Table of contents

For continuous phase transitions characterized by power-law divergences, Fisher renormalization prescribes how to obtain the critical exponents for a system under constraint from their ideal counterparts. In statistical mechanics, such ideal behaviour at phase transitions is frequently modified by multiplicative logarithmic corrections. Here, Fisher renormalization for the exponents of these logarithms is developed in a general manner. As for the leading exponents, Fisher renormalization at the logarithmic level is seen to be involutory and the renormalized exponents obey the same scaling relations as their ideal analogues. The scheme is tested in lattice animals and the Yang–Lee problem at their upper critical dimensions, where predictions for logarithmic corrections are made.

We study a gas of hard rods on a ring, driven by an external thermostat, with either elastic or inelastic collisions, which exhibits sub-diffusive behavior, . We show the validity of the usual fluctuation–dissipation (FD) relation, i.e. the proportionality between the response function and the correlation function, when the gas is elastic or diluted. In contrast, in strongly inelastic or dense cases, when the tracer velocity is no longer independent of the other degrees of freedom, the Einstein formula fails and must be replaced by a more general FD relation.

In this paper, we will discuss how to compactly express the Jarzynski identity for an open quantum system with dissipative dynamics. In quantum dynamics we must avoid explicitly measuring the work directly, which is tantamount to continuously monitoring the state of the system, and instead measure the heat flow from the environment. These measurements can be concisely represented with Hermitian map superoperators, which provide convenient and compact representations of correlation functions and sequential measurements of quantum systems.

We study experimentally the paths of an assembly of cracks growing in interaction in a heterogeneous two-dimensional elastic brittle material submitted to uniaxial stress. For a given initial crack assembly geometry, we observe two types of crack path. The first one corresponds to a repulsion followed by an attraction on one end of the crack and a tip-to-tip attraction on the other end. The second one corresponds to a pure attraction. Only one of the crack path types is observed in a given sample. Thus, selection between the two types appears as a statistical collective process.

A theoretical description of time correlation functions for electron properties in the presence of a positive ion of charge number Z is given. The simplest case of an electron gas distorted by a single ion is considered. A semi-classical representation with a regularized electron–ion potential is used to obtain a linear kinetic theory that is asymptotically exact at short times. This Markovian approximation includes all initial (equilibrium) electron–electron and electron–ion correlations through renormalized pair potentials. The kinetic theory is solved in terms of single-particle trajectories of the electron–ion potential and a dielectric function for the inhomogeneous electron gas. The results are illustrated by a calculation of the autocorrelation function for the electron field at the ion. The dependence on charge number Z is shown to be dominated by the bound states of the effective electron–ion potential. On this basis, a very simple practical representation of the trajectories is proposed and shown to be accurate over a wide range including strong electron–ion coupling. This simple representation is then used for a brief analysis of the dielectric function for the inhomogeneous electron gas.

In this work we investigate the stability of synchronized states for the Kuramoto model on scale-free (SF) and Erdös–Rényi (ER) random networks in the presence of white noise forcing. We show that for a fixed coupling constant, the robustness of the globally synchronized state against the noise depends on the noise intensity on both kinds of networks. At low noise intensities ER networks are more robust against losing the coherency but upon increasing the noise, synchronization among the population vanishes suddenly at a specific noise strength. In contrast, on SF networks the global synchronization disappears continuously at a much larger critical noise intensity with respect to ER networks.

We investigate the performance of the recently proposed stationary Fokker–Planck sampling method, considering a combinatorial optimization problem from statistical physics. The algorithmic procedure relies upon the numerical solution of a linear second-order differential equation that depends on a diffusion-like parameter D. We apply it to the problem of finding ground states of 2D Ising spin glasses for the ± J model. We consider square lattices with side length up to L = 24 with boundary conditions of two different types and compare the results to those obtained by exact methods. A particular value of D is found that yields an optimal performance of the algorithm. We compare the situation with this optimal value of D to the case of a percolation transition, which is found when studying the connected clusters of spins flipped by the algorithm. However, even for moderate lattice sizes, it becomes more and more difficult to find the exact ground states with the algorithm. This means that the approach, at least in its standard form, seems to be inferior to other approaches like parallel tempering.

We derive critical noise levels for Gallager codes on asymmetric channels as a function of the input bias and the temperature. Using a statistical mechanics approach we study the space of codewords and the entropy in the various decoding regimes. We further discuss the relation of the convergence of the message passing algorithm with the endogenous property and complexity, characterizing solutions of recursive equations of distributions for cavity fields.

We study experimentally and theoretically the probability density functions of the injected and dissipated energy in a system of a colloidal particle trapped in a double-well potential periodically modulated by an external perturbation. The work done by the external force and the dissipated energy are measured close to the stochastic resonance where the injected power is maximum. We show a good agreement between the probability density functions exactly computed from a Langevin dynamics and the measured ones. The probability density function of the work done on the particle satisfies the fluctuation theorem.

