Probabilistic models for stochastic elliptic partial differential equations

Mathematical requirements that the random coefficients of stochastic elliptical partial differential equations must satisfy such that they have unique solutions have been studied extensively. Yet, additional constraints that these coefficients must satisfy to provide realistic representations for physical quantities, referred to as physical requirements, have not been examined systematically.     

Frequency dependence of the photonic noise spectrum in an absorbing or amplifying diffusive medium

A theory is presented for the frequency dependence of the power spectrum of photon current fluctuations originating from a disordered medium. Both the cases of an absorbing medium (“grey body”) and of an amplifying medium (“random laser”) are considered in a waveguide geometry. The semiclassical approach (based on a Boltzmann-Langevin equation) is shown to be in complete agreement with a fully quantum mechanical theory, provided that the effects of wave localization can be neglected. The width of the peak in the power spectrum around zero frequency is much smaller than the inverse coherence time, characteristic for black-body radiation. Simple expressions for the shape of this peak are obtained, in the absorbing case, for waveguide lengths large compared to the absorption length, and, in the amplifying case, close to the laser threshold. Received 8 August 2000     

The Dilute Potts Model on Random Surfaces

We present a new solution of the asymmetric two-matrix model in the large-N limit which only involves a saddle point analysis. The model can be interpreted as Ising in the presence of a magnetic field, on random dynamical lattices with the topology of the sphere (resp. the disk) for closed (resp. open) surfaces; we elaborate on the resulting phase diagram. The method can be equally well applied to a more general (Q+1)-matrix model which represents the dilute Potts model on random dynamical lattices. We discuss in particular duality of boundary conditions for open random surfaces.     

Dynamical linked cluster expansions with applications to disordered systems

Dynamical linked cluster expansions are linked cluster expansions with hopping parameter terms endowed with their own dynamics. We discuss physical applications to systems with annealed and quenched disorder. Examples are the bond-diluted Ising model and the Sherrington-Kirkpatrick spin glass. We derive the rules and identify the full set of graphs that contribute to the series in the quenched case. This way it becomes possible to avoid the vague extrapolation from positive integer n to n = 0, that usually goes along with an application of the replica trick. Received 13 December 2001 Published online 25 June 2002     

Local Thermodynamic Equilibrium for some Stochastic Models of Hamiltonian Origin

We consider a class of 1-D stochastic models that are realizations of Hamiltonian models of heat conduction and prove that in the infinite volume limit local thermodynamic equilibrium is attained with linear energy profile.     

A nonextensive approach to the dynamics of financial observables

We present results about financial market observables, specifically returns and traded volumes. They are obtained within the current nonextensive statistical mechanical framework based on the entropy . More precisely, we present stochastic dynamical mechanisms which mimic probability density functions empirically observed. These mechanisms provide possible interpretations for the emergence of the entropic indices q in the time evolution of the corresponding observables. In addition to this, through multi-fractal analysis of return time series, we verify that the dual relation q_stat+q_sens=2 is numerically satisfied, q_stat and q_sens being associated to the probability density function and to the sensitivity to initial conditions respectively. This type of simple relation, whose understanding remains ellusive, has been empirically verified in various other systems.     

The Renormalization Group and Optimization of Entropy

We illustrate the possible connection that exists between the extremal properties of entropy expressions and the renormalization group (RG) approach when applied to systems with scaling symmetry. We consider three examples: (1) Gaussian fixed-point criticality in a fluid or in the capillary-wave model of an interface; (2) Lévy-like random walks with self-similar cluster formation; and (3) long-ranged bond percolation. In all cases we find a decreasing entropy function that becomes minimum under an appropriate constraint at the fixed point. We use an equivalence between random-walk distributions and order-parameter pair correlations in a simple fluid or magnet to study how the dimensional anomaly at criticality relates to walks with long-tailed distributions.     

Swollen-collapsed transition in random hetero-polymers

A lattice model of a hetero-polymer with random hydrophilic-hydrophobic charges interacting with the solvent is introduced, whose continuum counterpart has been proposed by Garel, Leibler and Orland [#!GLO!#]. The transfer matrix technique is used to study various constrained annealed systems which approximate at various degrees of accuracy the original quenched model. For highly hydrophobic chains an ordinary -point transition is found from a high temperature swollen phase to a low temperature compact phase. Depending on the type of constrained averages, at very low temperatures a swollen phase or a coexistence between compact and swollen phases are found. The results are carefully compared with the corresponding ones obtained in the continuum limit, and various improvements in the original calculations are discussed. Received: 10 April 1998 / Revised: 4 June 1998 / Accepted: 1st July 1998     

Quenched Averages for Self-Avoiding Walks and Polygons on Deterministic Fractals

We study rooted self avoiding polygons and self avoiding walks on deterministic fractal lattices of finite ramification index. Different sites on such lattices are not equivalent, and the number of rooted open walks W _n (S), and rooted self-avoiding polygons P _n (S) of n steps depend on the root S. We use exact recursion equations on the fractal to determine the generating functions for P _n (S), and W _n(S) for an arbitrary point S on the lattice. These are used to compute the averages ,, and over different positions of S. We find that the connectivity constant μ, and the radius of gyration exponent are the same for the annealed and quenched averages. However, , and , where the exponents and , take values different from the annealed case. These are expressed as the Lyapunov exponents of random product of finite-dimensional matrices. For the 3-simplex lattice, our numerical estimation gives and , to be compared with the known annealed values and .     

Effective coefficients of quasi-steady Maxwell’s equations with multiscale isotropic random conductivity

The effective coefficients in the quasi-steady Maxwell’s equations are calculated for a multiscale isotropic medium by using a subgrid modeling approach. The conductivity is mathematically represented by a Kolmogorov multiplicative continuous cascade with a lognormal probability distribution. The scale of the solution domain is assumed to be large as compared with the scale of heterogeneities of the medium. The theoretical results obtained in the paper are compared with the results of a direct 3D numerical simulation and the results of the conventional perturbation theory.