The evolution of the cluster size distribution in a coagulation system

The coagulation equation with kernelK _ij =A+B(i+j)+C _ij and arbitrary initial conditions is studied analytically and a simple expression for the solution is found. For monodisperse initial conditions, we recover the known size distribution expressed in terms of a degeneracy factorN _k, which is determined by a recursion relation. For polydisperse initial conditions, a similar solution form is found, which includes a degeneracy factorN _kl, also determined by a recursion relation. The physical meaning ofN _kl and the recursion relation is given. A method to get explicit expressions forN _k andN _kl is illustrated. Finally, the pre-gel solution is given explicitly and a general method to get the post-gel solution is proposed.     

Variance of the range of a random walk

T _n, the expectation of the square of the number of distinct sites occupied by a random walk in steps 1 throughn, is obtained from its relation to the dual first occupancy probabilityF _ij(x, x), and the latter quantity is obtained from a recursion with the first occupancy probabilityF _k (x). The varianceV _n of the number of distinct sites occupied is calculated directly from T_n; the procedure is illustrated by the calculation ofV _n (4096 /n) and the derivation of asymptotic expansions forV _n for a particular random walk in dimensions 1 through 3.Work completed under the auspices of the United States Department of Energy.     

On a coupled random walk process specified by a Hamiltonian

A coupled random walk process specified by an effective Hamiltonian in a potential field is proposed. The Hamiltonian is expressed in terms of a set of jumping probabilities which characterize the random walk processes. The steady state is expressed by the Hamiltonian. Conditions for the Hamiltonian to be reduced to the Ginzburg-Landau type are discussed.     

Environment-dependent continuous time random walk

A generalized continuous time random walk model which is dependent on environmental damping is proposed in which the two key parameters of the usual random walk theory: the jumping distance and the waiting time, are replaced by two new ones: the pulse velocity and the flight time. The anomalous diffusion of a free particle which is characterized by the asymptotical mean square displacement <x²(t)>～t^α is realized numerically and analysed theoretically, where the value of the power index α is in a region of 0 < α < 2. Particularly, the damping leads to a sub-diffusion when the impact velocities are drawn from a Gaussian density function and the super-diffusive effect is related to statistical extremes, which are called rare-though-dominant events.     

From random to self-avoiding walks

A brief review will be given of the current situation in the theory of self-avoiding walks (SAWs). The Domb-Joyce model first introduced in 1972 consists of a random walk on a lattice in which eachN step configuration has a weighting factor Π _i=0 ^N?2 Π_j=i+2/N(1?ωδ_ij). Herei andj are the lattice sites occupied by the ith and jth points of the walk. When ω=0 the model reduces to a standard random walk, and when ω=1 it is a self-avoiding walk. The universality hypothesis of critical phenomena will be used to conjecture the behavior of the model as a function ofω for largeN. The implications for the theory of dilute polymer solutions will be indicated.     

Theory of sparse random matrices and vibrational spectra of amorphous solids

The random matrix theory has been used for analyzing vibrational spectra of amorphous solids. The random dynamical matrix M = AA ^T with nonnegative eigenvalues ɛ = ω² has been investigated. The matrix A is an arbitrary square (N-by-N) real sparse random matrix with n nonzero elements in each row, mean values 〈A _ij 〉 = 0, and finite variance 〈A _ij ²〉 = V ². It has been demonstrated that the density of vibrational states g(ω) of this matrix at N, n ≫ 1 is described by the Wigner quarter-circle law with the radius independent of N. For n ≪ N, this representation of the dynamical matrix M = AA ^T makes it possible in a number of cases to adequately describe the interaction of atoms in amorphous solids. The statistics of levels (eigenfrequencies) of the matrix M is adequately described by the Wigner surmise formula and indicates the repulsion of vibrational terms. The participation ratio of the vibrational modes is approximately equal to 0.2–0.3 almost over the entire range of frequencies. The conclusions are in qualitative and, frequently, quantitative agreement with the results of numerical calculations performed by molecular dynamics methods for real amorphous systems.     

The statistical error of green's function Monte Carlo

The statistical error in the ground state energy as calculated by Green's Function Monte Carlo (GFMC) is analyzed and a simple approximate formula is derived which relates the error to the number of steps of the random walk, the variational energy of the trial function, and the time step of the random walk. Using this formula it is argued that as the thermodynamic limit is approached withN identical molecules, the computer time needed to reach a given error per molecule increases asN ^h where 0.5 <b < 1.5 and as the nuclear chargeZ of a system is increased the computer time necessary to reach a given error grows asZ ^5.5. Thus GFMC simulations will be most useful for calculating the properties of lowZ elements. The implications for choosing the optimal trial function from a series of trial functions is also discussed.     

