Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · cond-mat/9404101

Manifolds in random media: A variational approach to the spatial probability distribution

Yadin Y. Goldschmidt
View original source ↗Read source document ↗Export metadata ↓

Abstract

We develop a new variational scheme to approximate the position dependent spatial probability distribution of a zero dimensional manifold in a random medium. This celebrated 'toy-model' is associated via a mapping with directed polymers in 1+1 dimension, and also describes features of the commensurate-incommensurate phase transition. It consists of a pointlike 'interface' in one dimension subject to a combination of a harmonic potential plus a random potential with long range spatial correlations. The variational approach we develop gives far better results for the tail of the spatial distribution than the hamiltonian version, developed by Mezard and Parisi, as compared with numerical simulations for a range of temperatures. This is because the variational parameters are determined as functions of position. The replica method is utilized, and solutions for the variational parameters are presented. In this paper we limit ourselves to the replica symmetric solution.

Explore related subjects

cond-mat

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yadin Y. Goldschmidt. 1994-05-02. Manifolds in random media: A variational approach to the spatial probability distribution. https://doi.org/10.1051/jp1%3A1994209

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2021

Large amplitude behavior of the Grinfeld instability, Part I: High-order weakly nonlinear analysis

Amplitude expansions are used to determine steady states of a semi-infinite solid subject to the Grinfeld instability in systems with a fixed (wave)length. We present two methods to obtain high-order weakly nonlinear results. Using the system size as a control parameter, we circumvent the problem that there is no instability threshold for an extended system in the absence of gravity. This way, the case without gravity becomes accessible to a weakly nonlinear treatment. The dependence of the branch structure of solution space on the level of gravity (or density difference) is exhibited. In the zero-gravity limit, we recover the solution branch obtained by Spencer and Meiron. A transition from a supercritical to a subcritical bifurcation is observed as gravity is increased or the nonhydrostatic stress is decreased at fixed gravity. At given values of the system parameters, we find a discrete, possibly infinite, set of solution branches. This is reminiscent of dendritic or eutectic growth, where similar solution sets exist, of which only a particular one is linearly stable. Despite the high order of our expansions, the approach is restricted to relatively small nondimensional amplitudes ($\lesssim 0.2$), a disadvantage we can overcome by a variational approach that will be discussed in a companion paper. At the critical point, we find that not only the first Landau coefficient is negative but all of them up to the highest amplitude order (15) we could compute so far.

cond-mat↗
arXiv2021

Electronic Structure, Correlation Effects and Physical Properties of d- and f-Metals and Their Compounds

The book includes all main physical properties of d- and f-transition-metal systems and corresponding theoretical concepts. Especial attention is paid to the theory of magnetism and transport phenomena. Some examples of non-traditional questions which are treated in detail in the book: the influence of density of states singularities on electron properties; many-electron description of strong itinerant magnetism; mechanisms of magnetic anisotropy; microscopic theory of anomalous transport phenomena in ferromagnets. Besides considering classical problems of solid state physics as applied to transition metals, modern developments in the theory of correlation effects in d- and f-compounds are considered within many-electron models. The book contains, where possible, a simple physical discussion. More difficult questions are considered in Appendices.

cond-mat↗
arXiv2021

Quasiperiodic functions theory and the superlattice potentials for a two-dimensional electron gas

We consider Novikov problem of the classification of level curves of quasiperiodic functions on the plane and its connection with the conductivity of two-dimensional electron gas in the presence of both orthogonal magnetic field and the superlattice potentials of special type. We show that the modulation techniques used in the recent papers on the 2D heterostructures permit to obtain general quasiperiodic potentials for 2D electron gas and consider the asymptotic limit of conductivity when $τ\rightarrow \infty$. Using the theory of quasiperiodic functions we introduce here the topological characteristics of such potentials observable in the conductivity. The corresponding characteristics are the direct analog of the "topological numbers" introduced previously in the conductivity of normal metals.

cond-mat↗