随机多孔介质逾渗模型渗透率的临界标度性质

研究了一类非零键渗透率满足均匀分布的随机多孔介质逾渗模型-数值计算了该模型系统渗透率在临界点处的标度指数-结果表明该指数并不能看作是普适常数,而与均匀分布的参数有关-这意味着即使非零键渗透率值的概率密度函数满足负一阶矩存在条件,系统渗透率在逾渗临界点处的标度指数仍然依赖于分布函数的具体参数,并不是常数-这一数值结果与Sahimi对此问题的结论不同- 关键词： 逾渗 随机多孔介质 标度指数 渗透率     

Complex scaling behavior of nonconserved self-organized critical systems

The Olami-Feder-Christensen earthquake model is often considered the prototype dissipative self-organized critical model. It is shown that the size distribution of events in this model results from a complex interplay of several different phenomena, including limited floating-point precision. Parallels between the dynamics of synchronized regions and those of a system with periodic boundary conditions are pointed out, and the asymptotic avalanche size distribution is conjectured to be dominated by avalanches of size 1, with the weight of larger avalanches converging towards zero as the system size increases.     

Some mathematical aspects of the scaling limit of critical two-dimensional systems

It has been observed long ago that many systems from statistical physics behave randomly on macroscopic level at their critical temperature. In two dimensions, these phenomena have been classified by theoretical physicists thanks to conformal field theory, that led to the derivation of the exact value of various critical exponents that describe their behavior near the critical temperature. In the last couple of years, combining ideas of complex analysis and probability theory, mathematicians have constructed and studied a family of random fractals (called ‘Schramm-Loewner evolutions’ or SLE) that describe the only possible conformally invariant limits of the interfaces for these models. This gives a concrete construction of these random systems, puts various predictions on a rigorous footing, and leads to further understanding of their behavior. The goal of this paper is to survey some of these recent mathematical developments, and to describe a couple of basic underlying ideas. We will also briefly describe some very recent and ongoing developments relating SLE, Brownian loop soups and conformal field theory.     

Boundary effects in the critical scaling of entanglement entropy in 1D systems

We present exact diagonalization and density matrix renormalization group results for the entanglement entropy of critical spin-1/2 XXZ chains. We find that open boundary conditions induce an alternating term in both the energy density and the entanglement entropy which are approximately proportional, decaying away from the boundary with a power law. The power varies with anisotropy along the critical line and is corrected by a logarithmic factor, which we calculate analytically, at the isotropic point. A heuristic resonating valence bond explanation is suggested.     

Statistical mechanical transformation theory for quantum systems with discrete spectra

The superoperator transformation theory developed by the Brussels school is applied to quantum systems with discrete spectra. In the case of nondegeneracy of the spectra, there is no difficulty in obtaining explicit expressions for the most important superoperators in terms of the unitary operator which diagonalizes the Hamiltonian. The degenerate case presents special problems which are studied in detail.     

Quantum critical scaling in graphene

We show that the emergent relativistic symmetry of electrons in graphene near its quantum critical point (QCP) implies a crucial importance of the Coulomb interaction. We derive scaling laws, valid near the QCP, that dictate the nontrivial magnetic and charge response of interacting graphene. Our analysis yields numerous predictions for how the Coulomb interaction will be manifested in experimental observables such as the diamagnetic response and electronic compressibility.     

Variation of discrete spectra

A formula [see (1) below] estimating collectively the variation of eigenvalues of a symmetric matrix under a perturbation is extended to the case of discrete eigenvalues of a selfadjoint operator in Hilbert space, under the assumption that the perturbation is compact. For this purpose, the notion of an extended enumeration of discrete eigenvalues is introduced.     

Autocorrelation exponent of conserved spin systems in the scaling regime following a critical quench

We study the autocorrelation function of a conserved spin system following a quench at the critical temperature. Defining the correlation length L(t) approximately t(1/z), we find that for times t' and t satisfying L(t')infinity limit, we show that lambda(')(c)=d+2 and phi=z/2. We give a heuristic argument suggesting that this result is, in fact, valid for any dimension d and spin vector dimension n. We present numerical simulations for the conserved Ising model in d=1 and d=2, which are fully consistent with the present theory.     

A new mean field theory on critical scaling in spin-glasses

The critical exponents γ, δ in the non-linear susceptibility are given by making use of the free energy simply consisting of order parameter, magnetization and what was previously named an internal field parameter. The crossover exponent ø also is obtained. It is found that they satisfy the scaling relations proposed independently by Suzuki et al.     

