Systematic approximations in disordered systems

We discuss two systematic approximation schemes for disordered systems (random lattices) by extending the single site CPA. In the first scheme all self energy diagrams up to those includingν scattering centers are summed up. Thisν-center approximation has been discussed before and is here derived using a straightforward projection operator formalism. The second scheme is a “true” cluster-CPA in which the scattering from an arbitrarily large cluster of neighbouring atoms (ν-cluster) embedded in a random crystal is treated just in the same way as in the single site CPA. In contrast to other cluster formulations our theory preserves translational invariance and determines the self energy in a natural way. The two schemes are compared with each other concerning their practical applicability.     

Simple inequality for the free energy of disordered systems

It is proved that the free energy of a disordered system described by a quadratic form in Bose or Fermi operators with random coefficients, calculated in the simplest approximation for the associated eigenvalue problem, gives the upper (Bose case) and lower (Fermi case) bounds for the exact free energy.     

Bounds on the density of states in disordered systems

For a class of tight-binding models governed by short-range one-particle Hamiltonians with site-diagonal and/or off-diagonal disorder and continuous distribution of the matrix elements it is proven that the averaged density of states does neither vanish nor diverge inside the band. This refutes for these models conjectures that the density of states vanishes or diverges at the mobility edge.Work supported in part by the Sonderforschungsbereich 123 Stochastic Mathematical Models of the Deutsche Forschungsgemeinschaft     

Bounds on the density of states in disordered systems

In magnetic systems the radio frequency fieldH ₁ seen by the nuclei is often enhanced by a factor . If the atomic spins are distributed in multi directions, as in Bloch walls or spin glasses, one has to consider a spatial distribution (r) of enhancement factors. The usual way to define a mean enhancement factor is shown to lead to inconsistent results giving rise to errors up to more than 100%. One has to distinguish between two mean factors, and , depending on measuring conditions. Their deviations from the true value are discussed.     

Phonon-assisted hopping-conductivity of disordered systems II: Discussion of approximations and results

A theory of the frequency dependent hopping-conductivity of disordered systems which has been developed in a previous paper is shortly reviewed in connection with other approximations. In lowest order, the probability of finding a carrier on the same site some time later (Verweil-function) is calculated and discussed. After deriving an explicit expression for the lowest order conductivity its dependence on the three variables temperature, density of sites, and frequency is studied. In particular, the characteristic exponents for vanishing temperature and density differ from those predicted by Mott. The results are also compared with a nearest-neighbour-treatment and show good agreement in certain limiting cases.     

Singularity in the density of states of quasi-one-dimensional disordered systems

We have calculated the correction to the density-of-states in two-dimensional weakly disordered systems. It is shown that the electron-electron interaction gives rise to a singularity in the density-of-states at the Fermi level, the form of which is very sensitive to the width of the sample. The transition from a two-dimensional behaviour to the quasi-one-dimensional regime is discussed. The effects should be observable by electron tunnelling measurements.     

A statistical approach to direct density of states measurements in disordered systems

A statistical method for measuring the modal density of elastic waves through direct mode counting in strongly scattering disordered systems is presented. To illustrate this approach, the results of ultrasonic experiments in a highly porous sintered glass bead network are reported. This method is shown to yield a reliable and robust measurement of the density of states, enabling mode-counting techniques to be applied to increasingly complex systems, where modal overlap and sensitivity to experimental conditions have previously hampered definitive results.     

Critical behavior of disordered systems with a free surface

The behavior of homogeneous and disordered systems with a free boundary is described on the basis of group theory in the two-loop approximation directly in three-dimensional space. The effect of the free boundary on the regime of the bulk critical behavior is revealed. It is shown that the boundedness of the system slightly affects the regime of the bulk critical behavior in the case of the ordinary transition, whereas this effect is more noticeable in the case of the special transition. Surface critical phenomena are described for homogeneous and disordered systems, and the critical exponents are calculated in the two-loop approximation. It is shown that the effect of impurities is insignificant in the special phase transition, whereas it is more noticeable in the ordinary phase transition. The derived critical exponents are compared with the computer-simulation results.     

On spectral density of electronic states in disordered systems

In the present paper, the averaged spectral density of electronic states, 〈?(k, E)〉, has been computed for three-dimensional disordered systems with spherical well type atomic potentials. The formula for 〈?(k, E)〉 is derived by using a realistic perturbational approach that takes propagator modification into account, together with a handy approximation due toBallentine. 〈?(k, E)〉 has been computed for various sets of potential strength and concentration, and the dependence of 〈?(k, E)〉 upon these parameters has been discussed. The motivation behind the computations reported here, is to examine the circumstances under which the electron motion in disordered systems is expected to be almost free electronic. Our principal finding is that a simultaneous existence of weak electron-scatterer interaction-potential and high density of scatterers would tend to make the motion of an electron in a disordered system almost like that of a free one.     

Low energy density of states in several disordered systems

Disordered systems of arbitrary dimension consisting of randomly distributed scatterers are studied in the low energy limit (Roger Waxler, Ph.D. thesis, Columbia University). We evaluate the density of electronic energy levels for long range, attractive, and electric dipole scatterers.     