Effects of random biquadratic couplings in a spin-1 spin-glass model

A spin-1 model, appropriated to study the competition between bilinear (J _ij S _i S _j ) and biquadratic (K _ij S _i ² S _j ²) random interactions, both of them with zero mean, is investigated. The interactions are infinite-ranged and the replica method is employed. Within the replica-symmetric assumption, the system presents two phases, namely, paramagnetic and spin-glass, separated by a continuous transition line. The stability analysis of the replica-symmetric solution yields, besides the usual instability associated with the spin-glass ordering, a new phase due to the random biquadratic couplings between the spins. Received 18 May 1999 and Received in final form 20 October 1999     

Exact Solution for a Class of Random Walk on the Hypercube

A class of families of Markov chains defined on the vertices of the n-dimensional hypercube, Ω _n ={0,1} ⁿ , is studied. The single-step transition probabilities P _n,ij , with i,j∈Ω _n , are given by P_n,ij=\frac(1-a)^d_ij(2-a)ⁿP_{n,ij}=\frac{(1-{\alpha})^{d_{ij}}}{(2-{\alpha})^{n}}, where α∈(0,1) and d _ij is the Hamming distance between i and j. This corresponds to flip independently each component of the vertex with probability \frac1-a2-a\frac{1-{\alpha}}{2-{\alpha}}. The m-step transition matrix P_n,ij^mP_{n,ij}^{m} is explicitly computed in a close form. The class is proved to exhibit cutoff. A model-independent result about the vanishing of the first m terms of the expansion in α of P_n,ij^mP_{n,ij}^{m} is also proved.     

Random layered frustration models

We study inhomogeneous two-dimensional Ising models with a random distribution of ferro- and antiferromagnetic couplings,K _ij =±K, or equivalently a random distribution of frustrations. In particular, we considerRandom Layered Frustration models (RLF) where randomness is confined to the vertical direction. These RLF-models are solved exactly, i.e., partition function and free energy are obtained in closed form for an arbitrary random distribution of finite period. The phase transition is of Ising type. A simple formula for the transition temperature is derived which depends only on the mean coupling , but not on other details of the distribution. Both cases,T _c =0 andT _c 0, are possible. Groundstate energy and groundstate degeneracy, or equivalently the rest entropy, are determined. It is found that both the occurence or absence of a phase transition may be accompanied with vanishing or nonvanishing rest entropy. We also show that for the RLF-models a phase transition is excluded when all groundstates are connected with one another by local transformations which presumably holds generally. A remarkable result is that the transition of the ferromagnetic Ising model can be destroyed completely if one replaces an arbitrarily small fraction of ferromagnetic couplings by antiferromagnetic ones in a suitable way.Work performed within the research program of the Sonderforschungsbereich 125 Aachen-Jülich-Köln     

Conditional entropy in algebraic statistical physics

The notion of conditional entropy as entropy of conditional state on C^*-algebra with respect to its C^*-subalgebra ₁ is introduced. It is proved that for a compatible state σ on (which admits the conditional expectation of Umegaki-Takesaki) the mean conditional entropy in an a priori state σ₁ on ₁ is equal to the difference of the entropy of the state σ on and the entropy of the state σ₁ on ₁. The conditional entropy enables us to define the input-output information of a quantum communication channel in analogy to the classical Shannon formula.     

Phonon-assisted spin diffusion in solids

The spin flip-flop transition rate is calculated for the case of spectral spin diffusion within a system of dipolarly coupled spins in a solid where the lattice vibrations are present. Long-wavelength acoustic phonons time-modulate the interspin distance r_ij and enhance the transition rate via the change of the 1/r³_ij term in the coupling dipolar Hamiltonian. The phonon-assisted spin diffusion rate is calculated by the golden rule in the Debye approximation of the phonon density of states. The coupling of the spins to the phonons introduces temperature dependence into the transition rate, in contrast to the spin diffusion in a rigid lattice, where the rate is temperature-independent. The direct (one-phonon absorption or emission) processes introduce a linear temperature dependence into the rate at temperatures not too close to T = 0. Two-phonon processes introduce a more complicated temperature dependence that again becomes simple analytical for temperatures higher than the Debye temperature, where the rate is proportional to T², and in the limit T → 0, where the rate varies as T⁷. Raman processes (one-phonon absorption and another phonon emission) dominate by far the phonon-assisted spin flip-flop transitions.     

