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Introduction to the dynamics of disordered systems: Equilibrium and gradient descent ; Manacorda, Alessandro ; in Physica A. Statistical Mechanics and its Applications (in press) This paper contains the lecture notes of the short courses given by one of us (F.Z.) at the summer school Fundamental Problems in Statistical Physics XV, held in Brunico, Italy, in July 2021, and, just ... [more ▼] This paper contains the lecture notes of the short courses given by one of us (F.Z.) at the summer school Fundamental Problems in Statistical Physics XV, held in Brunico, Italy, in July 2021, and, just before that, at the summer school Glassy Systems and Inter-Disciplinary Applications, held in Cargese, France, in June 2021. The course was a short introductory overview of the dynamics of disordered systems, focused in particular on the equilibrium dynamics (with the associated glass transition), and on the simplest case of off-equilibrium dynamics, namely gradient descent. A few selected topics (and references) are chosen, based on the authors’ own taste and competences, and on pedagogical reasons, without aiming at a complete review of the subject. [less ▲] Detailed reference viewed: 27 (0 UL)Gradient descent dynamics and the jamming transition in infinite dimensions Manacorda, Alessandro ; in Journal of Physics. A, Mathematical and Theoretical (2022) Gradient descent dynamics in complex energy landscapes, i.e. featuring multiple minima, finds application in many different problems, from soft matter to machine learning. Here, we analyze one of the ... [more ▼] Gradient descent dynamics in complex energy landscapes, i.e. featuring multiple minima, finds application in many different problems, from soft matter to machine learning. Here, we analyze one of the simplest examples, namely that of soft repulsive particles in the limit of infinite spatial dimension d. The gradient descent dynamics then displays a jamming transition: at low density, it reaches zero-energy states in which particles' overlaps are fully eliminated, while at high density the energy remains finite and overlaps persist. At the transition, the dynamics becomes critical. In the d → ∞ limit, a set of self-consistent dynamical equations can be derived via mean field theory. We analyze these equations and we present some partial progress towards their solution. We also study the random Lorentz gas in a range of d = 2...22, and obtain a robust estimate for the jamming transition in d → ∞. The jamming transition is analogous to the capacity transition in supervised learning, and in the appendix we discuss this analogy in the case of a simple one-layer fully-connected perceptron. [less ▲] Detailed reference viewed: 31 (1 UL)Active matter in infinite dimensions: Fokker–Planck equation and dynamical mean-field theory at low density Manacorda, Alessandro in Journal of Chemical Physics (2021) We investigate the behavior of self-propelled particles in infinite space dimensions by comparing two powerful approaches in many-body dynamics: the Fokker-Planck equation and dynamical mean-field theory ... [more ▼] We investigate the behavior of self-propelled particles in infinite space dimensions by comparing two powerful approaches in many-body dynamics: the Fokker-Planck equation and dynamical mean-field theory. The dynamics of the particles at low densities and infinite persistence time is solved in the steady-state with both methods, thereby proving the consistency of the two approaches in a paradigmatic out-of-equilibrium system. We obtain the analytic expression for the pair distribution function and the effective self-propulsion to first order in the density, confirming the results obtained in a previous paper and extending them to the case of a non-monotonous interaction potential. Furthermore, we obtain the transient behavior of active hard spheres when relaxing from equilibrium to the nonequilibrium steady-state. Our results show how collective dynamics is affected by interactions to first order in the density, and point out future directions for further analytical and numerical solutions of this problem. [less ▲] Detailed reference viewed: 69 (6 UL)Numerical solution of the dynamical mean field theory of infinite-dimensional equilibrium liquids Manacorda, Alessandro ; ; in Journal of Chemical Physics (2020) We present a numerical solution of the dynamical mean field theory of infinite-dimensional equilibrium liquids established in [Phys. Rev. Lett. 116, 015902 (2016)]. For soft sphere interactions, we obtain ... [more ▼] We present a numerical solution of the dynamical mean field theory of infinite-dimensional equilibrium liquids established in [Phys. Rev. Lett. 116, 015902 (2016)]. For soft sphere interactions, we obtain the numerical solution by an iterative algorithm and a straightforward discretization of time. We also discuss the case of hard spheres, for which we first derive analytically the dynamical mean field theory as a non-trivial limit of the soft sphere one. We present numerical results for the memory function and the mean square displacement. Our results reproduce and extend kinetic theory in the dilute or short-time limit, while they also describe dynamical arrest towards the glass phase in the dense strongly-interacting regime. [less ▲] Detailed reference viewed: 25 (1 UL)Lattice Models for Fluctuating Hydrodynamics in Granular and Active Matter Manacorda, Alessandro Book published by Springer (2018) Detailed reference viewed: 32 (0 UL)Lattice model to derive the fluctuating hydrodynamics of active particles with inertia Manacorda, Alessandro in Physical Review Letters (2017) We derive the hydrodynamic equations with fluctuating currents for the density, momentum, and energy fields for an active system in the dilute limit. In our model, nonoverdamped self-propelled particles ... [more ▼] We derive the hydrodynamic equations with fluctuating currents for the density, momentum, and energy fields for an active system in the dilute limit. In our model, nonoverdamped self-propelled particles (such as grains or birds) move on a lattice, interacting by means of aligning dissipative forces and excluded volume repulsion. Our macroscopic equations, in a specific case, reproduce a transition line from a disordered phase to a swarming phase and a linear dispersion law accounting for underdamped wave propagation. Numerical simulations up to