Elastic Moduli in Nano-Size Samples of Amorphous Solids: System Size Dependence

This Letter is motivated by some recent experiments on pan-cake shaped nano-samples of metallic glass that indicate a decline in the measured shear modulus upon decreasing the sample radius. Similar measurements on crystalline samples of the same dimensions showed a much more modest change. In this Letter we offer a theory of this phenomenon; we argue that such results are generically expected for any amorphous solid, with the main effect being related to the increased contribution of surfaces with respect to bulk when the samples get smaller. We employ exact relations between the shear modulus and the eigenvalues of the system's Hessian matrix to explore the role of surface modes in affecting the elastic moduli

Conformal Theory of the Dimensions of Diffusion Limited Aggregates

We employ the recently introduced conformal iterative construction of Diffusion Limited Aggregates (DLA) to study the multifractal properties of the harmonic measure. The support of the harmonic measure is obtained from a dynamical process which is complementary to the iterative cluster growth. We use this method to establish the existence of a series of random scaling functions that yield, via the thermodynamic formalism of multifractals, the generalized dimensions D(q) of DLA for q >= 1. The scaling function is determined just by the last stages of the iterative growth process which are relevant to the complementary dynamics. Using the scaling relation D(3) = D(0)/2 we estimate the fractal dimension of DLA to be D(0) = 1.69 +- 0.03.Comment: 5 pages, 3 figures, submitted to Phys. Rev. Let

The need for speed : Maximizing random walks speed on fixed environments

We study nearest neighbor random walks on fixed environments of $\mathbb{Z}$ composed of two point types : $(1/2,1/2)$ and $(p,1-p)$ for $p>1/2$. We show that for every environment with density of $p$ drifts bounded by $\lambda$ we have $\limsup_{n\rightarrow\infty}\frac{X_n}{n}\leq (2p-1)\lambda$, where $X_n$ is a random walk on the environment. In addition up to some integer effect the environment which gives the best speed is given by equally spaced drifts