Energy landscapes, lowest gaps, and susceptibility of elastic manifolds at zero temperature

We study the effect of an external field on (1 + 1) and (2 + 1) dimensional elastic manifolds, at zero temperature and with random bond disorder. Due to the glassy energy landscape the configuration of a manifold changes often in abrupt, “first order”-type of large jumps when the field is applied. First the scaling behavior of the energy gap between the global energy minimum and the next lowest minimum of the manifold is considered, by employing exact ground state calculations and an extreme statistics argument. The scaling has a logarithmic prefactor originating from the number of the minima in the landscape, and reads ΔE ₁∼L ^θ[ln(L _z L ^{- ζ})]^-1/2, where ζ is the roughness exponent and θ is the energy fluctuation exponent of the manifold, L is the linear size of the manifold, and L_z is the system height. The gap scaling is extended to the case of a finite external field and yields for the susceptibility of the manifolds ∼L ^{2D + 1 - θ}[(1 - ζ)ln(L)]^1/2. We also present a mean field argument for the finite size scaling of the first jump field, h ₁∼L ^{d - θ}. The implications to wetting in random systems, to finite-temperature behavior and the relation to Kardar-Parisi-Zhang non-equilibrium surface growth are discussed. Received December 2000 and Received in final form April 2001     

Nonsymmetric first-order transitions: Finite-size scaling and tests for infinite-range models

Finite-size rounding of first-order transitions is studied for the general case of nonsymmetric phases and nonperiodic boundary conditions. The main features include the surface-induced shift of the rounded transition on the scale 1/L, while the order parameter discontinuity is rounded on the scale 1/L ^d. This rounding is described by the universal scaling forms with scaling functions identical to those for the periodic, symmetric case. The proposed formalism applies to scalar-order-parameter, single-domain systems. It is tested by exact calculations for a class of infinite-range models.     

Finite-Size Scaling in the Energy-Entropy Plane for the 2D ± Ising Spin Glass

For L × L square lattices with L ≤ 20 the 2D Ising spin glass with +1 and −1 bonds is found to have a strong correlation between the energy and the entropy of its ground states. A fit to the data gives the result that each additional broken bond in the ground state of a particular sample of random bonds increases the ground state degeneracy by approximately a factor of 10/3. For x=0.5 (where x is the fraction of negative bonds), over this range of L, the characteristic entropy defined by the energy-entropy correlation scales with size as L ^1.78(2). Anomalous scaling is not found for the characteristic energy, which essentially scales as L ². When x=0.25, a crossover to L ² scaling of the entropy is seen near L=12. The results found here suggest a natural mechanism for the unusual behavior of the low temperature specific heat of this model, and illustrate the dangers of extrapolating from small L. PACS numbers: 75.10.Nr, 75.40.Mg, 75.50.Lk     

From independent particle towards collective motion for two polarized electrons on a square lattice

The two dimensional crossover from independent particle towards collective motion is studied using 2 polarized electrons (spinless fermions) interacting via a U/r Coulomb repulsion in a L×L square lattice with periodic boundary conditions and nearest neighbor hopping t. Three regimes characterize the ground state when U/t increases. Firstly, when the fluctuation Δr of the spacing r between the two particles is larger than the lattice spacing a, there is a scaling length L ₀ = π²(t/U) such that the relative fluctuation Δr/〈r〉 is a universal function of the dimensionless ratio L/L ₀, up to finite size corrections of order L^-2. L < L ₀ and L > L ₀ are respectively the limits of the free particle Fermi motion and of the correlated motion of a Wigner molecule. Secondly, when U/t exceeds a threshold U ^*(L)/t, Δr becomes smaller than a, giving rise to a correlated lattice regime where the previous scaling breaks down and analytical expansions in powers of t/U become valid. A weak random potential reduces the scaling length and favors the correlated motion. Received 28 March 2002 Published online 19 November 2002     

