Simulating Coulomb and Log-Gases with Hybrid Monte Carlo Algorithms

Chafäı, Djalil; Ferré, Grégoire

doi:10.1007/s10955-018-2195-6

Cited by 12 publications

(22 citation statements)

References 78 publications

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“…The Kolmogorov-Smirnov distances [35] between the empirical distributions of the spectrum of L, and each of these curves (after fitting β) are listed in Table I. The spacings for the Coulomb gas are obtained by generating points with the distribution (4) by using the Metropolis algorithm, following [36], and then determining the spacing numerically. Fig.…”

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confidence: 99%

Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems

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We study the transition between integrable and chaotic behaviour in dissipative open quantum systems, exemplified by a boundary driven quantum spin-chain. The repulsion between the complex eigenvalues of the corresponding Liouville operator in radial distance s is used as a universal measure. The corresponding level spacing distribution is well fitted by that of a static two-dimensional Coulomb gas with harmonic potential at inverse temperature β ∈ [0, 2]. Here, β = 0 yields the two-dimensional Poisson distribution, matching the integrable limit of the system, and β = 2 equals the distribution obtained from the complex Ginibre ensemble, describing the fully chaotic limit. Our findings generalise the results of Grobe, Haake and Sommers who derived a universal cubic level repulsion for small spacings s. We collect mathematical evidence for the universality of the full level spacing distribution in the fully chaotic limit at β = 2. It holds for all three Ginibre ensembles of random matrices with independent real, complex or quaternion matrix elements. arXiv:1910.03520v4 [cond-mat.stat-mech]

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confidence: 99%

Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems

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“…Moreover, our problem is made harder by the singularity of the pair interaction in the Hamiltonian (1.9). It is known that Hybrid Monte Carlo schemes (relying on a second order discretization of an underdamped Langevin dynamics with a Metropolis-Hastings acceptance rule) provide efficient methods for sampling such probability distributions, see [11] and references therein. An issue when combining a Metropolis-Hastings rule with a projection on a submanifold is that reversibility may be lost, which introduces a bias.…”

Section: Description Of the Algorithm The Description Of The Constramentioning

confidence: 99%

“…Let us mention that the long time convergence of the law of this process towards P n (a difficult problem due to the singularity of the Hamiltonian) can be proved through Lyapunov function techniques, see [38] for a recent account. In practice, the singularity of g also makes the numerical integration of (4.10) difficult, and a Metropolis-Hastings selection rule can be used to stabilize the numerical discretization, see [11] and references therein. The algorithm described below makes precise how to adapt this strategy to sample measures constrained to the submanifold M.…”

Section: Constrained Langevin Dynamicsmentioning

confidence: 99%

“…When considering linear statistics, we prove in Section 3.2 that conditioning P n boils down to modifying the confinement potential V . In Section 3.3 we consider the case of quadratic statistics, which amounts to modifying the interaction kernel g. In order to validate our theoretical results and explore cases in which explicit solutions are not available, we also propose a method for sampling the law of Y n for a fixed n. Based on the Hamiltonian Monte Carlo (HMC) method used in [11] for sampling Gibbs measures associated to Coulomb and Log-gases, we describe and implement the generalized Hamiltonian Monte Carlo algorithm proposed in [35] for sampling probability measures on submanifolds. The method, detailed in Section 4.1, shows remarkable performance, and the results presented in Section 4.2 are in agreement with the theory.…”

Section: Introductionmentioning

confidence: 99%

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Coulomb Gases Under Constraint: Some Theoretical and Numerical Results

Chafäı¹,

Ferré²,

Stoltz³

2021

SIAM J. Math. Anal.

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We consider Coulomb gas models for which the empirical measure typically concentrates, when the number of particles becomes large, on an equilibrium measure minimizing an electrostatic energy. We study the behavior when the gas is conditioned on a rare event. We first show that the special case of quadratic confinement and linear constraint is exactly solvable due to a remarkable factorization, and that the conditioning has then the simple effect of shifting the cloud of particles without deformation. To address more general cases, we perform a theoretical asymptotic analysis relying on a large deviations technique known as the Gibbs conditioning principle. The technical part amounts to establishing that the conditioning ensemble is an Icontinuity set of the energy. This leads to characterizing the conditioned equilibrium measure as the solution of a modified variational problem. For simplicity, we focus on linear statistics and on quadratic statistics constraints. Finally, we numerically illustrate our predictions and explore cases in which no explicit solution is known. For this, we use a Generalized Hybrid Monte Carlo algorithm for sampling from the conditioned distribution for a finite but large system. Contents 12 4.1. Description of the algorithm 12 4.2. Numerical results 17 Acknowledgements 20 Appendix A.

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“…The case λ < 1 is useless since Lemma 1.1 tells us in this case that Z n = ∞. We used the algorithm from [17] with dt=.5 and T=10e6. About 10 independent copies were simulated and merged and we retained only the last 10% the trajectories.…”

Section: Lemma 11 (Confinement or Integrability Condition) We Havementioning

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Macroscopic and edge behavior of a planar jellium

Chafäı

García-Zelada

Jung

2020

Journal of Mathematical Physics

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We consider a planar Coulomb gas in which the external potential is generated by a smeared uniform background of opposite-sign charge on a disc. This model can be seen as a two-dimensional Wigner jellium, not necessarily charge neutral, and with particles allowed to exist beyond the support of the smeared charge. The full space integrability condition requires low enough temperature or high enough total smeared charge. This condition does not allow at the same time, total charge neutrality and determinantal structure. The model shares similarities with both the complex Ginibre ensemble and the Forrester-Krishnapur spherical ensemble of random matrix theory. In particular, for a certain regime of temperature and total charge, the equilibrium measure is uniform on a disc as in the Ginibre ensemble, while the modulus of the farthest particle has heavy-tailed fluctuations as in the Forrester-Krishnapur spherical ensemble. We also touch on a higher temperature regime producing a crossover equilibrium measure, as well as a transition to Gumbel edge fluctuations. More results in the same spirit on edge fluctuations are explored by the second author together with Raphael Butez.Contents P →, convergence in law and in probability, respectively, and we annotate X ∼ µ to mean that the law of the random variable X is given by the probability distribution µ. We denote by P(C) the set of probability measures on C, equipped with the topology of weak convergence with respect to continuous and bounded test functions, and its associated Borel σ-field. This topology is metrized by the bounded-Lipschitz metric

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Simulating Coulomb and Log-Gases with Hybrid Monte Carlo Algorithms

Cited by 12 publications

References 78 publications

Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems

Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems

Coulomb Gases Under Constraint: Some Theoretical and Numerical Results

Macroscopic and edge behavior of a planar jellium

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