Static correlations functions and domain walls in glass-forming liquids: the case of a sandwich geometry

The problem of measuring nontrivial static correlations in deeply supercooled liquids made recently some progress thanks to the introduction of amorphous boundary conditions, in which a set of free particles is subject to the effect of a different set of particles frozen into their (low temperature) equilibrium positions. In this way, one can study the crossover from nonergodic to ergodic phase, as the size of the free region grows and the effect of the confinement fades. Such crossover defines the so-called point-to-set correlation length, which has been measured in a spherical geometry, or cavity. Here, we make further progress in the study ofcorrelations under amorphous boundary conditions by analyzing the equilibrium properties of a glass-forming liquid, confined in a planar ("sandwich") geometry. The mobile particles are subject to amorphous boundary conditions with the particles in the surrounding walls frozen into their low temperature equilibrium configurations. Compared to the cavity, the sandwich geometry has three main advantages: i) the width of the sandwich is decoupled from its longitudinal size, making the thermodynamic limit possible; ii) for very large width, the behaviour off a single wall can be studied; iii) we can use "anti-parallel" boundary conditions to force a domain wall and measure its excess energy. Our results confirm that amorphous boundary conditions are indeed a very useful new tool inthe study of static properties of glass-forming liquids, but also raise some warning about the fact that not all correlation functions that can be calculated in this framework give the same qualitative results.Comment: Submited to JCP special issue on the glass transisio

Coexistence of supersymmetric and supersymmetry-breaking states in spherical spin-glasses

The structure of states of the perturbed p-spin spherical spin-glass is analyzed. At low enough free energy metastable states have a supersymmetric structure, while at higher free energies the supersymmetry is broken. The transition between the supersymmetric and the supersymmetry-breaking phase is triggered by a change in the stability of states

The bounce of the body in hopping, running and trotting: different machines with the same motor

The bouncing mechanism of human running is characterized by a shorter duration of the brake after ‘landing’ compared with a longer duration of the push before ‘takeoff’. This landing–takeoff asymmetry has been thought to be a consequence of the force–velocity relation of the muscle, resulting in a greater force exerted during stretching after landing and a lower force developed during shortening before takeoff. However, the asymmetric lever system of the human foot during stance may also be the cause. Here, we measure the landing–takeoff asymmetry in bouncing steps of running, hopping and trotting animals using diverse lever systems. We find that the duration of the push exceeds that of the brake in all the animals, indicating that the different lever systems comply with the basic property of muscle to resist stretching with a force greater than that developed during shortening. In addition, results show both the landing–takeoff asymmetry and the mass-specific vertical stiffness to be greater in small animals than in large animals. We suggest that the landing–takeoff asymmetry is an index of a lack of elasticity, which increases with increasing the role of muscle relative to that of tendon within muscle–tendon units

Numerical simulations of liquids with amorphous boundary conditions

It has recently become clear that simulations under amorphpous boundary conditions (ABCs) can provide valuable information on the dynamics and thermodynamics of disordered systems with no obvious ordered parameter. In particular, they allow to detect a correlation length that is not measurable with standard correlation functions. Here we explain what exactly is meant by ABCs, discuss their relation with point-to-set correlations and briefly describe some recent results obtained with this technique.Comment: Presented at STATPHYS 2

Geometric approach to the dynamic glass transition

We numerically study the potential energy landscape of a fragile glassy system and find that the dynamic crossover corresponding to the glass transition is actually the effect of an underlying geometric transition caused by a qualitative change in the topological properties of the landscape. Furthermore, we show that the potential energy barriers connecting local glassy minima increase with decreasing energy of the minima, and we relate this behaviour to the fragility of the system. Finally, we analyze the real space structure of activated processes by studying the distribution of particle displacements for local minima connected by simple saddles

Numerical study of metastable states in Ising spin glasses

We study numerically the structure of metastable states in the Sherrington-Kirkpatrick spin glass. We find that all non-paramagnetic stationary points of the free energy are organized into pairs, consisting in a minimum and a saddle of order one, which coalesce in the thermodynamic limit. Within the annealed approximation, the entropic contribution of these states, that is the complexity, is compatible with the supersymmetry-breaking calculation performed in [A.J. Bray and M.A. Moore, J. Phys. C, 13 L469 (1980)]. This result indicates that the supersymmetry is spontaneously broken in the Sherrington-Kirkpatrick model

Topological Signature of First Order Phase Transitions

We show that the presence and the location of first order phase transitions in a thermodynamic system can be deduced by the study of the topology of the potential energy function, V(q), without introducing any thermodynamic measure. In particular, we present the thermodynamics of an analytically solvable mean-field model with a k-body interaction which -depending on the value of k- displays no transition (k=1), second order (k=2) or first order (k>2) phase transition. This rich behavior is quantitatively retrieved by the investigation of a topological invariant, the Euler characteristic, of some submanifolds of the configuration space. Finally, we conjecture a direct link between the Euler characteristic and the thermodynamic entropy.Comment: 6 pages, 2 figure

Spin-Glass Theory for Pedestrians

In these notes the main theoretical concepts and techniques in the field of mean-field spin-glasses are reviewed in a compact and pedagogical way, for the benefit of the graduate and undergraduate student. One particular spin-glass model is analyzed (the p-spin spherical model) by using three different approaches. Thermodynamics, covering pure states, overlaps, overlap distribution, replica symmetry breaking, and the static transition. Dynamics, covering the generating functional method, generalized Langevin equation, equations for the correlation and the response, the Mode Coupling approximation, and the dynamical transition. And finally complexity, covering the mean-field (TAP) free energy, metastable states, entropy crisis, threshold energy, and saddles. Particular attention has been paid on the mutual consistency of the results obtained from the different methods.Comment: Lecture notes of the school: "Unifying Concepts in Glassy Physics III", Bangalore, June 200

Statistical mechanics of the mixed majority-minority game with random external information

We study the asymptotic macroscopic properties of the mixed majority-minority game, modeling a population in which two types of heterogeneous adaptive agents, namely ``fundamentalists'' driven by differentiation and ``trend-followers'' driven by imitation, interact. The presence of a fraction f of trend-followers is shown to induce (a) a significant loss of informational efficiency with respect to a pure minority game (in particular, an efficient, unpredictable phase exists only for f<1/2), and (b) a catastrophic increase of global fluctuations for f>1/2. We solve the model by means of an approximate static (replica) theory and by a direct dynamical (generating functional) technique. The two approaches coincide and match numerical results convincingly.Comment: 19 pages, 3 figure