A novel approach to solve the Generalized Hydrodynamics equation
King's College London
Abstract: The generalized hydrodynamics (GHD) equation is a complicated non-linear transport equation. At first it might seem impossible to find analytical solutions to the GHD equation. In 2018 Doyon, Spohn and Yoshimura (Nuclear Physics B 926, 570–583) developed a functional equation that allows to directly find the solution of the GHD equation at arbitrarily large times $t$, without any need to solve it for times in between. I would like to present a novel approach that goes even beyond that: Given any time $t$ and any point $x$, one can write down a functional fixed point problem. The solution to this fixed point problem directly determines a solution to the GHD equation at $t$ and $x$. Furthermore, I will show that for the Lieb-Liniger model the fixed point map is contracting, which implies the existence and uniqueness of a global solution to the GHD equation.
Thermal state and thermalization of hard rods in integrability-breaking confining traps
International Centre for Theoretical Sciences, Bengaluru
Abstract: We examine the effects of integrability-breaking in a 1D gas of hard rods when confined to external harmonic and quartic traps. We first find the equilibrium properties, in particular the density profile, of the confined system. Then to understand the chaos properties and thermalization, we compute several diagnostic tools, such as the maximal Lyapunov exponent, Poincare sections, density and kinetic temperature profiles. Our numerical results reveal that the hard-rods gas in a quartic trap equilibrates to a Gibbs state, as is expected of a nonintegrable system. However, in a harmonic trap and depending on the system size, the hard rods gas shows intriguing chaos properties, ergodicity breakdown, and a lack of thermalization even for extremely long times.
Dynamical correlation of the Gibbs measure of a gas in low density scaling
Corentin Le Bihan
ENS de Lyon
Abstract: We consider a gas of N particles in a box of dimension 3, interacting pairwise with a potential $\alpha V(r/\epsilon)$. We want to understand the behavior of the system in the limit $N \to \infty$, with a suitable scaling for $\alpha$ and $\epsilon$. If we choose $\epsilon=1$, $\alpha=1/N$, it is the mean field limit: particles interact weakly at long distance. We are interested in the low density limit $\epsilon=N-1/2$, $\alpha=1$. Then the distance crossed by a particle is constant. This limit is well understood since the work of Lanford: the empirical law of the system converges to a solution of the Boltzmann equation. It is a kind of Law of Large Numbers. Sadly this convergence occurs only for a short time. In order to go to longer times, we study the fluctuations around the equilibrium, which follow a linearized version of the Boltzmann equation. The talk will present the idea of the proof of Bodineau, Gallagher, Saint Raymond and Simonella for hard sphere potentials and the idea of an improvement in case of realistic interaction potentials.
Integrable Systems and Random Matrices
The Royal Institute of Technology, Stockholm
Abstract: Computing the density of states for a given integrable system with random initial data, it is a crucial step to apply the theory of Generalized Hydrodynamics to this model. In recent years, H. Spohn was able to compute the density of states of the Toda lattice mapping this model to the Gaussian $\beta$ ensemble, a well known random matrix model.
In this contribution, I show that it is possible to apply this technique to other integrable systems and random matrix models. In particular, I show how to compute the density of states for some integrable systems when the initial data is sample according to some Generalized Gibbs ensemble by connecting these model to some well known random matrix ones.
This talk is based on the following articles:
- M. Gisonni, T. Grava, G. Gubbiotti, G. M.: Discrete integrable systems and random Lax matrices. Journal of Statistical Physics 190, Article number: 10 (2023). DOI: 10.1007/s10955-022-03024-z
- G. M., and T. Grava: Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, circular $\beta$-ensemble and double confluent Heun equation. To appear in Communication in Mathematical Physics.
- G. M., and R. Memin: Large Deviations for Ablowitz-Ladik lattice, and the Schur flow. ArXiv e-print 2201.03429 (2022).
Thermalization and hydrodynamics in an interacting integrable system:
the case of hard rods
Sahil Kumar Singh
International Centre for Theoretical Sciences, Bengaluru
Abstract: A classical Hamiltonian many-body system will generally thermalize to Gibbs Ensemble (GE) if left alone for a long time. However, there may exist systems that do not thermalize to GE, because of the existence of extra conservation laws which restrict their motion in the phase space. Thus, dynamical many-body systems can be thought of as constituting a spectrum, with systems having only Hamiltonian as the conserved quantity at one end of the spectrum, and systems having infinitely many conservation laws at the other end. The latter end consists of integrable many-body systems, which are believed to thermalize to the Generalized Gibbs Ensemble (GGE). They have a number of conservation laws equal to the number of degrees of freedom, and thus an infinity of them in the thermodynamic limit. Their non-equilibrium states close to local GGE is described by generalised hydrodynamics (GHD). In this talk, we will study thermalization to GGE of an interacting integrable system, which is that of hard rods, starting from an initial non-equilibrium state. We will also solve the GHD equations at the Euler level exactly by mapping it to a free particle Euler equation. We will compare our analytical results with those of molecular dynamics simulations.
Crossover scaling functions in the asymmetric avalanche process
Abstract: We consider the particle current in the asymmetric avalanche process on a ring. It is known to exhibit a transition from the intermittent to continuous flow at the critical density of particles. We present the exact expressions for the first two scaled cumulants of the particle current and explain how they were obtained in the large time limit $t\to\infty$ via the Bethe ansatz and a perturbative solution of the TQ-equation. The results are presented in an integral form suitable for the asymptotic analysis in the large system size limit $N\to\infty$. In the thermodynamic limit we demonstrate the scaling exponents for both the average current per site and the diffusion coefficient in subcritical, critical and supercritical regimes. Also, we identify the crossover regime and obtain the scaling functions for the uniform asymptotics unifying these three regimes. The talk is based on our joint work with Alexander Povolotsky, J. Phys. A: Math. Theor. 55 025202, arXiv: 2109.06318.
Hydrodynamic and hydrostatic limit for a generalized contact process
with mixed boundary conditions
Université Paris Cité
Abstract: We consider an interacting particle system which models the sterile insect technique. It is the superposition of a generalized contact process with exchanges of particles on a finite cylinder with open boundaries. We show that when the system is in contact with reservoirs at different slow-down rates, the hydrodynamic limit is a set of coupled non linear reaction-diffusion equations with mixed boundary conditions. We also prove the hydrostatic limit when the macroscopic equations exhibit a unique attractor.