We study the directed polymer of length t in a random potential with fixed endpoints in dimension 1+1 in the continuum and on the square lattice, by analytical and numerical methods. The universal regime of high temperature T is described, upon scaling 'time' t ∼ T 5 /κ and space x = T 3 /κ (with κ = T for the discrete model) by a continuum model with δ-function disorder correlation. Using the Bethe Ansatz solution for the attractive boson problem, we obtain all positive integer moments of the partition function. The lowest cumulants of the free energy are predicted at small time and found in agreement with numerics. We then obtain the exact expression at any time for the generating function of the free energy distribution, in terms of a Fredholm determinant. At large time we find that it crosses over to the Tracy Widom distribution (TW) which describes the fixed T infinite t limit. The exact free energy distribution is obtained for any time and compared with very recent results on growth and exclusion models. PACS numbers:The directed polymer (DP) in a random potential provides the simplest example of a glass phase induced by quenched disorder [1] and has numerous applications, e.g. vortex lines [2], domain walls [3], biophysics [4]. It is closely related to much studied growth models in the KPZ class [5], such as asymmetric exclusion processes (ASEP) [6,7], and to Burgers turbulence [8]. It belongs to the broader class of disordered elastic manifolds, known to exhibit statistically scale invariant ground states. Within the functional RG (FRG) [9] these were described in a dimensional expansion by T = 0 fixed points, where the ratio temperature/disorder is irrelevant and scales with internal size with exponent −θ.Exact results were obtained in dimension d = 1 + 1 [1]. Johansson proved [10,11] that (i) the minimal energy path of length t on a square lattice with fixed endpoints has transverse roughness x ∼ t ζ with ζ = 2 3 (ii) the fluctuation of the ground state energy grows as t θ with θ = 1 3 and its scaled distribution coincides with the one of the smallest eigenvalue of a hermitian random matrix, the GUE Tracy Widom (TW) distribution [12]. The TW distribution was found in many other related models, polynuclear growth [13], TASEP [6], random subsequences [14,15] and others [16][17][18]. The unifying concept of determinantal space-time process and edge scaling was studied to account for such universality [19]. An exact result for the space-time scaling function of the two-point correlator of the height in KPZ was obtained [20].On the other hand, in d = 1 + 1 the model can be mapped onto the quantum mechanics of n attractive bosons in the limit n = 0, where t plays the role of (imaginary) time. It can be solved with the Bethe Ansatz (BA) for δ-function interactions. Until now only the ground state energy E 0 (n) was studied, i.e. the limit t → ∞ first. Pioneering attempts at its direct analytical continuation at n = 0 for a system of transverse size L = ∞ led to scaling behavior [21,22], but not to free energy...

Yield stress materials flow if a sufficiently large shear stress is applied. Although such materials are ubiquitous and relevant for industry, there is no accepted microscopic description of how they yield, even in the simplest situations in which temperature is negligible and in which flow inhomogeneities such as shear bands or fractures are absent. Here we propose a scaling description of the yielding transition in amorphous solids made of soft particles at zero temperature. Our description makes a connection between the Herschel-Bulkley exponent characterizing the singularity of the flow curve near the yield stress Σ c , the extension and duration of the avalanches of plasticity observed at threshold, and the density P(x) of soft spots, or shear transformation zones, as a function of the stress increment x beyond which they yield. We argue that the critical exponents of the yielding transition may be expressed in terms of three independent exponents, θ, d f , and z, characterizing, respectively, the density of soft spots, the fractal dimension of the avalanches, and their duration. Our description shares some similarity with the depinning transition that occurs when an elastic manifold is driven through a random potential, but also presents some striking differences. We test our arguments in an elastoplastic model, an automaton model similar to those used in depinning, but with a different interaction kernel, and find satisfying agreement with our predictions in both two and three dimensions. M any solids will flow and behave as fluids if a sufficiently large shear stress is applied. In crystals, plasticity is governed by the motion of dislocations (1, 2). In amorphous solids, there is no order, and conserved defects cannot be defined. However, as noticed by Argon (3), plasticity consists of elementary events localized in space, called shear transformations, in which a few particles rearrange. This observation supports that there are special locations in the sample, called shear transformation zones (STZs) (4), in which the system lies close to an elastic instability. Several theoretical approaches of plasticity, such as STZ theory (4) or soft glassy rheology (5), assume that such zones relax independently or are coupled to each other via an effective temperature. However, at zero temperature and small applied strain rate _ γ, computer experiments (6-11) and very recent experiments (12, 13) indicate that local rearrangements are not independent: plasticity occurs via avalanches in which many shear transformations are involved, forming elongated structures in which plasticity localizes. If conditions are such that flow is homogeneous (as may occur, for example, in foams or emulsions), one finds that the flow curves are singular at small strain rate and follow a Herschel-Bulkley law Σ − Σ c ∼ _ γ 1=β (14, 15). These features are reproduced qualitatively by elasto-plastic models (16)(17)(18)(19)(20) in which space is discretized. In these models, a site that yields plastically affects the stress in its surroundi...