Heat transport in low-dimensional systems

Recent results on theoretical studies of heat conduction in low-dimensional systems are presented. These studies are on simple, yet non-trivial, models. Most of these are classical systems, but some quantum-mechanical work is also reported. Much of the work has been on lattice models corresponding to phononic systems, and some on hard-particle and hard-disc systems. A recently developed approach, using generalized Langevin equations and phonon Green's functions, is explained and several applications to harmonic systems are given. For interacting systems, various analytic approaches based on the Green–Kubo formula are described, and their predictions are compared with the latest results from simulation. These results indicate that for momentum-conserving systems, transport is anomalous in one and two dimensions, and the thermal conductivity κ diverges with system size L as κ ～ L ^α. For one-dimensional interacting systems there is strong numerical evidence for a universal exponent α = 1/3, but there is no exact proof for this so far. A brief discussion of some of the experiments on heat conduction in nanowires and nanotubes is also given.     

Domain Branching in Uniaxial Ferromagnets: A Scaling Law for the Minimum Energy

We address the branching of magnetic domains in a uniaxial ferromagnet. Our thesis is that branching is required by energy minimization. To show this, we consider the nonlocal, nonconvex variational problem of micromagnetics. We identify the scaling law of the minimum energy by proving a rigorous lower bound which matches the already-known upper bound. We further show that any domain pattern achieving this scaling law must have average width of order L ^2/3, where L is the length of the magnet in the easy direction. Finally we argue that branching is required, by considering the constrained variational problem in which branching is prohibited and the domain structure is invariant in the easy direction. Its scaling law is different. Received: 15 April 1998 / Accepted: 14 August 1998     

Design considerations and scaling laws for high power convective cooled CW CO2 lasers

Various criteria for designing high power convective cooled CO₂ lasers have been discussed. Considering the saturation intensity, optical damage threshold of the optical resonator components and the small-signal gain, the scaling laws for designing high power CW CO₂ lasers have been established. In transverse flow CO₂ lasers having discharge of square cross-section, the discharge lengthL and its widthW for a specific laser powerP (Watt) and gas flow velocityV (cm/s) can be given byL = 1.4 x 10⁴ p ^1/2 V ^-1 cms andW = 0.04P ^1/2 cms. The optimum transmitivity of the output coupler is found to be almost constant (about 60%), independent of the small signal gain and laser power. In fast axial flow CO₂ lasers the gas flow should be divided into several discharge tubes to maintain the flow velocity within sonic limit. The discharge length in this type of laser does not depend explicitly on the laser power, instead it depends on the input power density in the discharge and the gas flow velocity. Various considerations for ensuring better laser beam quality are also discussed.     

Exponential bounds and semi-finiteness of point spectrum forN-body Schrödinger operators

For a large class ofN-body Schrödinger operatorsH, we prove that eigenvalues ofH cannot accumulate from above at any threshold ofH. Our proof relies onL ² exponential upper bounds for eigenfunctions ofH with explicit constants obtained by modifying methods of Froese and Herbst.Bantrell Research Fellow in Mathematical Physics     

Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces

The Airy distribution function describes the probability distribution of the area under a Brownian excursion over a unit interval. Surprisingly, this function has appeared in a number of seemingly unrelated problems, mostly in computer science and graph theory. In this paper, we show that this distribution function also appears in a rather well studied physical system, namely the fluctuating interfaces. We present an exact solution for the distribution P(h_m,L) of the maximal height h_m (measured with respect to the average spatial height) in the steady state of a fluctuating interface in a one dimensional system of size L with both periodic and free boundary conditions. For the periodic case, we show that P(h_m,L)=L^−1/2f(h_m L^−1/2) for all L>0 where the function f(x) is the Airy distribution function. This result is valid for both the Edwards–Wilkinson (EW) and the Kardar–Parisi–Zhang interfaces. For the free boundary case, the same scaling holds P(h_m,L)=L^−1/2F(h_m L^−1/2), but the scaling function F(x) is different from that of the periodic case. We compute this scaling function explicitly for the EW interface and call it the F-Airy distribution function. Numerical simulations are in excellent agreement with our analytical results. Our results provide a rather rare exactly solvable case for the distribution of extremum of a set of strongly correlated random variables. Some of these results were announced in a recent Letter [S.N. Majumdar and A. Comtet, Phys. Rev. Lett. 92: 225501 (2004)].     

