), namely the AB-terminated surface (AB model), the AB-terminated with Fe_A vacancy (AB-Fe_A vac model), the AB-terminated with Fe_B vacancy (AB-Fe_B vac model), the B-terminated surface (B model), the B-terminated surface with Fe_B vacancy (B-Fe_B vac model), and the B-terminated surface with O vacancy (B-O vac model). Here, A and B denoted the Fe cations in tetrahedrally (Fe_A) and octahedrally (Fe_B) coordinated interstices. The stability, the electronic structure and the magnetic properties of the six surface models were also calculated. The results predict that the B-O vac model is more stable than other surface models. The half-metallic property remains in the AB and B models, while the other four surface models exhibit metallic properties. At the same time, the AB, AB-Fe_A vac, AB-Fe_B vac, B and the B-Fe_B vac models have ferrimagnetic properties, while the B-O vac model has antiferromagnetic property.     

First-principle studies on the electronic structure of Fe3O4(110) surface

The first-principle was employed to study the six possible models for the Fe₃O₄(110) surface, namely the AB-terminated surface (AB model), the AB-terminated with Fe_A vacancy (AB-Fe_A vac model), the AB-terminated with Fe_B vacancy (AB-Fe_B vac model), the B-terminated surface (B model), the B-terminated surface with Fe_B vacancy (B-Fe_B vac model) and the B-terminated surface with O vacancy (B-O vac model). The stability, the electronic structure and the magnetic properties of the six surface models were also calculated. The results predict that the B-O vac model is more stable than other types of surface models. The half-metallic property remain in the AB and B models, while the other four surface models exhibit metallic properties. At the same time, the AB, AB-Fe_A vac, AB-Fe_B vac, B and the B-Fe_B vac models have ferrimagnetic properties, while the B-O vac model has antiferromagnetic property.      

The size distribution for theA g RB f−g Model of polymerization

This paper gives the equilibrium distribution of polymer sizes for Flory'sA _g RB _f–g model of polymerization. In this model, the polymers are composed of structural units withg functional groups of the typeA and (f-g) functional groups of the typeB. Reaction is subject to three conditions: (1) Functional groups of the typeA react only with those of typeB, and vice versa. (2) Intramolecular reactions do not occur [and therefore only branched-chain (noncyclic) polymers and formed]. (3) Subject to conditions (1) and (2), all functional groups are equally reactive. The derivation employs Stockmayer's statistical mechanical method (first used on Flory'sRA _f model), coupled with a recursion giving the number of distinct polymers which may be assembled fromk units of theA _g RB _f–g type. We also give distributions for a limiting case of theA _g RB _f–g model, the so-calledA _g RB model. This paper completes the solution of the Smoluchowski coagulation equation (monodisperse case) for the kernelsa _ij =A + B(i +j)+ Cij. The proof will be given in another publication.     

Flows in Complex Networks: Theory,Algorithms, and Application to Lennard–Jones Cluster Rearrangement

A set of analytical and computational tools based on transition path theory (TPT) is proposed to analyze flows in complex networks. Specifically, TPT is used to study the statistical properties of the reactive trajectories by which transitions occur between specific groups of nodes on the network. Sampling tools are built upon the outputs of TPT that allow to generate these reactive trajectories directly, or even transition paths that travel from one group of nodes to the other without making any detour and carry the same probability current as the reactive trajectories. These objects permit to characterize the mechanism of the transitions, for example by quantifying the width of the tubes by which these transitions occur, the location and distribution of their dynamical bottlenecks, etc. These tools are applied to a network modeling the dynamics of the Lennard–Jones cluster with 38 atoms ( \(\mathrm{LJ}_{38}\) ) and used to understand the mechanism by which this cluster rearranges itself between its two most likely states at various temperatures.     

A non-thermal lattice gas model for a dimer-trimer reaction on a catalytic surface: A computer simulation study

The kinetics of an irreversible dimer-trimer reaction of the type 3A₂ + 2B₃ → 6AB have been studied using a non-thermal (precursor mechanism) model on a square as well as on a hexagonal lattice surface by Monte Carlo simulation. When the range of the precursors (A atoms) is increased, the model gives production rates (reactive window widths) that are quite large as compared with those for thermal (Langmuir-Hanshelwood mechanism) model. The phase diagrams qualitatively resemble with the standard ZGB model except that the continuous transition point is eliminated when the range of the precursors is extended up to the third nearest neighbourhood. The diffusion of A atoms on the surface as well as their desorption from the surface with a certain probability is also considered to see their effects on the reaction mechanism.     

DTO分子反应碰撞的非对称性

基于DTO分子（X¹A₁）的氢同位素效应,得到修正的Born-Oppenheimer（B-O）理论下多体展式分析势能函数.用准经典的Monte-Carlo轨迹法研究了O+DT（0,0）的分子反应动力学过程.结果表明：在碰撞能量较低时（<209.2?kJ·mol^-1）,O+DT（0,0）可以生成长寿命络合物DTO（X¹A₁）,并且该络合反应是无阈能的,这一结论与多体项展式理论计算的DTO分子势能曲线结果一致.碰撞能大于209·2?kJ·mol^-1时,逐渐出现置换产物TO和DO,随碰撞能进一步增大,分子将被完全碰散成D,T,O原子.反应O+DT（0,0）→OD+T和O+DT（0,0）→OT+D是有阈能的反应,置换产物TO和DO轨线存在非对称性. 关键词： DTO 势能函数 分子反应碰撞 轨线非对称性     

Statistical mechanics analysis of the continuous number partitioning problem

The number partitioning problem consists of partitioning a sequence of positive numbers {a₁,a₂,…,a_N} into two disjoint sets, A and B, such that the absolute value of the difference of the sums of a_j over the two sets is minimized. We use statistical mechanics tools to study analytically the linear programming relaxation of this NP-complete integer programming. In particular, we calculate the probability distribution of the difference between the cardinalities of A and B and show that this difference is not self-averaging.     

