laboratoire de physique statistique
 
 
laboratoire de physique statistique

Publications

Rechercher
 
2016
Quantum walks and non-Abelian discrete gauge theory - Arnault, Pablo and Di Molfetta, Giuseppe and Brachet, Marc and Debbasch, Fabrice
PHYSICAL REVIEW A 94 (2016) 
LPS


Abstract : A family of discrete-time quantum walks (DTQWs) on the line with an exact discrete U(N) gauge invariance is introduced. It is shown that the continuous limit of these DTQWs, when it exists, coincides with the dynamics of a Dirac fermion coupled to usual U(N) gauge fields in two-dimensional spacetime. A discrete generalization of the usual U(N) curvature is also constructed. An alternate interpretation of these results in terms of superimposed U(1) Maxwell fields and SU(N) gauge fields is discussed in the Appendix. Numerical simulations are also presented, which explore the convergence of the DTQWs towards their continuous limit and which also compare the DTQWs with classical (i.e., nonquantum) motions in classical SU(2) fields. The results presented in this paper constitute a first step towards quantum simulations of generic Yang-Mills gauge theories through DTQWs.
Helicity, topology, and Kelvin waves in reconnecting quantum knots - Clark di Leoni, P. and Mininni, P. D. and Brachet, M. E.
PHYSICAL REVIEW A 94 (2016) 
LPS


Abstract : Helicity is a topological invariant that measures the linkage and knottedness of lines, tubes, and ribbons. As such, it has found myriads of applications in astrophysics, fluid dynamics, atmospheric sciences, and biology. In quantum flows, where topology-changing reconnection events are a staple, helicity appears as a key quantity to study. However, the usual definition of helicity is not well posed in quantum vortices, and its computation based on counting links and crossings of centerline vorticity can be downright impossible to apply in complex and turbulent scenarios. We present a definition of helicity which overcomes these problems and which gives the expected result in the large-scale limit. With it, we show that certain reconnection events can excite Kelvin waves and other complex motions of the centerline vorticity, which slowly deplete helicity as they interact nonlinearly, thus linking the theory of vortex knots with observations of quantum fluids. This process also results in the depletion of helicity in a fully turbulent quantum flow, in a way reminiscent of the decay of helicity in classical fluids.
Large-scale instabilities of helical flows - Cameron, Alexandre and Alexakis, Alexandros and Brachet, Marc-Etienne
PHYSICAL REVIEW FLUIDS 1 (2016) 
LPS


Abstract : Large-scale hydrodynamic instabilities of periodic helical flows of a given wave number K are investigated using three-dimensional Floquet numerical computations. In the Floquet formalism the unstable field is expanded in modes of different spacial periodicity. This allows us (i) to clearly distinguish large from small scale instabilities and (ii) to study modes of wave number q of arbitrarily large-scale separation q << K. Different flows are examined including flows that exhibit small-scale turbulence. The growth rate sigma of the most unstable mode is measured as a function of the scale separation q/K << 1 and the Reynolds number Re. It is shown that the growth rate follows the scaling s. q if an AKA effect [Frisch et al., Physica D: Nonlinear Phenomena 28, 382 (1987)] is present or a negative eddy viscosity scaling sigma alpha q(2) in its absence. This holds both for the Re << 1 regime where previously derived asymptotic results are verified but also for Re = O(1) that is beyond their range of validity. Furthermore, for values of Re above a critical value Re-S(c) beyond which small-scale instabilities are present, the growth rate becomes independent of q and the energy of the perturbation at large scales decreases with scale separation. The nonlinear behavior of these large-scale instabilities is also examined in the nonlinear regime where the largest scales of the system are found to be the most dominant energetically. These results are interpreted by low-order models.
Sticking transition in a minimal model for the collisions of active particles in quantum fluids - Shukla, Vishwanath and Brachet, Marc and Pandit, Rahul
PHYSICAL REVIEW A 94 (2016) 
LPS


Abstract : Particles of low velocity, traveling without dissipation in a superfluid, can interact and emit sound when they collide. We propose a minimal model in which the equations of motion of the particles, including a short-range repulsive force, are self-consistently coupled with the Gross-Pitaevskii equation. We show that this model generates naturally an effective superfluid-mediated attractive interaction between the particles; and we study numerically the collisional dynamics of particles as a function of their incident kinetic energy and the length scale of the repulsive force. We find a transition from almost elastic to completely inelastic (sticking) collisions as the parameters are tuned. We find that aggregation and clustering result from this sticking transition in multiparticle systems.
Statistical theory of reversals in two-dimensional confined turbulent flows - Shukla, Vishwanath and Fauve, Stephan and Brachet, Marc
PHYSICAL REVIEW E 94 (2016) 
LPS