Renormalization group treatment of the scaling properties of finite systems with subleading long-range interaction

The finite size behavior of the susceptibility, Binder cumulant and some even moments of the magnetization of a fully finite O(n) cubic system of size L are analyzed and the corresponding scaling functions are derived within a field-theoretic ɛ-expansion scheme under periodic boundary conditions. We suppose a van der Waals type long-range interaction falling apart with the distance r as r ^{- (d + σ)}, where 2 < σ < 4, which does not change the short-range critical exponents of the system. Despite that the system belongs to the short-range universality class it is shown that above the bulk critical temperature T _c the finite-size corrections decay in a power-in-L, and not in an exponential-in-L law, which is normally believed to be a characteristic feature for such systems. Received 8 August 2001     

On finite-size scaling in the presence of dangerous irrelevant variables

A scaling hypothesis on finite-size scaling in the presence of a dangerous irrelevant variable is formulated for systems with long-range interaction and general geometryL ^d–d× ^d . A characteristic length which obeys a universal finite-size scaling relation is defined. The general conjectures are based on exact results for the mean spherical model with inverse power law interaction.     

A scaling law of high-order harmonic generation

Using a nonperturbative quantum electrodynamics theory of high-order harmonic generation (HHG), a scaling law of HHG is established. The scaling law states that when the atomic binding energy E_b, the wavelength λ and the intensity I of the laser field change simultaneously to kE_b, λ/k, and k³I, respectively. The characteristics of the HHG spectrum remain unchanged, while the harmonic yield is enhanced k³ times. That HHG obeys the same scaling law with above-threshold ionization is a solid proof of the fact that the two physical processes have similar physical mechanisms. The variation of integrated harmonic yields is also discussed.     

The corrections to scaling within Mazenko’s theory in the limit of low and high dimensions

We consider corrections to scaling within an approximate theory developed by Mazenko for nonconserved order parameter in the limit of low (d → 1) and high (d → ∞) dimensions. The corrections to scaling considered here follows from the departures of the initial condition from the scaling morphology. Including corrections to scaling, the equal time correlation function has the form: C(r, t) = f ₀(r/L)+L ^−ω f ₁(r/L)+…, where L is a characteristic length scale (i.e. domain size). The correction-to-scaling exponent ω and the correction-to-scaling functions f ₁(x) are calculated for both low and high dimensions. In both dimensions the value of ω is found to be ω = 4 similar to 1D Glauber model and OJK theory (the theory developed by Ohta, Jasnow and Kawasaki).     

Poincaré–Lelong Approach to Universality and Scaling of Correlations Between Zeros

This note is concerned with the scaling limit as N→∞ of n-point correlations between zeros of random holomorphic polynomials of degree N in m variables. More generally we study correlations between zeros of holomorphic sections of powers L ^N of any positive holomorphic line bundle L over a compact K?hler manifold. Distances are rescaled so that the average density of zeros is independent of N. Our main result is that the scaling limits of the correlation functions and, more generally, of the “correlation forms” are universal, i.e. independent of the bundle L, manifold M or point on M. Received: 17 March 1999 / Accepted: 5 August 1999     

