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arXiv2007

Fractional Diffusion Processes: Probability Distributions and Continuous Time Random Walk

A physical-mathematical approach to anomalous diffusion may be based on fractional diffusion equations and related random walk models. The fundamental solutions of these equations can be interpreted as probability densities evolving in time of peculiar self-similar stochastic processes: an integral representation of these solutions is here presented. A more general approach to anomalous diffusion is known to be provided by the master equation for a continuous time random walk (CTRW). We show how this equation reduces to our fractional diffusion equation by a properly scaled passage to the limit of compressed waiting times and jump widths. Finally, we describe a method of simulation and display (via graphics) results of a few numerical case studies.

cond-mat.stat-mech↗
arXiv2007

Hard Squares for z = -1

The hard square model in statistical mechanics has been investigated for the case when the activity z is -1. For cyclic boundary conditions, the characteristic polynomial of the transfer matrix has an intriguingly simple structure, all the eigenvalues $x$ being zero, roots of unity, or solutions of x^3 = 4 cos^2 (pi*m/N). Here we tabulate the results for lattices of up to 12 columns with cyclic or free boundary conditions and the two obvious orientations. We remark that they are all unexpectedly simple and that for the rotated lattice with free or fixed boundary conditions there are obvious likely generalizations to any lattice size.

cond-mat.stat-mech↗
arXiv2008

Density profiles in the raise and peel model with and without a wall. Physics and combinatorics

We consider the raise and peel model of a one-dimensional fluctuating interface in the presence of an attractive wall. The model can also describe a pair annihilation process in a disordered unquenched media with a source at one end of the system. For the stationary states, several density profiles are studied using Monte Carlo simulations. We point out a deep connection between some profiles seen in the presence of the wall and in its absence. Our results are discussed in the context of conformal invariance ($c = 0$ theory). We discover some unexpected values for the critical exponents, which were obtained using combinatorial methods. We have solved known (Pascal's hexagon) and new (split-hexagon) bilinear recurrence relations. The solutions of these equations are interesting on their own since they give information on certain classes of alternating sign matrices.

cond-mat.stat-mech↗
arXiv2008

Kinetic Equationins in the Theory of Normal Fermi Liquid

On the bases of the improved approximation for the spectral function of one-particle states the Landau-Silin kinetic equations for the normal Fermi liquids of neutral and electrically charged particles are shown to be valid at finite temperature above the temperature of superfluid transition.

cond-mat.stat-mech↗
arXiv2007

The Student ensemble of correlation matrices: eigenvalue spectrum and Kullback-Leibler entropy

We study a new ensemble of random correlation matrices related to multivariate Student (or more generally elliptic) random variables. We establish the exact density of states of empirical correlation matrices that generalizes the Marcenko-Pastur result. The comparison between the theoretical density of states in the Student case and empirical financial data is surprisingly good, even if we are still able to detect systematic deviations. Finally, we compute explicitely the Kullback-Leibler entropies of empirical Student matrices, which are found to be independent of the true correlation matrix, as in the Gaussian case. We provide numerically exact values for these Kullback-Leibler entropies.

q-fin.ST↗
arXiv2007

Grain Boundary Diffusion in a Peierls-Nabarro Potential

We investigate the diffusion of a grain boundary in a crystalline material. We consider in particular the case of a regularly spaced low-angle grain boundary schematized as an array of dislocations that interact with each other through long-range stress fields and with the crystalline Peierls-Nabarro potential. The methodology employed to analyze the dynamics of the center of mass of the grain boundary and its spatio-temporal fluctuations is based on over-damped Langevin equations. The generality and the efficiency of this technique is proved by the agreement with molecular dynamics simulations.

cond-mat.stat-mech↗
arXiv2012

Topological phase transition in complex networks

Preferential attachment is a central paradigm in the theory of complex networks. In this contribution we consider various generalizations of preferential attachment including for example node removal and edge rewiring. We demonstrate that generalized preferential attachment networks can undergo a topological phase transition. This transition separates networks having a power-law tail degree distribution from those with an exponential tail. The appearance of the phase transition is closely related to the breakdown of the continuous variable description of the network dynamics.

