Large-distance and long-time properties of a randomly stirred fluid

Forster, Dieter; Nelson, David R.; Stephen, Michael J.

doi:10.1103/physreva.16.732

Cited by 1,291 publications

(1,349 citation statements)

References 34 publications

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“…These connections lead us to powerful heuristic pictures for entanglement growth, in both 1D and higher dimensions. The KPZ universality class embraces the statistical mechanics of a directed polymer in a disordered potential landscape [47] and 1D hydrodynamics with noise (the noisy Burgers equation [48]). These problems, together with surface growth, are sometimes known as the "KPZ triumvirate" [49].…”

Section: Introductionmentioning

confidence: 99%

Quantum Entanglement Growth under Random Unitary Dynamics

et al. 2017

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Characterizing how entanglement grows with time in a many-body system, for example, after a quantum quench, is a key problem in nonequilibrium quantum physics. We study this problem for the case of random unitary dynamics, representing either Hamiltonian evolution with time-dependent noise or evolution by a random quantum circuit. Our results reveal a universal structure behind noisy entanglement growth, and also provide simple new heuristics for the "entanglement tsunami" in Hamiltonian systems without noise. In 1D, we show that noise causes the entanglement entropy across a cut to grow according to the celebrated KardarParisi-Zhang (KPZ) equation. The mean entanglement grows linearly in time, while fluctuations grow like ðtimeÞ 1=3 and are spatially correlated over a distance ∝ ðtimeÞ 2=3 . We derive KPZ universal behavior in three complementary ways, by mapping random entanglement growth to (i) a stochastic model of a growing surface, (ii) a "minimal cut" picture, reminiscent of the Ryu-Takayanagi formula in holography, and (iii) a hydrodynamic problem involving the dynamical spreading of operators. We demonstrate KPZ universality in 1D numerically using simulations of random unitary circuits. Importantly, the leading-order time dependence of the entropy is deterministic even in the presence of noise, allowing us to propose a simple coarse grained minimal cut picture for the entanglement growth of generic Hamiltonians, even without noise, in arbitrary dimensionality. We clarify the meaning of the "velocity" of entanglement growth in the 1D entanglement tsunami. We show that in higher dimensions, noisy entanglement evolution maps to the wellstudied problem of pinning of a membrane or domain wall by disorder.

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Section: Introductionmentioning

confidence: 99%

Quantum Entanglement Growth under Random Unitary Dynamics

et al. 2017

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“…The statistics of v, b is completely determined by both the non-linear equations (1,2) and the statistics of the external inter-correlated large-scale random forces f v , f b . The dissipation is controlled by the parameter of kinematic viscosity ν 0 , and u 0 denotes inverse Prandtl number (hereafter all parameters with a subscript 0 denote bare parameters of unrenormalized theory; see below).…”

Section: Functional Formulation Of Double Expansion Modelmentioning

confidence: 99%

“…The renormalization group (RG) methods have been widely used to the analysis of fully developed hydrodynamic (HD) turbulence beginning from pioneering papers [1,2] based on [3,4]. It gives possibility to reply upon some principal questions, e.g., on the fundamental description of the infrared (IR) scale invariance, as well as it is useful for calculation of many quantities, e.g., critical dimensions of the fields and their gradients, viscosity, etc.…”

Section: Introductionmentioning

confidence: 99%

D-dimensional developed MHD turbulence: double expansion model

Jurčišin¹,

Stehlík²

2006

J. Phys. A: Math. Gen.

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Abstract. Developed magnetohydrodynamic turbulence near two dimensions d up to three dimensions has been investigated by means of renormalization group approach and double expansion regularization. A modification of standard minimal subtraction scheme has been used to analyze the stability of the Kolmogorov scaling regime which is governed by the renormalization group fixed point. The exact analytical expressions have been obtained for the fixed points. The continuation of the universal value of the inverse Prandtl number u = 1.562 determined at d = 2 up to d = 3 restores the value of u = 1.393 which is known in the kinetic fixed point from usual ǫ-expansion. The magnetic stable fixed point has been calculated and its stability region has been also examined. This point losses stability: (1) below critical value of dimension dc = 2.36 (independently on the a-parameter of a magnetic forcing) and, (2) below the value of ac = 0.146 (independently on the dimension).

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“…However, the logarithmic correction cannot be seen from the their simulation. This decay rate was also obtained by Forster, Nelson and Stephen [5] by a renormalization group method. The later simulation by van der Hoef and Frenkel [6] confirmed that there is a discrepancy to the pure algebraic decay t −1 , but is far from being able to determine the precise logarithmic correction.…”

Section: Introductionmentioning

confidence: 99%

Superdiffusivity of Two Dimensional Lattice Gas Models

2005

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Abstract. It was proved [4,13] that stochastic lattice gas dynamics converge to the Navier-Stokes equations in dimension d = 3 in the incompressible limits. In particular, the viscosity is finite. We proved that, on the other hand, the viscosity for a two dimensional lattice gas model diverges faster than log log t. Our argument indicates that the correct divergence rate is (log t) 1/2 . This problem is closely related to the logarithmic correction of the time decay rate for the velocity auto-correlation function of a tagged particle.

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Large-distance and long-time properties of a randomly stirred fluid

Cited by 1,291 publications

References 34 publications

Quantum Entanglement Growth under Random Unitary Dynamics

Quantum Entanglement Growth under Random Unitary Dynamics

D-dimensional developed MHD turbulence: double expansion model

Superdiffusivity of Two Dimensional Lattice Gas Models

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