Dimensional scaling of flame propagation in discrete particulate clouds

The critical dimension necessary for a flame to propagate in suspensions of fuel particles in oxidiser is studied analytically and numerically. Two types of models are considered: First, a continuum model, wherein the individual particulate sources are not resolved and the heat release is assumed spatially uniform, is solved via conventional finite difference techniques. Second, a discrete source model, wherein the heat diffusion from individual sources is modelled via superposition of the Green's function of each source, is employed to examine the influence of the random, discrete nature of the media. Heat transfer to cold, isothermal walls and to a layer of inert gas surrounding the reactive medium are considered as the loss mechanisms. Both cylindrical and rectangular (slab) geometries of the reactive medium are considered, and the flame speed is measured as a function of the diameter and thickness of the domains, respectively. In the continuum model with inert gas confinement, a universal scaling of critical diameter to critical thickness near 2:1 is found. In the discrete source model, as the time scale of heat release of the sources is made small compared to the interparticle diffusion time, the geometric scaling between cylinders and slabs exhibits values greater than 2:1. The ability of the flame in the discrete regime to propagate in thinner slabs than predicted by continuum scaling is attributed to the flame being able to exploit local fluctuations in concentration across the slab to sustain propagation. As the heat release time of the sources is increased, the discrete source model reverts back to results consistent with the continuum model. Implications of these results for experiments are discussed.     

Multicomponent loschmidt echo and mixing in the quantum dynamics of systems with dense discrete spectra

A dynamic problem for a system coupled to a reservoir possessing a dense discrete spectrum of states has been analytically solved under two simplifying assumptions proposed by Zwanzig [15], according to which the reservoir spectrum is equidistant and all the system-reservoir coupling matrix elements are identical (i.e., independent of reservoir states). It is demonstrated that a multicomponent Loschmidt echo arises in each recurrence cycles, the number of components being equal to the cycle number. At a certain critical cycle number, the components of neighboring cycles exhibit mixing. As a result, the dynamics of the system transforms from a regular to stochastic-like dynamics, in which an arbitrarily small coarse graining (inherent in any real system) of the results of measurements or uncertainty in the initial conditions leads to (i) the loss of one-to-one correspondence between the discrete spectrum of eigenvalues and the state of the system and (ii) the loss of invariance with respect to the time reversal. Interrelation between the mixing of cycles with the entanglement of trajectories, on the one hand, and the overlap of resonances in classical systems with mixing, on the other hand, is discussed. The properties of the proposed model are consistent with a variety of the kinetic regimes of vibrational relaxation (from exponential decay to irregular, weakly damping oscillations) observed in various objects. Common features are average distances between neighboring levels on the order of 1–10 cm^?1 and recurrence cycles on a time scale of 10^–13–10^?11 s, which is studied using femtosecond spectroscopy techniques.     

Microcanonical scaling in small systems

A microcanonical finite-size scaling ansatz is discussed. It exploits the existence of a well-defined transition point for systems of finite size in the microcanonical ensemble. The best data collapse obtained for small systems yields values for the critical exponents in good agreement with other approaches. The exact location of the infinite system critical point is not needed when extracting critical exponents from the microcanonical finite-size scaling theory.     

Dynamical scaling and kinetic equations for heisenberg spin systems just below the critical temperature

We propose an approximate set of kinetic equations to describe the time evolution of spin correlation functions below T_c in a system with long-range interactions. In the ordered region, these equations lead to weakly damped spin waves, in exact agreement with the previous work of Vaks, Lakkin and Pikin. When T_c is approached, spin waves persist for q ? κ_, where κ_is the inverse correlation length but, for q ? κ_, these kinetic equations continuously transform into the non-Markoffian equation previously derived by one of the authors and De Leener above F_c. Moreover, dynamical scaling is microscopically justified in our model. Beyond the assumption of a large number of interacting neighbours, we also need the phenomenological hypothesis that, close to T_c, the equilibrium properties appearing in our kinetic equations can be described by their correct (nonclassical) critical indices.     

A scaling analysis of the critical conditions for a deflagration-to-detonations transition

In this paper an intuitive criterion for the deflagration-to-detonations transition (DDT) by turbulent mixing was deduced. An analysis was first carried out to determine the critical conditions for detonation initiation within a reactive mixture, which is non-uniformly heated. It was demonstrated that the critical conditions depend on both the size and the characteristic heating time of the energy source. If the characteristic heating time is larger than a critical value, no detonation initiation can be expected, no matter how large the total energy of the heat source is. The critical parameters obtained were then applied to determine the critical conditions for detonation initiation via turbulent mixing. It was found that the DDT depends on both the mixing time and length scales. According to the Damkohler number, there are two regimes. (a) When the Damkohler number is smaller than unity, the critical integral length scale for DDT decreases as the inverse of the turbulent velocity. (b) When the Damkohler number is larger than unity, the critical integral length scale increases according to the cube of the turbulent velocity. These critical conditions were explained in a turbulent phase diagram of Borghi.(Some figures in this article appear in colour in the electronic version; see www.iop.org)