cond-mat.stat-mech↗
arXiv2007

q-deformed statistical-mechanical structure in the dynamics of the Feigenbaum attractor

We show that the two complementary parts of the dynamics associated to the Feigenbaum attractor, inside and towards the attractor, form together a q -deformed statistical-mechanical structure. A time-dependent partition function produced by summing distances between neighboring positions of the attractor leads to a q-entropy that measures the fraction of ensemble trajectories still away at a given time from the attractor (and the repellor). The values of the q-indexes are given by the attractor's universal constants, while the thermodynamic framework is closely related to that first developed for multifractals.

cond-mat.stat-mech↗
arXiv2007

Dynamically correlated regions and configurational entropy in supercooled liquids

When a liquid is cooled below its melting temperature, if crystallization is avoided, it forms a glass. This phenomenon, called glass transition, is characterized by a marked increase of viscosity, about 14 orders of magnitude, in a narrow temperature interval. The microscopic mechanism behind the glass transition is still poorly understood. However, recently, great advances have been made in the identification of cooperative rearranging regions, or dynamical heterogeneities, i.e. domains of the liquid whose relaxation is highly correlated. The growth of the size of these domains is now believed to be the driving mechanism for the increase of the viscosity. Recently a tool to quantify the size of these domains has been proposed. We apply this tool to a wide class of materials to investigate the correlation between the size of the heterogeneities and their configurational entropy, i.e. the number of states accessible to a correlated domain. We find that the relaxation time of a given system, apart from a material dependent pre-factor, is a universal function of the configurational entropy of a correlated domain. As a consequence, we find that at the glass transition temperature, the size of the domains and the configurational entropy per unit volume are anti-correlated, as originally predicted by the Adam-Gibbs theory. Finally, we use our data to extract some exponents defined in the framework of the Random First Order Theory, a recent quantitative theory of the glass transition.

cond-mat.dis-nn↗
arXiv2007

Percolation in the Sherrington-Kirkpatrick Spin Glass

We present extended versions and give detailed proofs of results concerning percolation (using various sets of two-replica bond occupation variables) in Sherrington-Kirkpatrick spin glasses (with zero external field) that were first given in an earlier paper by the same authors. We also explain how ultrametricity is manifested by the densities of large percolating clusters. Our main theorems concern the connection between these densities and the usual spin overlap distribution. Their corollaries are that the ordered spin glass phase is characterized by a unique percolating cluster of maximal density (normally coexisting with a second cluster of nonzero but lower density). The proofs involve comparison inequalities between SK multireplica bond occupation variables and the independent variables of standard Erdos-Renyi random graphs.

cond-mat.dis-nn↗
arXiv2007

Solving Infinite Kolam in Knot Theory

In south India, there are traditional patterns of line-drawings encircling dots, called ``Kolam'', among which one-line drawings or the ``infinite Kolam'' provide very interesting questions in mathematics. For example, we address the following simple question: how many patterns of infinite Kolam can we draw for a given grid pattern of dots? The simplest way is to draw possible patterns of Kolam while judging if it is infinite Kolam. Such a search problem seems to be NP complete. However, it is certainly not. In this paper, we focus on diamond-shaped grid patterns of dots, (1-3-5-3-1) and (1-3-5-7-5-3-1) in particular. By using the knot-theory description of the infinite Kolam, we show how to find the solution, which inevitably gives a sketch of the proof for the statement ``infinite Kolam is not NP complete.'' Its further discussion will be given in the final section.

cs.DM↗
arXiv2008

On the Equivalent Theory of the Generalized τ^{(2)}-model and the Chiral Potts Model with two Alternating Vertical Rapidities

