Thermal fluctuations in pinned elastic systems: field theory of rare events and droplets

Using the functional renormalization group (FRG) we study the thermal fluctuations of elastic objects (displacement field u, internal dimension d) pinned by a random potential at low temperature T, as prototypes for glasses. A challenge is how the field theory can describe both typical (minimum energy T = 0) configurations, as well as thermal averages which, at any non-zero T as in the phenomenological droplet picture, are dominated by rare degeneracies between low lying minima. We show that this occurs through an essentially non-perturbative thermal boundary layer (TBL) in the (running) effective action Γ [u] at T > 0 for which we find a consistent scaling ansatz to all orders. The TBL describes how temperature smoothes the singularities of the T = 0 theory and contains the physics of rare thermal excitations (droplets). The formal structure of this TBL, which involves all cumulants of the coarse grained disorder, is first explored around d = 4 using a one-loop Wilson RG. Next, a more systematic exact RG (ERG) method is employed, and first tested on d = 0 models where it can be pushed quite far. There we obtain precise relations between TBL quantities and droplet probabilities (those are constrained by exact identities which are then checked against recent exact results). Our analysis is then extended to higher d, where we illustrate how the TBL scaling remains consistent to all orders in the ERG and how droplet picture results can be retrieved. Since correlations are determined deep in the TBL (by derivatives of Γ [u] at u = 0), it remains to be understood (in any d) how they can be retrieved (as u = 0⁺ limits in the non-analytic T = 0 effective action), i.e., how to recover a T = 0 critical theory. This formidable “matching problem” is solved in detail for d = 0, N = 1 by studying the (partial) TBL structure of higher cumulants when points are brought together. We thereby obtain the β-function at T = 0, all ambiguities removed, displayed here up to four loops. A discussion of the d > 4 case and an exact solution at large d are also provided.     

Nonlinear Double-well Schrödinger Equations in the Semiclassical Limit

We consider time-dependent Schrödinger equations with a double well potential and an external nonlinear, both local and non-local, perturbation. In the semiclassical limit, the finite dimensional eigenspace associated to the lowest eigenvalues of the linear operator is almost invariant for times of the order of the beating period and the dominant term of the wavefunction is given by means of the solutions of a finite dimensional dynamical system. In the case of local nonlinear perturbation, we assume the spatial dimension d=1 or d=2.     

Lower bounds for the maximal Lyapunov exponent

Upper bounds for the maximal Lyapunov exponent,E, of a sequence of matrix-valued random variables are easy to come by asE is the infimum of a real-valued sequence. We shall show that under irreducibility conditions similar to those needed to prove the Perron-Frobenius theorem, one can find sequences which increase toE. As a byproduct of the proof we shall see that we may replace the matrix norm with the spectral radius when computingE in such cases. Finally, a sufficient condition for transience of random walk in a random environment is given.     

Continuity properties of the normal cone to the level sets of a quasiconvex function

Two main properties of the subgradient mapping of convex functions are transposed for quasiconvex ones. The continuity of the functionxf(x)^–1f(x) on the domain where it is defined is deduced from some continuity properties of the normal coneN to the level sets of the quasiconvex functionf. We also prove that, under a pseudoconvexity-type condition, the normal coneN(x) to the set {x:f(x)f(x)} can be expressed as the convex hull of the limits of type {N(x _n)}, where {x _n} is a sequence converging tox and contained in a dense subsetD. In particular, whenf is pseudoconvex,D can be taken equal to the set of points wheref is differentiable.This research was completed while the second author was on a sabbatical leave at the University of Montreal and was supported by a NSERC grant. It has its origin in the doctoral thesis of the first author (Ref. 1), prepared under the direction of the second author.The authors are grateful to an anonymous referee and C. Zalinescu for their helpful remarks on a previous version of this paper.     

关于γ—可重排列方面的一个计数公式

本文给也了规格为２＾ｍ的２ｍ个元的γ－可重排列的个数的一个计算公式。     

δ势垒对多臂量子环中持续电流的影响

提出了含δ势垒的多臂量子环模型.研究发现总磁通为零时,持续电流随半导体环增大发生非周期性振荡,下臂因含δ势垒而获得最小的平均持续电流.AB磁通增强时,持续电流会发生周期性等幅振荡,并与电极的磁矩方向以及隧穿电子的自旋方向相关.两电极磁矩方向平行时,Rashba自旋轨道耦合具有改变持续电流相位和相位差的效应;两电极磁矩方向反平行时,Rashba自旋轨道耦合具有改变持续电流振幅的效应.各臂之间持续电流的不同与臂长和磁通分布的差异相关.在一定条件下,两种波函数所对应的持续电流是可分离的. 关键词： 多臂量子环 持续电流 δ势垒 Rashba自旋轨道耦合     

一类非线性Schr(o)dinger方程的非协调有限元分析

在半离散格式下研究一类带幂次非线性项的Schr(o)dinger方程的非协调矩形EQrot1元方法.直接利用插值技巧和该单元的两个特殊性质(相容误差比插值误差高一阶及其插值算子与传统的Ritz投影是一致的),给出相应的收敛性分析及误差估计.     

An old conjecture of Erdos-Turán on additive bases

There is a 1941 conjecture of Erdos and Turán on what is now called additive basis that we restate:

Conjecture 0.1(Erdos and Turán). Suppose that is an increasing sequence of integers and

Suppose that

If 0$"> for all , then is unbounded.

Our main purpose is to show that the sequence cannot be bounded by . There is a surprisingly simple, though computationally very intensive, algorithm that establishes this.

     

关于正定厄米特矩阵的Schur补的迹和特征值的不等式

本文研究了正定厄米特矩阵Schur补的迹和特征值的性质,通过一个不等式的证明,得到了正定厄米特矩阵和的Schur补与正定厄米特矩阵Schur补的和的迹和特征值之间的不等式.     

关于一种新模型对软件总体可靠度的估计

本文在对文[4]中软件可靠度的估计进行研究的基础上,提出了软件总体可靠性的一个新模型,对软件总体的可靠度进行了讨论,利用点估计方法,得到软件可靠度的点估计值,证明了软件总体失效率λ估计解的唯一性,并对缺陷数进行了估计,从而给出了新模型下软体总体可靠性的估计.