Jean-Yves Fortin

Universal fluctuations in correlated systems

The probability density function (PDF) of a global measure in a large class of highly correlated systems has been suggested to be of the same functional form. Here, we identify the analytical form of the PDF of one such measure, the order parameter in the low temperature phase of the 2D XY model. We demonstrate that this function describes the fluctuations of global quantities in other correlated equilibrium and nonequilibrium systems. These include a coupled rotor model, Ising and percolation models, models of forest fires, sandpiles, avalanches, and granular media in a self-organized critical state. We discuss the relationship with both Gaussian and extremal statistics.

Magnetic fluctuations in the classical XY model: The origin of an exponential tail in a complex system

We study the probability density function for the fluctuations of the magnetic order parameter in the low-temperature phase of the XY model of finite size. In two dimensions, this system is critical over the whole of the low-temperature phase. It is shown analytically and without recourse to the scaling hypothesis that, in this case, the distribution is non-Gaussian and of universal form, independent of both system size and critical exponent eta. An exact expression for the generating function of the distribution is obtained, which is transformed and compared with numerical data from high-resolution molecular dynamics and Monte Carlo simulations. The asymptotes of the distribution are calculated and found to be of exponential and double exponential form. The calculated distribution is fitted to three standard functions: a generalization of Gumbels first asymptote distribution from the theory of extremal statistics, a generalized log-normal distribution, and a chi(2) distribution. The calculation is extended to general dimension and an exponential tail is found in all dimensions less than 4, despite the fact that critical fluctuations are limited to D=2. These results are discussed in the light of similar behavior observed in models of interface growth and for dissipative systems driven into a nonequilibrium steady state.

Induced random fields in the LiHoxY1-xF4 quantum Ising magnet in a transverse magnetic field.

The LiHoxY1-xF4 magnetic material in a transverse magnetic field Bx x perpendicular to the Ising spin direction has long been used to study tunable quantum phase transitions in a random disordered system. We show that the Bx-induced magnetization along the x direction, combined with the local random dilution-induced destruction of crystalline symmetries, generates, via the predominant dipolar interactions between Ho3+ ions, random fields along the Ising z direction. This identifies LiHoxY1-xF4 in Bx as a new random field Ising system. The random fields explain the rapid decrease of the critical temperature in the diluted ferromagnetic regime and the smearing of the nonlinear susceptibility at the spin-glass transition with increasing Bx and render the Bx-induced quantum criticality in LiHoxY1-xF4 likely inaccessible.

Criterion for universality-class-independent critical fluctuations: Example of the two-dimensional Ising model

Order parameter fluctuations for the two-dimensional Ising model in the region of the critical temperature are presented. A locus of temperatures T(*) (L) and a locus of magnetic fields B(*) (L) are identified, for which the probability density function is similar to that for the two-dimensional XY model in the spin wave approximation. The characteristics of the fluctuations along these points are largely independent of universality class. We show that the largest range of fluctuations relative to the variance of the distribution occurs along these loci of points, rather than at the critical temperature itself and we discuss this observation in terms of intermittency. Our motivation is the identification of a generic form for fluctuations in correlated systems in accordance with recent experimental and numerical observations. We conclude that a universality-class-dependent form for the fluctuations is a particularity of critical phenomena related to the change in symmetry at a phase transition.

How skew distributions emerge in evolving systems

Despite the ubiquitous emergence of skew distributions such as power law, log-normal, and Weibull distributions, there still lacks proper understanding of the mechanism as well as relations between them. It is studied how such distributions emerge in general evolving systems and what makes the difference between them. Beginning with a master equation for general evolving systems, we obtain the time evolution equation for the size distribution function. Obtained in the case of size changes proportional to the current size are the power law stationary distribution with an arbitrary exponent and the evolving distribution, which is of either log-normal or Weibull type asymptotically, depending on production and growth in the system. This master equation approach thus gives a unified description of those three types of skew distribution observed in a variety of systems, providing physical derivation of them and disclosing how they are related.

Analytical treatment of the de Haas–van Alphen frequency combination due to chemical potential oscillations in an idealized two-band Fermi liquid

de Haas-van Alphen oscillation spectrum is studied for an idealized two-dimensional Fermi liquid with two parabolic bands in the case of canonical (fixed number of quasiparticles) and grand canonical (fixed chemical potential) ensembles. As already reported in the literature, oscillations of the chemical potential in magnetic field yield frequency combinations that are forbidden in the framework of the semiclassical theory. Exact analytical calculation of the Fourier components is derived at zero temperature and an asymptotic expansion is given for the high temperature and low magnetic field range. A good agreement is obtained between analytical formulae and numerical computations.

Emergence of skew distributions in controlled growth processes.

Starting from a master equation, we derive the evolution equation for the size distribution of elements in an evolving system, where each element can grow, divide into two, and produce new elements. We then probe general solutions of the evolution equation, to obtain such skew distributions as power-law, log-normal, and Weibull distributions, depending on the growth or division and production. Specifically, repeated production of elements of uniform size leads to power-law distributions, whereas production of elements with the size distributed according to the current distribution as well as no production of new elements results in log-normal distributions. Finally, division into two, or binary fission, bears Weibull distributions. Numerical simulations are also carried out, confirming the validity of the obtained solutions.

Alternative description of the 2D Blume–Capel model using Grassmann algebra

We use Grassmann algebra to study the phase transition in the two-dimensional ferromagnetic Blume-Capel model from a fermionic point of view. This model presents a phase diagram with a second order critical line which becomes first order through a tricritical point, and was used to model the phase transition in specific magnetic materials and liquid mixtures of He

Emergence of criticality in the transportation passenger flow: scaling and renormalization in the Seoul bus system.

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Exact correlations in the one-dimensional coagulation–diffusion process investigated by the empty-interval method

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