It is well known that an arbitrary graphical model of statistical inference defined on a tree, i.e. on a graph without loops, is solved exactly and efficiently by an iterative belief propagation (BP) algorithm convergent to the unique minimum of the so-called Bethe free energy functional. For a general graphical model on a loopy graph, the functional may show multiple minima, the iterative BP algorithm may converge to one of the minima or may not converge at all, and the global minimum of the Bethe free energy functional is not guaranteed to correspond to the optimal maximum likelihood (ML) solution in the zero-temperature limit. However, there are exceptions to this general rule, discussed by Kolmogorov and Wainwright (2005) and by Bayati et al (2006, 2008) in two different contexts, where the zero-temperature version of the BP algorithm finds the ML solution for special models on graphs with loops. These two models share a key feature: their ML solutions can be found by an efficient linear programming (LP) algorithm with a totally uni-modular (TUM) matrix of constraints. Generalizing the two models, we consider a class of graphical models reducible in the zero-temperature limit to LP with TUM constraints. Assuming that a gedanken algorithm, g-BP, for finding the global minimum of the Bethe free energy is available, we show that in the limit of zero temperature, g-BP outputs the ML solution. Our consideration is based on equivalence established between gapless linear programming (LP) relaxation of the graphical model in the T → 0 limit and the respective LP version of the Bethe free energy minimization.

In this work we have developed an appropriate and novel model for computing the contribution of all traps at the Si–SiO₂ interface to the low frequency (LF) noise power spectrum. In our approach we have used theoretical and experimental convex 'U'-shape curves to model the density of states of traps distributed at the interface, predicting the qualitative and quantitative behavior for the reduction of LF noise. In the model development, basic discrete device physics quantities are used. The low frequency noise behavior of metal–oxide–semiconductor devices under cyclo-stationary excitation is evaluated through analytical and numerical calculations, and a comparison to relevant experimental data found in the literature is provided.

We study an agent-based model, as a special type of opinion dynamics, of the spreading of innovations in socio-economic systems varying the topology of agents' social contacts. The agents are organized on a square lattice where the connections are rewired with a certain probability. We show that the degree polydispersity and long range connections of agents can facilitate, but can also hinder the spreading of new technologies, depending on the amount of advantages provided by the innovation. We determine the critical fraction of innovative agents required to initiate spreading and to obtain a significant technological progress. As the fraction of innovative agents approaches the critical value, the spreading process slows down analogously to the critical slowing down observed at continuous phase transitions. The characteristic timescale at the critical point proved to have the same scaling as the average shortest path of the underlying social network. The model captures some relevant features of the spreading of innovations in telecommunication technologies.

I investigate a disordered version of a simplified model of protein folding, with binary degrees of freedom, applied to an ideal β-hairpin structure. Disorder is introduced by assuming that the contact energies are independent and identically distributed random variables. The equilibrium free energy of the model is studied, by performing the exact calculation of its quenched value and proving the self-averaging feature.

We develop front spreading models for several jump distance probability distributions (dispersion kernels). We derive expressions for a cohabitation model (cohabitation of parents and children) and a non-cohabitation model, and apply them to the Neolithic using data from real human populations. The speeds that we obtain are consistent with observations of the Neolithic transition. The correction due to the cohabitation effect is up to 38%.

The stochastic motion of an arbitrarily shaped rotor, free to rotate around a fixed axis as a result of dissipative collisions with a surrounding thermalized gas, is investigated. A Boltzmann master equation is derived, starting from the elementary gas–rotor collisions. Analytical expressions for the moments of the rotational speed and the rotational temperature are obtained in the form of a series expansion, using the mass ratio of the gas particle and the rotor as the expansion parameter. We discuss the general features of the motion, which is largely determined by the geometry of the rotor. In particular, a stationary non-zero rotation is observed for rotationally asymmetric rotors. Molecular dynamics simulations support the analytical results.

Let n points be chosen randomly and independently in the unit disk. 'Sylvester's question' concerns the probability p_n that they are the vertices of a convex n-sided polygon. Here we establish the link with another problem. We show that for large n this polygon, when suitably parameterized by a function r(ϕ) of the polar angle ϕ, satisfies the equation of the random acceleration process (RAP), d²r/dϕ² = f(ϕ), where f is Gaussian noise. On the basis of this relation we derive the asymptotic expansion log p_n = –2n log n + n log(2π²e²)–c₀n^1/5 + · · ·, of which the first two terms agree with a rigorous result due to Bárány. The non-analyticity in n of the third term is a new result. The value of the exponent follows from recent work on the RAP due to Györgyi et al (2007 Phys. Rev. E 75 021123). We show that the n-sided polygon is effectively contained in an annulus of width ∼n^−4/5 along the edge of the disk. The distance δ_n of closest approach to the edge is exponentially distributed with average (2n)⁻¹.