Amplification in one-dimensional random active medium near the lasing threshold

We studied the transmittance of a random amplifying medium near the lasing threshold by using the convergence criterion proposed by Nam and Zhang [Phys. Rev. B 66 73101, 2002] that allows separating the physical solutions of the time-independent Maxwell equations from the unphysical ones and describing critical size L _c of a random system in statistical terms. We found that the dependence of the critical gain ∈ ^″ _c (at which the lasing threshold occurs) as a function of number of layers is configuration-dependent and thus the lasing condition for random media is described by means of the probability of finding of physical solutions and evaluated by averaging over the ensemble of random configurations. By employing this approach we inspect the validity of the two-parameter scaling model by Zhang [Phys. Rev. B 52 7960, 1995], according to which the behavior of the random system with gain is described by relation 1/ξ = 1/ξ ₀ + 1/l _g, where ξ and ξ ₀ are localization length with and without gain, respectively, and l _g = 2/ω∈ ^″, is gain length, where ∈ ^″ is imaginary part of the dielectric constant that represents gain. We show that the range of validity of this relation depends on the ratio of both lengths and also affects the slope of the ln Λ_c(q) (where Λ_c ≡ L _c/ξ ₀ is normalized critical size and q ^?1 ≡ l _g/ξ ₀ is dimensionless gain length). We extend the study of the relation for the critical size L _c by inspecting the dependence of the slope of the ln Λ_c(q) on the strength of the randomness. We interpret its behavior in terms of the statistical properties of the localized states. Namely, by studying of the variance of the Lyapunov exponent λ (the inverse of the localization length ξ ₀) we have found that the slope of the ln Λ_c(q)) reflects the transition between two different regimes of localization with Anderson and Lifshits-like behavior that is known to be indicated by peak in var(λ). We discuss the generalization of two-parameter scaling model by implementing revisited single parameter scaling (SPS) theory by Deych et al. [Phys. Rev. Lett. 84 2678, 2000] which allows describing non-SPS regime in terms of a new scale l _s.

     

Periodic structures and scaling Laws in a magnetic colloid

Summary Phase separation was found in an eicosane-based ferrofluid in zero applied field. Upon applying a magnetic field, the condensed phase breaks up to form a lattice of periodic columns. The periodicity λ scales with the layer thicknessL. Two scaling regimes were found. When thickness is small, where no branching is observed, λ ∞L ^1/2. In the large-thickness regime, a tree type of structure develops at the tail of each column, and the columns are separated by side bands while the scaling relation crosses over to λ ∞L ^2/3. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994. this work is in collaboration with Hao Wang, Yun Zhu, C. Boyd, A. Cebers and R. E. Rosensweig.     

Fluctuation Bounds in the Exponential Bricklayers Process

This paper is the continuation of our earlier paper (Balázs et al. in Ann. Inst. Henri Poincaré Probab. Stat. 48(1):151–187, 2012), where we proved t ^1/3-order of current fluctuations across the characteristics in a class of one dimensional interacting systems with one conserved quantity. We also claimed two models with concave hydrodynamic flux which satisfied the assumptions which made our proof work. In the present note we show that the totally asymmetric exponential bricklayers process also satisfies these assumptions. Hence this is the first example with convex hydrodynamics of a model with t ^1/3-order current fluctuations across the characteristics. As such, it further supports the idea of universality regarding this scaling.     

Universal crossing probabilities and incipient spanning clusters in directed percolation

Shape-dependent universal crossing probabilities are studied, via Monte Carlo simulations, for bond and site directed percolation on the square lattice in the diagonal direction, at the percolation threshold. In a dynamical interpretation, the crossing probability is the probability that, on a system with size L, an epidemic spreading without immunization remains active at time t. Since the system is strongly anisotropic, the shape dependence in space-time enters through the effective aspect ratio r _eff = ct/L ^z, where c is a non-universal constant and z the anisotropy exponent. A particular attention is paid to the influence of the initial state on the universal behaviour of the crossing probability. Using anisotropic finite-size scaling and generalizing a simple argument given by Aizenman for isotropic percolation, we also obtain the behaviour of the probability to find n incipient spanning clusters on a finite system at time t. The numerical results are in good agreement with the conjecture. Received 10 February 2003 Published online 20 June 2003 RID="a" ID="a"e-mail: turban@lpm.u-nancy.fr RID="b" ID="b"UMR CNRS 7556