Abstract : It is shown that the truncated Euler equation (TEE), i.e., a finite set of ordinary differential equations for the amplitude of the large-scale modes, can correctly describe the complex transitional dynamics that occur within the turbulent regime of a confined two-dimensional flow obeying Navier-Stokes equation (NSE) with bottom friction and a spatially periodic forcing. The random reversals of the NSE large-scale circulation on the turbulent background involve bifurcations of the probability distribution function of the large-scale circulation. We demonstrate that these NSE bifurcations are described by the related TEE microcanonical distribution which displays transitions from Gaussian to bimodal and broken ergodicity. A minimal 13-mode model reproduces these results.
 
2015
Self-truncation and scaling in Euler-Voigt-alpha and related fluid models - Di Molfetta, Giuseppe and Krstlulovic, Giorgio and Brachet, Marc
PHYSICAL REVIEW E 92 (2015) 
LPS


Abstract : A generalization of the 3D Euler-Voigt-alpha model is obtained by introducing derivatives of arbitrary order beta (instead of 2) in the Helmholtz operator. The beta -> infinity limit is shown to correspond to Galerkin truncation of the Euler equation. Direct numerical simulations (DNS) of the model are performed with resolutions up to 20483 and Taylor-Green initial data. DNS performed at large beta demonstrate that this simple classical hydrodynamical model presents a self-truncation behavior, similar to that previously observed for the Gross-Pitaeveskii equation in Krstulovic and Brachet [Phys. Rev. Lett. 106, 115303 (2011)]. The self-truncation regime of the generalized model is shown to reproduce the behavior of the truncated Euler equation demonstrated in Cichowlas et al. [Phys. Rev. Lett. 95, 264502 (2005)]. The long-time growth of the self-truncation wave number k(st) appears to be self-similar. Two related alpha-Voigt versions of the eddy-damped quasinormal Markovian model and the Leith model are introduced. These simplified theoretical models are shown to reasonably reproduce intermediate time DNS results. The values of the self-similar exponents of these models are found analytically.
Nonlinear optical Galton board: Thermalization and continuous limit - Di Molfetta, Giuseppe and Debbasch, Fabrice and Brachet, Marc
PHYSICAL REVIEW E 92 (2015) 
LPS


Abstract : The nonlinear optical Galton board (NLOGB), a quantum walk like (but nonlinear) discrete time quantum automaton, is shown to admit a complex evolution leading to long time thermalized states. The continuous limit of the Galton board is derived and shown to be a nonlinear Dirac equation (NLDE). The (Galerkin-truncated) NLDE evolution is shown to thermalize toward states qualitatively similar to those of the NLOGB. The NLDE conserved quantities are derived and used to construct a stochastic differential equation converging to grand canonical distributions that are shown to reproduce the (microcanonical) NLDE thermalized statistics. Both the NLOGB and the Galerkin-truncated NLDE are thus demonstrated to exhibit spontaneous thermalization.
Spatiotemporal detection of Kelvin waves in quantum turbulence simulations - Clark di Leoni, P. and Mininni, P. D. and Brachet, M. E.
PHYSICAL REVIEW A 92 (2015) 
LPS


Abstract : We present evidence of Kelvin excitations in space-time resolved spectra of numerical simulations of quantum turbulence. Kelvin waves are transverse and circularly polarized waves that propagate along quantized vortices, for which the restitutive force is the tension of the vortex line, and which play an important role in theories of superfluid turbulence. We use the Gross-Pitaevskii equation to model quantum flows, letting an initial array of well-organized vortices develop into a turbulent bundle of intertwined vortex filaments. By achieving high spatial and temporal resolution we are able to calculate space-time resolved mass density and kinetic energy spectra. Evidence of Kelvin and sound waves is clear in both spectra. Identification of the waves allows us to extract the spatial spectrum of Kelvin waves, clarifying their role in the transfer of energy.
 