By the Baxter's $Q_{72}$-operator method, we demonstrate the equivalent theory between the generalized $τ^{(2)}$-model (other than two special cases with a pseudovacuum state) and the $N$-state chiral Potts model with two alternating vertical rapidities, where the degenerate models are included. As a consequence, the theory of the XXZ chain model associated to cyclic representations (with the parameter $ς$) of $U_{\sf q}(sl_2)$ with ${\sf q}^N=1$ for odd $N$ is identified with either (for $ς^N=1$) the chiral Potts model with two superintegrable vertical rapidities, or (for $ς^N \neq 1$) the degenerate model for the selfdual solution of the star-triangle relation. In all these identifications, the transfer matrices $T, \hat{T}$ of the chiral Potts model (including the degenerate ones) serve as the $Q_R, Q_L$-operators of the corresponding $τ^{(2)}$-model, so that the functional relations hold as in the solvable $N$-state chiral Potts model.

cond-mat.stat-mech↗
arXiv2008

Does a particle swept by a turbulent liquid diffuse?

Since the famous 1926 paper by Richardson, the relative diffusion of two particles in a turbulent liquid has attracted a lot of interest. The motion of a single particle on the other hand is usually considered not to be especially interesting. The widely accepted picture is that the velocity of the particle has short-range correlations in time, resulting in motion that is diffusive on time scales large compared to the correlation time. We find, however, that the correlation time is infinite and that the square displacement, F, is not linear in the traversed time, T, which would correspond to diffusion, but rather F is proportional to T^6/5. Namely, the motion is slightly super diffusive.

cond-mat.stat-mech↗
arXiv2007

On the effects of conformal degrees of freedom inside a neutron star

In this paper a neutron star with an inner core which undergoes a phase transition, which is characterized by conformal degrees of freedom on the phase boundary, is considered. Typical cases of such a phase transition are e.g. quantum Hall effect, superconductivity and superfluidity. Assuming the mechanical stability of this system the effects induced by the conformal degrees of freedom on the phase boundary will be analyzed. We will see that the inclusion of conformal degrees of freedom is not always consistent with the staticity of the phase boundary. Indeed also in the case of mechanical equilibrium there may be the tendency of one phase to swallow the other. Such a shift of the phase boundary would not imply any compression or decompression of the core. By solving the Israel junction conditions for the conformal matter, we have found the range of physical parameters which can guarantee a stable equilibrium of the phase boundary of the neutron star. The relevant parameters turn out to be not only the density difference but also the difference of the slope of the density profiles of the two phases. The values of the parameters which guarantee the stability turn out to be in a phenomenologically reasonable range. For the parameter values where the the phase boundary tends to move, a possible astrophysical consequence related to sudden small changes of the moment of inertia of the star is briefly discussed.

gr-qc↗
arXiv2007

Continuous Abelian Sandpile Model in Two Dimensional Lattice

We define a new version of sandpile model which is very similar to Abelian Sandpile Model (ASM), but the height variables are continuous ones. With the toppling rule we define in our model, we show that the model can be mapped to ASM, so the general properties of the two models are identical. Yet the new model allows us to investigate some problems such as effect of very small mass on the height probabilities, different boundary conditions, etc.

cond-mat.stat-mech↗
arXiv2008

Local power fluctuations in two-dimensional turbulence

The statistics of power fluctuations are studied in simulations of two-dimensional turbulence in both inverse (energy) and direct (enstrophy) cascade regimes from both Lagrangian and Eulerian perspectives. The probability density function (PDF) of the appropriately defined dimensionless power is strongly non-gaussian with asymmetric exponential tails. This distribution can be modeled by the distribution of the product of correlated normal variables allowing a derivation of the asymptotics of the tails. The PDF of the dimensionless power is shown to exhibit an empirical Fluctuation Relation. An expression for the entropy production rate is deduced from the asymptotic form of the power PDF and is found to agree very well with the measured entropy rate.

cond-mat.stat-mech↗
arXiv2007

Inefficient quantum walks on networks: the role of the density of states

We show by general arguments that networks whose density of states contains few highly degenerate eigenvalues result in inefficient performances of continuous-time quantum walks (CTQW) over these networks, while systems whose eigenvalues all have the same degeneracy lead to very efficient transport. We exemplify our results by considering CTQW and, for comparison, its classical counterpart, continuous-time random walks, over simple structures, whose eigenvalues and eigenstates can be calculated analytically. Extensions to more complicated, hyper-branched networks are discussed.

quant-ph↗