Difussion in a titled periodic potential: Enhancement, universality, and scaling

An exact analytical expression for the effective diffusion coefficient of an overdamped Brownian particle in a tilted periodic potential is derived for arbitrary potentials and arbitrary strengths of the thermal noise. Near the critical tilt (threshold of deterministic running solutions) a scaling behavior for weak thermal noise is revealed and various universality classes are identified. In comparison with the bare (potential-free) thermal diffusion, the effective diffusion coefficient in a critically tilted periodic potential may be, in principle, arbitrarily enhanced. For a realistic experimental setup, an enhancement by 14 orders of magnitude is predicted so that thermal diffusion should be observable on a macroscopic scale at room temperature.

Magnetic field scaling of relaxation curves in small particle systems

We study the effects of the magnetic field on the relaxation of the magnetization of smallmonodomain noninteracting particles with random orientations and distribution of anisotropyconstants. Starting from a master equation, we build up an expression for the time dependence of themagnetization which takes into account thermal activation only over barriers separating energyminima, which, in our model, can be computed exactly from analytical expressions. Numericalcalculations of the relaxation curves for different distribution widths, and under different magneticfields H and temperatures T, have been performed. We show how a T ln(t/t0) scaling of the curves,at different T and for a given H, can be carried out after proper normalization of the data to theequilibrium magnetization. The resulting master curves are shown to be closely related to what wecall effective energy barrier distributions, which, in our model, can be computed exactly fromanalytical expressions. The concept of effective distribution serves us as a basis for finding a scalingvariable to scale relaxation curves at different H and a given T, thus showing that the fielddependence of energy barriers can be also extracted from relaxation measurements.

Finite-size scaling investigation of the liquid-liquid critical point in ST2 water and its stability with respect to crystallization.

The liquid-liquid critical point scenario of water hypothesizes the existence of two metastable liq- uid phases low-density liquid (LDL) and high-density liquid (HDL) deep within the supercooled region. The hypothesis originates from computer simulations of the ST2 water model, but the stabil- ity of the LDL phase with respect to the crystal is still being debated. We simulate supercooled ST2 water at constant pressure, constant temperature, and constant number of molecules N for N ≤ 729 and times up to 1 μs. We observe clear differences between the two liquids, both structural and dynamical. Using several methods, including finite-size scaling, we confirm the presence of a liquid-liquid phase transition ending in a critical point. We find that the LDL is stable with respect to the crystal in 98% of our runs (we perform 372 runs for LDL or LDL-like states), and in 100% of our runs for the two largest system sizes (N = 512 and 729, for which we perform 136 runs for LDL or LDL-like states). In all these runs, tiny crystallites grow and then melt within 1 μs. Only for N ≤ 343 we observe six events (over 236 runs for LDL or LDL-like states) of spontaneous crystal- lization after crystallites reach an estimated critical size of about 70 ± 10 molecules.

Solid phase crystallization under continuous heating: kinetic and microstructure scaling laws

The kinetics and microstructure of solid-phase crystallization under continuous heating conditions and random distribution of nuclei are analyzed. An Arrhenius temperature dependence is assumed for both nucleation and growth rates. Under these circumstances, the system has a scaling law such that the behavior of the scaled system is independent of the heating rate. Hence, the kinetics and microstructure obtained at different heating rates differ only in time and length scaling factors. Concerning the kinetics, it is shown that the extended volume evolves with time according to αex = [exp(κCt′)]m+1, where t′ is the dimensionless time. This scaled solution not only represents a significant simplification of the system description, it also provides new tools for its analysis. For instance, it has been possible to find an analytical dependence of the final average grain size on kinetic parameters. Concerning the microstructure, the existence of a length scaling factor has allowed the grain-size distribution to be numerically calculated as a function of the kinetic parameters

Entropic transport: kinetics, scaling, and control mechanisms

We show that transport in the presence of entropic barriers exhibits peculiar characteristics which makes it distinctly different from that occurring through energy barriers. The constrained dynamics yields a scaling regime for the particle current and the diffusion coefficient in terms of the ratio between the work done to the particles and available thermal energy. This interesting property, genuine to the entropic nature of the barriers, can be utilized to effectively control transport through quasi-one-dimensional structures in which irregularities or tortuosity of the boundaries cause entropic effects. The accuracy of the kinetic description has been corroborated by simulations. Applications to different dynamic situations involving entropic barriers are outlined.