Abstract. We compute the distribution of the partition functions for a class of one-dimensional Random Energy Models (REM) with logarithmically correlated random potential, above and at the glass transition temperature. The random potential sequences represent various versions of the 1/f noise generated by sampling the two-dimensional Gaussian Free Field (2dGFF) along various planar curves. Our method extends the recent analysis of [13] from the circular case to an interval and is based on an analytical continuation of the Selberg integral. In particular, we unveil a duality relation satisfied by the suitable generating function of free energy cumulants in the high-temperature phase. It reinforces the freezing scenario hypothesis for that generating function, from which we derive the distribution of extrema for the 2dGFF on the [0, 1] interval. We provide numerical checks of the circular and the interval case and discuss universality and various extensions. Relevance to the distribution of length of a segment in Liouville quantum gravity is noted.arXiv:0907.2359v2 [cond-mat.dis-nn]

The fractional Laplacian operator, −(−△) α 2 , appears in a wide class of physical systems, including Lévy flights and stochastic interfaces. In this paper, we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval. The implementation of boundary conditions is justified by appealing to two physical models, namely hopping particles and elastic springs. The eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions. Some analytical results concerning the structure of the eigenvalues spectrum are also obtained.

We combine an analytically solvable mean-field elasto-plastic model with molecular dynamics simulations of a generic glass former to demonstrate that, depending on their preparation protocol, amorphous materials can yield in two qualitatively distinct ways. We show that well-annealed systems yield in a discontinuous brittle way, as metallic and molecular glasses do. Yielding corresponds in this case to a first-order nonequilibrium phase transition. As the degree of annealing decreases, the first-order character becomes weaker and the transition terminates in a second-order critical point in the universality class of an Ising model in a random field. For even more poorly annealed systems, yielding becomes a smooth crossover, representative of the ductile rheological behavior generically observed in foams, emulsions, and colloidal glasses. Our results show that the variety of yielding behaviors found in amorphous materials does not necessarily result from the diversity of particle interactions or microscopic dynamics but is instead unified by carefully considering the role of the initial stability of the system.

In this paper, we compute the roughness exponent ζ of a long-range elastic string, at the depinning threshold, in a random medium with high precision, using a numerical method which exploits the analytic structure of the problem ('no-passing' theorem), but avoids direct simulation of the evolution equations. This roughness exponent has recently been studied by simulations, functional renormalization group calculations, and by experiments (fracture of solids, liquid meniscus in 4 He). Our result ζ = 0.390 ± 0.002 is significantly larger than what was stated in previous simulations, which were consistent with a one-loop renormalization group calculation. The data are furthermore incompatible with the experimental results for crack propagation in solids and for a 4 He contact line on a rough substrate. This implies that the experiments cannot be described by pure harmonic long-range elasticity in the quasi-static limit.The statics and dynamics of elastic manifolds in random media govern the physics of a variety of systems, ranging from vortices in type-II superconductors [1] and charge density waves [2] to interfaces in disordered magnets [3], contact lines of liquid menisci on a rough substrate [4] and to the propagation of cracks in solids [5].In most cases, the restoring elastic forces acting on a point on the manifold are local i.e. depend only on the deformation in its neighborhood. The corresponding short-range string has been the object of many theoretical and experimental studies. In the depinning limit, two different scenarios are possible: numerical simulations and analytical calculations [6,7] have established that a string with an elastic restoring force breaks at the depinning threshold, while percolation experiments and numerical studies on directed polymers in random media [8,9] agree that in those systems with stronger than harmonic restoring forces the roughness exponent is ζ = 0.63.It has also been shown [5,10] that for a contact line of a liquid meniscus or for crack propagation in a solid, the elastic force is long-range, rather than local. Nonlocal elasticity can be expected to modify the dynamic and static properties of these systems and to change the critical exponents. In this work, we compute one of these exponents, the roughness exponent ζ of a long-range elastic string at the depinning threshold f c .The theoretical approaches are up to now based on the assumption that the motion of the line at the threshold is quasi-static. This means that velocity-dependent terms in the equations of motion of the deformation field h(x, t) are taken to be irrelevant and can be derived from an energy function, which incorporates potential energy due to the driving force f and the disorder potential η(x, h), as well as an elastic energy. According to this hypothesis, the equation of motion of the deformation field at zero temperature is:The last term in this equation accounts for long-range restoring forces. Let us note that measurements of local velocities for a liquid 4 He contact line [4] have cast doub...

We study the stability of amorphous solids, focussing on the distribution P (x) of the local stress increase x that would lead to an instability. We argue that this distribution behaves as P (x) ∼ x θ , where the exponent θ is larger than zero if the elastic interaction between rearranging regions is non-monotonic, and increases with the interaction range. For a class of finite-dimensional models we show that stability implies a lower bound on θ, which is found to lie near saturation. For quadrupolar interactions these models yield θ ≈ 0.6 for d = 2 and θ ≈ 0.4 in d = 3 where d is the spatial dimension, accurately capturing previously unresolved observations in atomistic models, both in quasi-static flow and after a fast quench. In addition, we compute the Herschel-Buckley exponent in these models and show that it depends on a subtle choice of dynamical rules, whereas the exponent θ does not.

We study the dynamics of a single active Brownian particle (ABP) in two spatial dimensions. The ABP has an intrinsic time scale D −1 R set by the rotational diffusion constant DR. We show that, at short-times t D −1 R , the presence of 'activness' results in a strongly anisotropic and non-diffusive dynamics in the (xy) plane. We compute exactly the marginal distributions of the x and y position coordinates along with the radial distribution, which are all shown to be non-Brownian. In addition, we show that, at early times, the ABP has anomalous first-passage properties, characterized by non-Brownian exponents. arXiv:1804.09027v3 [cond-mat.stat-mech]