TiN多型体高压相变的第一性原理计算

基于密度泛函理论框架下的赝势平面波方法,计算了B1(氯化钠结构)、B2(氯化铯结构)、B3(闪锌矿结构)、B_k(六方氮化硼结构)、B_h(碳化钨结构)和B8₁(砷化镍结构)6种TiN多型体的晶体结构、体积弹性模量和相对稳定性.计算指出,不存在B4(纤锌矿)结构的TiN.通过不同外压下的晶格弛豫得到每种结构的焓,发现外压 关键词： 氮化钛 赝势 高压相变 密度泛函理论     

The magnetic properties of a finite superlattice with two disordered interfaces

Within the framework of the effective field theory, based on a probability distribution technique, we examine the critical and compensation behaviors of a ferrimagnetic alternating superlattice on a simple cubic structure. The superlattice consists of k unit cells each of which consists of L layers of spin-1/2 A atoms, L layers of spin-1 B atoms and a disordered interface with two layers in between that is characterized by a random arrangement of A and B atoms A_pB_1−pA_1−pB_p with a negative coupling A − B. Considering a finite and infinite superlattices, the effect of the thickness of the film and the surface exchange coupling on the magnetic properties are studied. The obtained results show a number of characteristic phenomena.     

Topological quantum characteristics observed in the investigation of the conductivity in normal metals

It is shown that the investigation of the conductivity in a single crystal of a normal metal with a complicated Fermi surface in strong magnetic fields B can reveal integral topological characteristics which are determined by the topology of open-ended quasiclassical electron trajectories. Specifically, in the case of open-ended trajectories of the general position there always exists a direction η orthogonal to B in which the conductivity approaches zero for large B, and this direction lies in some integral (i.e., generated by two reciprocal-lattice vectors) plane that remains stationary for small variations of the direction of B. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 10, 809–813 (25 May 1996)     

Exchangeability and Conditionally Identical Common Cause Systems

A pair (A, B) of events in a classical probability measure space (Ω, p) is called exchangeable iff p(A ) = p( B). Conditionally identical common cause system of size n for the correlation is an n-partition of Ω such that (i) any member of the partition screens the correlation off and (ii) for any member {C _i } _{i ∊ I} of the partition p(A|C _i ) = p(B|C _i ). The common cause system is called proper if p(A|C _i )≠(A|C _j ) for some i ≠ j. In the paper it is shown that exchangeable correlations be explained by proper conditionally identical common cause systems in the following sense. (i) Given a proper conditionally identical common cause system of size n for the two events A and B in Ω, then the pair (A, B) will be an exchangeable (positively) correlating pair. (ii) Given any exchangeable (positively) correlating pair of events in Ω and given any finite number n > 2, then the probability space can be embedded into a larger probability space in such a way that the larger space contains a proper conditionally identical common cause system of size n for the correlation.     

Evolving networks with bimodal degree distribution

Networks with bimodal degree distribution are most robust to targeted and random attacks. We present a model for constructing a network with bimodal degree distribution. The procedure adopted is to add nodes to the network with a probability p and delete the links between nodes with probability (1 − p). We introduce an additional constraint in the process through an immunity score, which controls the dynamics of the growth process based on the feedback value of the last few time steps. This results in bimodal nature for the degree distribution. We study the standard quantities which characterize the networks, like average path length and clustering coefficient in the context of our growth process and show that the resultant network is in the small world family. It is interesting to note that bimodality in degree distribution is an emergent phenomenon.     

The inviscid Burgers equation with initial data of Brownian type

The solutions to Burgers equation, in the limit of vanishing viscosity, are investigated when the initial velocity is a Brownian motion (or fractional Brownian motion) function, i.e. a Gaussian process with scaling exponent 0<h<1 (typeA) or the derivative thereof, with scaling exponent –1<h<0 (typeB). Largesize numerical experiments are performed, helped by the fact that the solution is essentially obtained by performing a Legendre transform. The main result is obtained for typeA and concerns the Lagrangian functionx(a) which gives the location at timet=1 of the fluid particle which started at the locationa. It is found to be a complete Devil's staircase. The cumulative probability of Lagrangian shock intervals a (also the distribution of shock amplitudes) follows a (a)^–h law for small a. The remaining (regular) Lagrangian locations form a Cantor set of dimensionh. In Eulerian coordinates, the shock locations are everywhere dense. The scaling properties of various statistical quantities are also found. Heuristic interpretations are provided for some of these results. Rigorous results for the case of Brownian motion are established in a companion paper by Ya. Sinai. For typeB initial velocities (e.g. white noise), there are very few small shocks and shock locations appear to be isolated. Finally, it is shown that there are universality classes of random but smooth (non-scaling) initial velocities such that the long-time large-scale behavior is, after rescaling, the same as for typeA orB.