Nonlinear principal and canonical directions from continuous extensions of multidimensional scaling

A continuous random variable is expanded as a sum of a sequence of uncorrelated random variables. These variables are principal dimensions in continuous scaling on a distance function, as an extension of classic scaling on a distance matrix. For a particular distance, these dimensions are principal components. Then some properties are studied and an inequality is obtained. Diagonal expansions are considered from the same continuous scaling point of view, by means of the chi-square distance. The geometric dimension of a bivariate distribution is defined and illustrated with copulas. It is shown that the dimension can have the power of continuum.

Semidiscretization and long-time asymptotics of nonlinear diffusion equations

We review several results concerning the long time asymptotics of nonlinear diffusion models based on entropy and mass transport methods. Semidiscretization of these nonlinear diffusion models are proposed and their numerical properties analysed. We demonstrate the long time asymptotic results by numerical simulation and we discuss several open problems based on these numerical results. We show that for general nonlinear diffusion equations the long-time asymptotics can be characterized in terms of fixed points of certain maps which are contractions for the euclidean Wasserstein distance. In fact, we propose a new scaling for which we can prove that this family of fixed points converges to the Barenblatt solution for perturbations of homogeneous nonlinearities for values close to zero.

An alternative way to model merit good arguments

Besley (1988) uses a scaling approach to model merit good arguments in commodity tax policy. In this paper, I question this approach on the grounds that it produces 'wrong' recommendations--taxation (subsidisation) of merit (demerit) goods--whenever the demand for the (de)merit good is inelastic. I propose an alternative approach that does not suffer from this deficiency, and derive the ensuing first and second best tax rules, as well as the marginal cost expressions to perform tax reform analysis.

Sensitivity of simulation results to competing SAM updates

Recently there has been a renewed research interest in the properties of non survey updates of input-output tables and social accounting matrices (SAM). Along with the venerable and well known scaling RAS method, several alternative new procedures related to entropy minimization and other metrics have been suggested, tested and used in the literature. Whether these procedures will eventually substitute or merely complement the RAS approach is still an open question without a definite answer. The performance of many of the updating procedures has been tested using some kind of proximity or closeness measure to a reference input-output table or SAM. The first goal of this paper, in contrast, is the proposal of checking the operational performance of updating mechanisms by way of comparing the simulation results that ensue from adopting alternative databases for calibration of a reference applied general equilibrium model. The second goal is to introduce a new updatin! g procedure based on information retrieval principles. This new procedure is then compared as far as performance is concerned to two well-known updating approaches: RAS and cross-entropy. The rationale for the suggested cross validation is that the driving force for having more up to date databases is to be able to conduct more current, and hopefully more credible, policy analyses.

Cat(0) and Cat (-1) dimensions of torsion free

We show that a particular free-by-cyclic group has CAT(0) dimension equal to 2, but CAT(-1) dimension equal to 3. We also classify the minimal proper 2-dimensional CAT(0) actions of this group; they correspond, up to scaling, to a 1-parameter family of locally CAT(0) piecewise Euclidean metrics on a fixed presentation complex for the group. This information is used to produce an infinite family of 2-dimensional hyperbolic groups, which do not act properly by isometries on any proper CAT(0) metric space of dimension 2. This family includes a free-by-cyclic group with free kernel of rank 6.

Generalized multiresolution analyses with given multiplicity functions

Generalized multiresolution analyses are increasing sequences of subspaces of a Hilbert space H that fail to be multiresolution analyses in the sense of wavelet theory because the core subspace does not have an orthonormal basis generated by a fixed scaling function. Previous authors have studied a multiplicity function m which, loosely speaking, measures the failure of the GMRA to be an MRA. When the Hilbert space H is L2(Rn), the possible multiplicity functions have been characterized by Baggett and Merrill. Here we start with a function m satisfying a consistency condition which is known to be necessary, and build a GMRA in an abstract Hilbert space with multiplicity function m.

Point-occurrence self-similarity in crackling-noise systems and in other complex systems

It has been recently found that a number of systems displaying crackling noise also show a remarkable behavior regarding the temporal occurrence of successive events versus their size: a scaling law for the probability distributions of waiting times as a function of a minimum size is fulfilled, signaling the existence on those systems of self-similarity in time-size. This property is also present in some non-crackling systems. Here, the uncommon character of the scaling law is illustrated with simple marked renewal processes, built by definition with no correlations. Whereas processes with a finite mean waiting time do not fulfill a scaling law in general and tend towards a Poisson process in the limit of very high sizes, processes without a finite mean tend to another class of distributions, characterized by double power-law waiting-time densities. This is somehow reminiscent of the generalized central limit theorem. A model with short-range correlations is not able to escape from the attraction of those limit distributions. A discussion on open problems in the modeling of these properties is provided.