Mixed spin- and spin- Ising models with random nearest-neighbour interactions

Using the effective field theory with correlations, we study mixed spin?3/2 and spin?1/2 Ising models with random bonds and crystal-field interactions on the honeycomb lattice. The nearest-neighbour couplings J_ij are taken as random variables with distribution P(J_ij) = pδ(J_ij ? J)+(1 ? p)δ(J_ij ? αJ), where J > 0 and |α| ≤ 1. In a certain range of negative values of α, the phase diagrams exhibit re-entrant behaviour. In detail, we investigate separately two kinds of disorder: Bond dilution (α = 0) and random ±J interactions (α = ?1). In both cases, the influence of the an-isotropy on the phase diagrams shows some new outstanding features.     

Non-Equilibrium Steady States of Finite¶Quantum Systems Coupled to Thermal Reservoirs

We study the non-equilibrium statistical mechanics of a 2-level quantum system, ?, coupled to two independent free Fermi reservoirs ?₁, ?₂, which are in thermal equilibrium at inverse temperatures β₁≠β₂. We prove that, at small coupling, the combined quantum system ?+?₁+?₂ has a unique non-equilibrium steady state (NESS) and that the approach to this NESS is exponentially fast. We show that the entropy production of the coupled system is strictly positive and relate this entropy production to the heat fluxes through the system. A part of our argument is general and deals with spectral theory of NESS. In the abstract setting of algebraic quantum statistical mechanics we introduce the new concept of the C-Liouvillean, L, and relate the NESS to zero resonance eigenfunctions of L ^*. In the specific model ?+?₁+?₂ we study the resonances of L ^* using the complex deformation technique developed previously by the authors in [JP1]. Dedicated to Jean Michel Combes on the occasion of his sixtieth birthday Received: 12 July 2001 / Accepted: 11 October 2001     

Local production rate of an “enzyme kinetics” based on coupled random walks

Based on the coupled random walk process, a local production rate of an enzyme kinetics is studied. A substrate as a walker is specified by a set of the jumping probabilities between the sites and between the modes. The modes represent the states of the substrate before, in and after the reaction. The local production rate is explicitly given and its position dependence is studied in terms of jumping probabilities. Qualitatively, the deviations from the Michaelis-Menten's expression are discussed.     

The Entropy Production of Diffusion Processes on Manifolds and Its Circulation Decompositions

In non-equilibrium statistical mechanics, the entropy production is used to describe flowing in or pumping out of the entropy of a time-dependent system. Even if a system is in a steady state (invariant in time), Prigogine suggested that there should be a positive entropy production if it is open. In 1979, the first author of this paper and Qian Min-Ping discovered that the entropy production describes the irreversibility of stationary Markov chains, and proved the circulation decomposition formula of the entropy production. They also obtained the entropy production formula for drifted Brownian motions on Euclidean space R ⁿ (see a report without proof in the Proc. 1st World Congr. Bernoulli Soc.). By the topological triviality of R ⁿ , there is no discrete circulation associated to the diffusion processes on $R^n$. In this paper, the entropy production formula for stationary drifted Brownian motions on a compact Riemannian manifold M is proved. Furthermore, the entropy production is decomposed into two parts – in addition to the first part analogous to that of a diffusion process on R ⁿ , some discrete circulations intrinsic to the topology of M appear! The first part is called the hidden circulation and is then explained as the circulation of a lifted process on M×S ¹ around the circle S ¹. The main result of this paper is the circulation decomposition formula which states that the entropy production of a stationary drifted Brownian motion on M is a linear sum of its circulations around the generators of the fundamental group of M and the hidden circulation. Received: 4 November 1998 / Accepted: 7 April 1999     

随机性对部分随机介质激光器阈值的影响

基于速率方程和Maxwell方程相结合的模型，采用时域有限差分法(FDTD)研究了介质的随机性和层数对部分随机介质激光器阈值的影响.模拟结果显示，当抽运速率超过阈值时，出现一个或者多个振荡模；随机性或者系统的尺度增加时，振荡模数量也增加；部分随机介质激光器的阈值在一定的随机强度和层数下将达到最小值，它与完全随机情况下的结论有所不同.对所得到的结论给出了物理解释.这些结果对于制作随机激光器和光集成潜在应用价值. 关键词： 激光物理 随机激光器 阈值 时域有限差分法     

Angular Momentum and Positive Mass Theorem

Total angular momentum for asymptotically flat manifolds is defined. Positive mass theorem for initial (spin) data set (M, g _ij , p _ij ) with nonsymmetric p _ij is proved. As an application, we establish positive mass theorems involving total linear momentum and total angular momentum. This gives an answer to a problem of S. T. Yau in his Problem Section [Ya2] and a partial answer to his recent conjecture on the relationship among total energy, total linear momentum, total angular momentum and entropy of black hole. Received: 26 October 1998 / Accepted: 10 March 1999