We consider a system of two-parameter deformed boson oscillators whose spectrum is given by a generalized Fibonacci sequence. In order to establish the role of the deformation parameters (q₁,q₂) in the thermostatistics of the system, we calculate several thermostatistical functions in the thermodynamical limit and investigate the low temperature behavior of the system. In this framework, we show that the thermostatistics of the (q₁,q₂)-bosons can be studied using the formalism of Fibonacci calculus which generalizes the recently proposed formalism of q-calculus. We also discuss the conditions under which the Bose–Einstein condensation would occur in the present two-parameter generalized boson gas. However, the ordinary boson gas results can be obtained by applying the limit q₁ = q₂ = 1.

We propose a simple method to extract the community structure of large networks. Our method is a heuristic method that is based on modularity optimization. It is shown to outperform all other known community detection methods in terms of computation time. Moreover, the quality of the communities detected is very good, as measured by the so-called modularity. This is shown first by identifying language communities in a Belgian mobile phone network of 2 million customers and by analysing a web graph of 118 million nodes and more than one billion links. The accuracy of our algorithm is also verified on ad hoc modular networks.

This paper presents a new formulation of the equilibrium and non-equilibrium statistical mechanical properties of a gas consisting of free particles and independent correlated pairs with any angular momentum , interacting via a non-local separable potential, in terms of the partial-wave scattering amplitude. Analytical expressions for some equilibrium thermodynamic properties, such as the Helmholtz free energy and second virial coefficient, as well as some non-equilibrium thermophysical properties, such as collision cross sections and transport collision integrals, are described.

A harmonic oscillator that evolves under the action of both a systematic time-dependent force and a random time-correlated force can do work w. This work is a random quantity, and Mai and Dhar have recently shown, using the generalized Langevin equation (GLE) for the oscillator's position x, that it satisfies a fluctuation theorem. In principle, the same result could have been derived from the Fokker–Planck equation (FPE) for the probability density function, P(x,w,t), for the oscillator being at x at time t, having done work w. Although the FPE equivalent to the above GLE is easily constructed and solved, one finds, unexpectedly, that its predictions for the mean and variance of w do not agree with the fluctuation theorem. We show that to resolve this contradiction, it is necessary to construct an FPE that includes the velocity of the oscillator, v, as an additional variable. The FPE for P(x,v,w,t) does indeed yield expressions for the mean and variance of w that agree with the fluctuation theorem.

The distribution of the ground-state energy of the Sherrington–Kirkpatrick model is studied numerically. The standard deviation scales with the number of spins as N^−ρ, with ρ = 5/6 favoured over the alternative theoretical value ρ = 3/4. The asymptotic behaviour of the probability distribution function is well described by a Gumbel distribution.

We compute the probability distribution of the number of metastable states at a given applied field in the mean-field random-field Ising model at T = 0. Remarkably, there is a non-zero probability in the thermodynamic limit of observing metastable states on the so-called 'unstable' branch of the magnetization curve. This implies that the branch can be reached when the magnetization is controlled instead of the magnetic field, in contrast to the situation for the pure system.

We perform a detailed study of the critical behavior of the mean field diluted Ising ferromagnet using analytical and numerical tools. We obtain self-averaging for the magnetization and write down an expansion for the free energy close to the critical line. The scaling of the magnetization is also rigorously obtained and compared with extensive Monte Carlo simulations. We explain the transition from an ergodic region to a non-trivial phase by commutativity breaking of the infinite volume limit and a suitable vanishing field. We find full agreement among theory, simulations and previous results.

We analyse a biased random walk on a 1D lattice with unequal step lengths. Such a walk was recently shown to undergo a phase transition from a state containing a single connected cluster of visited sites to one with several clusters of visited sites (fragments) separated by unvisited sites at a critical probability p_c (Anteneodo and Morgado 2007 Phys. Rev. Lett. 99 180602). The behaviour of ρ(l), the probability of formation of fragments of length l, is analysed. An exact expression for the generating function of ρ(l) at the critical point is derived. We prove that the asymptotic behaviour is of the form .

We establish the Bethe equation of the τ⁽²⁾ model in the N-state chiral Potts model (including the degenerate self-dual cases) with alternating vertical rapidities. The eigenvalues of a finite-size transfer matrix of the chiral Potts model are computed by use of functional relations. The significance of the 'alternating superintegrable' case of the chiral Potts model is discussed, and the degeneracy of the τ⁽²⁾ model found as in the homogeneous superintegrable chiral Potts model.

Commentary on `Universal distribution of random matrix eigenvalues near the ``birth of a cut'' transition', by B Eynard, 2006 J. Stat. Mech.P07005.