a packing fraction ∼ 10 % are in fair agreement with the theory, including the macroscopic noise amplitudes. At a higher packing fraction, a dense-diluted coexistence emerges. We underline the analogies with the granular kinetic theories, elucidating the relation between the active swarming phase and granular shear instability. [less ▲] Detailed reference viewed: 25 (1 UL)Lattice models for granular-like velocity fields: finite-size effects ; Manacorda, Alessandro ; et al in Journal of Statistical Mechanics: Theory and Experiment (2016) Long-range spatial correlations in the velocity and energy fields of a granular fluid are discussed in the framework of a 1d lattice model. The dynamics of the velocity field occurs through nearest ... [more ▼] Long-range spatial correlations in the velocity and energy fields of a granular fluid are discussed in the framework of a 1d lattice model. The dynamics of the velocity field occurs through nearest-neighbour inelastic collisions that conserve momentum but dissipate energy. A set of equations for the fluctuating hydrodynamics of the velocity and energy mesoscopic fields give a first approximation for (i) the velocity structure factor and (ii) the finite-size correction to the Haff law, both in the homogeneous cooling regime. At a more refined level, we have derived the equations for the two-site velocity correlations and the total energy fluctuations. First, we seek a perturbative solution thereof, in powers of the inverse of system size. On the one hand, when scaled with the granular temperature, the velocity correlations tend to a stationary value in the long time limit. On the other hand, the scaled standard deviation of the total energy diverges, that is, the system shows multiscaling. Second, we find an exact solution for the velocity correlations in terms of the spectrum of eigenvalues of a certain matrix. The results of numerical simulations of the microscopic model confirm our theoretical results, including the above described multiscaling phenomenon. [less ▲] Detailed reference viewed: 24 (0 UL)Lattice models for granular-like velocity fields: hydrodynamic description Manacorda, Alessandro ; ; et al in Journal of Statistical Physics (2016) A recently introduced model describing—on a 1d lattice—the velocity field of a granular fluid is discussed in detail. The dynamics of the velocity field occurs through next-neighbours inelastic collisions ... [more ▼] A recently introduced model describing—on a 1d lattice—the velocity field of a granular fluid is discussed in detail. The dynamics of the velocity field occurs through next-neighbours inelastic collisions which conserve momentum but dissipate energy. The dynamics is described through the corresponding Master Equation for the time evolution of the probability distribution. In the continuum limit, equations for the average velocity and temperature fields with fluctuating currents are derived, which are analogous to hydrodynamic equations of granular fluids when restricted to the shear modes. Therefore, the homogeneous cooling state, with its linear instability, and other relevant regimes such as the uniform shear flow and the Couette flow states are described. The evolution in time and space of the single particle probability distribution, in all those regimes, is also discussed, showing that the local equilibrium is not valid in general. The noise for the momentum and energy currents, which are correlated, are white and Gaussian. The same is true for the noise of the energy sink, which is usually negligible [less ▲] Detailed reference viewed: 27 (0 UL)Fluctuating hydrodynamics and mesoscopic effects of spatial correlations in dissipative systems with conserved momentum ; Manacorda, Alessandro ; et al in New Journal of Physics (2015) We introduce a model described in terms of a scalar velocity field on a 1D lattice, evolving through collisions that conserve momentum but do not conserve energy. Such a system possesses some of the main ... [more ▼] We introduce a model described in terms of a scalar velocity field on a 1D lattice, evolving through collisions that conserve momentum but do not conserve energy. Such a system possesses some of the main ingredients of fluidized granular media and naturally models them. We deduce non-linear fluctuating hydrodynamics equations for the macroscopic velocity and temperature fields, which replicate the hydrodynamics of shear modes in a granular fluid. Moreover, this Landau-like fluctuating hydrodynamics predicts an essential part of the peculiar behaviour of granular fluids, like the instability of homogeneous cooling state at large size or inelasticity. We also compute the exact shape of long range spatial correlations which, even far from the instability, have the physical consequence of noticeably modifying the cooling rate. This effect, which stems from momentum conservation, has not been previously reported in the realm of granular fluids. [less ▲] Detailed reference viewed: 34 (0 UL)Coulomb Friction Driving Brownian Motors Manacorda, Alessandro ; ; in Communications in Theoretical Physics (2014) We review a family of models recently introduced to describe Brownian motors under the influence of Coulomb friction, or more general non-linear friction laws. It is known that, if the heat bath is ... [more ▼] We review a family of models recently introduced to describe Brownian motors under the influence of Coulomb friction, or more general non-linear friction laws. It is known that, if the heat bath is modeled as the usual Langevin equation (linear viscosity plus white noise), additional non-linear friction forces are not sufficient to break detailed balance, i.e. cannot produce a motor effect. We discuss two possibile mechanisms to elude this problem. A first possibility, exploited in several models inspired to recent experiments, is to replace the heat bath's white noise by a "collisional noise", that is the effect of random collisions with an external equilibrium gas of particles. A second possibility is enlarging the phase space, e.g. by adding an external potential which couples velocity to position, as in a Klein—Kramers equation. In both cases, non-linear friction becomes sufficient to achieve a non-equilibrium steady state and, in the presence of an even small spatial asymmetry, a motor effect is produced. [less ▲] Detailed reference viewed: 36 (0 UL) |
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