Holes in the probability density of strongly colored noise driven systems

A qualitative change in the topology of the joint probability densityP(,x), which occurs for strongly colored noise in multistable systems, has recently been observed first by analog simulation (F. Moss and F. Marchesoni,Phys. Lett. A 131:322 (1988)) and confirmed by matrix continued fraction methods (Th. Leiber and H. Riskin, unpublished), and by analytic theory (P. Hänggi, P. Jung, and F. Marchesoni,J. Stat. Phys., this issue). Systems studied were of the classx=–U(x)/x+(t,), whereU(x) is a multistable potential and (t, ) is a colored, Gaussian noise of intensityD, for which =0, and (t) (s)=(D/)exp(–t–s/). When the noise correlation time is smaller than some critical value ₀, which depends onD, the two-dimensional densityP(,x) has the usual topology [P. Jung and H. Risken,Z. Phys. B 61:367 (1985); F. Moss and P. V. E. McClintock,Z. Phys. B 61:381 (1985)]: a pair of local maxima ofP(,x), which correspond to a pair of adjacent local minima ofU(x), are connected by a single saddle point which lies on thex axis. When >₀, however,the single saddle disappears and is replaced by a pair of off-axis saddles. A depression, or hole, which is bounded by the saddles and the local maxima thus appears. The most probable trajectory connecting the two potential wells therefore does not pass through the origin for >₀, but instead must detour around the local barrier. This observation implies that successful mean-first-passage-time theories of strongly colored noise driven systems must necessarily be two dimensional (Hänggiet al.). We have observed these holes for several potentialsU(x): (1)a soft, bistable potential by analog simulation (Moss and Marchesoni); (2) a periodic potential [Th. Leiber, F. Marchesoni, and H. Risken,Phys. Rev. Lett. 59:1381 (1987)] by matrix continued fractions; (3) the usual hard, bistable potential,U(x)=–ax ²/2+bx ⁴/4, by analog simulations only; and (4) a random potential for which the forcingf(x)=–U(x)/x is an approximate Gaussian with nonzero correlation length, i.e., colored spatiotemporal noise, by analog simulation. There is a critical curve ₀(D) in the versusD plane which divides the two topological behaviors. For a fixed value ofD, this curve is shifted toward larger values of ₀ for progressively weaker barriers between the wells. Therefore, strong barriers favor the observation of this topological transformation at smaller values of . Recently, an analytic expression for the critical curve, valid asymptotically in the small-D limit, has been obtained (Hänggiet al.).This paper will appear in a forthcoming issue of theJournal of Statistical Physics.     

随机系统的概率密度函数形状调节

针对受高斯白噪声激励的非线性随机系统,提出了使状态响应的概率密度函数形状跟踪期望形状的调节方法.首先,确立了非线性随机系统的多项式反馈机制,同时对系统中的非线性部分进行多项式展开;然后,以Fokker-Planck-Kolmogorov方程为工具,导出了与控制增益相关的各阶矩递推方程,并根据跟踪问题的要求,构造了矩逼近优化问题,用梯度搜索法求解该优化问题,获得了调节函数;再依据特征函数与概率密度函数构成Fourier对的关系,对状态响应的概率密度函数进行重构;最后,通过两个例子仿真,验证了本文方法的有效性.     

氢原子电子云密度分布分析

通过数值计算形象地分析了氢原子中电子概率密度的分布情况,对此做了一些有别于传统认识的讨论,并由此引出了对粒子运动和波函数概率解释的认识。     

Electromagnetic cavity to explore disordered systems

In the course of characterizing an electromagnetic cavity we have come to understand details of transmission and reflection traces, some of which may be useful as tools to investigate local modes in random media. We have demonstrated quantitative agreement of frequency shift observations with theory, have demonstrated that the spatial distribution of electric and magnetic fields can be measured using insulating and conducting spheres, and have used perturbation due to wires and disks to demonstrate the local direction of electric and magnetic fields. We have observed that not only frequency shifts but also spatial shifts of modes can be used to observe the intensity of interaction of modes with extended objects in the cavity.     

Error analysis of a practical energy density sensor

The investigation of an active control system based on acoustic energy density has led to the analysis and development of an inexpensive three-axes energy density sensor. The energy density sensor comprises six electret microphones mounted on the surface of a 0.025-m (1 in.) radius sphere. The bias errors for the potential, kinetic, and total energy density as well as the magnitude of intensity of a spherical sensor are compared to a sensor comprising six microphones suspended in space. Analytical, computer-modeled, and experimental data are presented for both sensor configurations in the case of traveling and standing wave fields, for an arbitrary incidence angle. It is shown that the energy density measurement is the most nearly accurate measurement of the four for the conditions presented. Experimentally, it is found that the spherical energy density sensor is within +/- 1.75 dB compared to reference measurements in the 110-400 Hz frequency range in a reverberant enclosure. The diffraction effects from the hard sphere enable the sensor to be made more compact by a factor of 3 compared to the sensor with suspended microphones.     

Analyticity of density of states in a gauge-invariant model for disordered electronic systems

Then-orbital gauge-invariant model of disordered electronic systems proposed by Wegner is studied in the regime of dominant diagonal disorder. Analyticity of the density of states is established in two cases: (a) when the number of orbitals is small, (b) when the number of orbitals is large and the energy is in the expected extended states region.     

On various representations of higher order approximations of the free oscillatory response of nonlinear dynamical systems

This paper investigates the flexibility afforded by the application of regular perturbation methods (in particular, the method of multiple scales) for the purpose of obtaining higher order approximations of the oscillatory response of a nonlinear dynamical system. It is shown that the non-uniqueness of these higher order approximations can be removed by enforcing additional conditions while the relationship between the frequency of oscillation and measurable quantities (the Hamiltonian, the time-averaged kinetic or stored energy) is unique and is thus not affected by these additional conditions.