Lecture notes on Generalised Hydrodynamics
TL;DR: The main aspects of generalised hydrodynamics, a hydrodynamic theory for quantum and classical many-body integrable systems, were discussed in a series of lectures at the Les Houches Summer School on Integrability in Atomic and Condensed Matter Physics as mentioned in this paper.
Abstract: These are lecture notes for a series of lectures given at the Les Houches Summer School on Integrability in Atomic and Condensed Matter Physics, 30 July to 24 August 2018. The same series of lectures has also been given at the Tokyo Institute of Technology, October 2019. I overview in a pedagogical fashion the main aspects of the theory of generalised hydrodynamics, a hydrodynamic theory for quantum and classical many-body integrable systems. Only very basic knowledge of hydrodynamics and integrable systems is assumed.
Citations
More filters
Journal Article•
TL;DR: In this paper, the authors show that a homogeneous 1D Bose gas with point-like collisional interactions is integrable, and that it is possible to construct a system with many degrees of freedom that does not reach thermal equilibrium even after thousands of collisions.
Abstract: It is a fundamental assumption of statistical mechanics that a closed system with many degrees of freedom ergodically samples all equal energy points in phase space. To understand the limits of this assumption, it is important to find and study systems that are not ergodic, and thus do not reach thermal equilibrium. A few complex systems have been proposed that are expected not to thermalize because their dynamics are integrable. Some nearly integrable systems of many particles have been studied numerically, and shown not to ergodically sample phase space. However, there has been no experimental demonstration of such a system with many degrees of freedom that does not approach thermal equilibrium. Here we report the preparation of out-of-equilibrium arrays of trapped one-dimensional (1D) Bose gases, each containing from 40 to 250 87Rb atoms, which do not noticeably equilibrate even after thousands of collisions. Our results are probably explainable by the well-known fact that a homogeneous 1D Bose gas with point-like collisional interactions is integrable. Until now, however, the time evolution of out-of-equilibrium 1D Bose gases has been a theoretically unsettled issue, as practical factors such as harmonic trapping and imperfectly point-like interactions may compromise integrability. The absence of damping in 1D Bose gases may lead to potential applications in force sensing and atom interferometry.
941 citations
TL;DR: In this article , the authors present a new playground for quantum many-body physics and a tractable setting to explore universal collective phenomena far from equilibrium. But their model is not suitable for quantum information and entanglement.
Abstract: Quantum circuits—built from local unitary gates and local measurements—are a new playground for quantum many-body physics and a tractable setting to explore universal collective phenomena far from equilibrium. These models have shed light on longstanding questions about thermalization and chaos, and on the underlying universal dynamics of quantum information and entanglement. In addition, such models generate new sets of questions and give rise to phenomena with no traditional analog, such as dynamical phase transitions in quantum systems that are monitored by an external observer. Quantum circuit dynamics is also topical in view of experimental progress in building digital quantum simulators that allow control of precisely these ingredients. Randomness in the circuit elements allows a high level of theoretical control, with a key theme being mappings between real-time quantum dynamics and effective classical lattice models or dynamical processes. Many of the universal phenomena that can be identified in this tractable setting apply to much wider classes of more structured many-body dynamics. Expected final online publication date for the Annual Review of Condensed Matter Physics, Volume 14 is March 2023. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
69 citations
TL;DR: It is established that the collision rate ansatz follows because (i) the charge-current susceptibility matrix is symmetric and (ii) the stretch current is proportional to the momentum, hence conserved.
Abstract: We consider a generalized Gibbs ensemble of the classical Toda lattice. We establish that the collision rate ansatz follows because (i) the charge-current susceptibility matrix is symmetric and (ii) the stretch current is proportional to the momentum, hence conserved. The method applies also to other integrable many-body systems, either classical or quantum, provided there is a self-conserved current.
37 citations
TL;DR: In this article, the generalized Gibbs ensemble of BBS soliton gas by thermodynamic Bethe ansatz and generalized hydrodynamics is studied and the results include the solution to the speed equation for solitons, an intriguing connection of the effective speed with the period matrix of the tropical Riemann theta function, an explicit description of the density plateaux that emerge from domain wall initial conditions including their diffusive corrections.
Abstract: Box-ball system (BBS) is a prominent example of integrable cellular automata in one dimension connected to quantum groups, Bethe ansatz, ultradiscretization, tropical geometry and so forth. In this paper we study the generalized Gibbs ensemble of BBS soliton gas by thermodynamic Bethe ansatz and generalized hydrodynamics. The results include the solution to the speed equation for solitons, an intriguing connection of the effective speed with the period matrix of the tropical Riemann theta function, an explicit description of the density plateaux that emerge from domain wall initial conditions including their diffusive corrections.
10 citations
References
More filters
TL;DR: In this article, the authors consider statistical mechanics as a form of statistical inference rather than as a physical theory, and show that the usual computational rules, starting with the determination of the partition function, are an immediate consequence of the maximum-entropy principle.
Abstract: Information theory provides a constructive criterion for setting up probability distributions on the basis of partial knowledge, and leads to a type of statistical inference which is called the maximum-entropy estimate. It is the least biased estimate possible on the given information; i.e., it is maximally noncommittal with regard to missing information. If one considers statistical mechanics as a form of statistical inference rather than as a physical theory, it is found that the usual computational rules, starting with the determination of the partition function, are an immediate consequence of the maximum-entropy principle. In the resulting "subjective statistical mechanics," the usual rules are thus justified independently of any physical argument, and in particular independently of experimental verification; whether or not the results agree with experiment, they still represent the best estimates that could have been made on the basis of the information available.It is concluded that statistical mechanics need not be regarded as a physical theory dependent for its validity on the truth of additional assumptions not contained in the laws of mechanics (such as ergodicity, metric transitivity, equal a priori probabilities, etc.). Furthermore, it is possible to maintain a sharp distinction between its physical and statistical aspects. The former consists only of the correct enumeration of the states of a system and their properties; the latter is a straightforward example of statistical inference.
12,099 citations
TL;DR: In this paper, an expression for the equilibrium free energy difference between two configurations of a system, in terms of an ensemble of finite-time measurements of the work performed in parametrically switching from one configuration to the other, is derived.
Abstract: An expression is derived for the equilibrium free energy difference between two configurations of a system, in terms of an ensemble of finite-time measurements of the work performed in parametrically switching from one configuration to the other. Two well-known identities emerge as limiting cases of this result.
4,496 citations
3,461 citations
IBM1
TL;DR: In this paper, the ground-state energy as a function of γ was derived for all γ, except γ = 0, and it was shown that Bogoliubov's perturbation theory is valid when γ is small.
Abstract: A gas of one-dimensional Bose particles interacting via a repulsive delta-function potential has been solved exactly. All the eigenfunctions can be found explicitly and the energies are given by the solutions of a transcendental equation. The problem has one nontrivial coupling constant, γ. When γ is small, Bogoliubov’s perturbation theory is seen to be valid. In this paper, we explicitly calculate the ground-state energy as a function of γ and show that it is analytic for all γ, except γ=0. In Part II, we discuss the excitation spectrum and show that it is most convenient to regard it as a double spectrum—not one as is ordinarily supposed.
2,230 citations