2014
Quantum walks in artificial electric and gravitational fields - Di Molfetta, Giuseppe and Brachet, Marc and Debbasch, Fabrice
PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS 397157-168 (2014) 
LPS


Abstract : The continuous limit of quantum walks (QWs) on the line is revisited through a new, recently developed method. In all cases but one, the limit coincides with the dynamics of a Dirac fermion coupled to an artificial electric and/or relativistic gravitational field. All results are carefully discussed and illustrated by numerical simulations. Possible experimental realizations are also addressed. (C) 2013 Elsevier B.V. All rights reserved.
DOI
10
Modeling quantum fluid dynamics at nonzero temperatures - Berloff, Natalia G. and Brachet, Marc and Proukakis, Nick P.
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA 1114675-4682 (2014) 
LPS


Abstract : The detailed understanding of the intricate dynamics of quantum fluids, in particular in the rapidly growing subfield of quantum turbulence which elucidates the evolution of a vortex tangle in a superfluid, requires an in-depth understanding of the role of finite temperature in such systems. The Landau two-fluidmodel is the most successful hydrodynamical theory of superfluid helium, but by the nature of the scale separations it cannot give an adequate description of the processes involving vortex dynamics and interactions. In our contribution we introduce a framework based on a nonlinear classical-field equation that is mathematically identical to the Landau model and provides a mechanism for severing and coalescence of vortex lines, so that the questions related to the behavior of quantized vortices can be addressed self-consistently. The correct equation of state as well as nonlocality of interactions that leads to the existence of the roton minimum can also be introduced in such description. We review and apply the ideas developed for finite-temperature description of weakly interacting Bose gases as possible extensions and numerical refinements of the proposed method. We apply this method to elucidate the behavior of the vortices during expansion and contraction following the change in applied pressure. We show that at low temperatures, during the contraction of the vortex core as the negative pressure grows back to positive values, the vortex line density grows through a mechanism of vortex multiplication. This mechanism is suppressed at high temperatures.
DOI
11
Dynamo action by turbulence in absolute equilibrium - Prasath, Srinivasa Gopalakrishnan Ganga and Fauve, Stephan and Brachet, Marc
EPL 106 (2014) 
LPS


Abstract : We consider the generation of a large-scale magnetic field by a turbulent flow driven by a small-scale helical forcing in a low magnetic Prandtl number fluid. We provide an estimate of the dynamo threshold that takes into account the presence of large-scale turbulent fluctuations by considering that the scales of the flow that mostly contribute to the dynamo process are roughly in absolute equilibrium. We show that turbulent flows in absolute equilibrium do generate dynamos and we compare their growth rates to their laminar counterparts. Finally, we show that the back reaction of the growing magnetic field modifies the statistical properties of turbulent flow by suppressing its kinetic helicity at large magnetic Reynolds number. Copyright (C) EPLA, 2014
DOI
12
Forced magnetohydrodynamic turbulence in three dimensions using Taylor-Green symmetries - Krstulovic, G. and Brachet, M. E. and Pouquet, A.
PHYSICAL REVIEW E 89 (2014) 
LPS


Abstract : We examine the scaling laws of magnetohydrodynamic (MHD) turbulence for three different types of forcing functions and imposing at all times the fourfold symmetries of the Taylor-Green (TG) vortex generalized to MHD; no uniform magnetic field is present and the magnetic Prandtl number is equal to unity. We also include pumping in the induction equation, and we take the three configurations studied in the decaying case in Lee et al. [Phys. Rev. E 81, 016318 (2010)]. To that effect, we employ direct numerical simulations up to an equivalent resolution of 20483 grid points. We find that, similarly to the case when the forcing is absent, different spectral indices for the total energy spectrum emerge, corresponding to either a Kolmogorov law, an Iroshnikov-Kraichnan law that arises from the interactions of turbulent eddies and Alfven waves, or to weak turbulence when the large-scale magnetic field is strong. We also examine the inertial range dynamics in terms of the ratios of kinetic to magnetic energy, and of the turnover time to the Alfven time, and analyze the temporal variations of these quasiequilibria.
 
2013
DOI
13
Ideal evolution of magnetohydrodynamic turbulence when imposing Taylor-Green symmetries - Brachet, M. E. and Bustamante, M. D. and Krstulovic, G. and Mininni, P. D. and Pouquet, A. and Rosenberg, D.
PHYSICAL REVIEW E 87 (2013) 
LPS


Abstract : We investigate the ideal and incompressible magnetohydrodynamic (MHD) equations in three space dimensions for the development of potentially singular structures. The methodology consists in implementing the fourfold symmetries of the Taylor-Green vortex generalized to MHD, leading to substantial computer time and memory savings at a given resolution; we also use a regridding method that allows for lower-resolution runs at early times, with no loss of spectral accuracy. One magnetic configuration is examined at an equivalent resolution of 6144(3) points and three different configurations on grids of 4096(3) points. At the highest resolution, two different current and vorticity sheet systems are found to collide, producing two successive accelerations in the development of small scales. At the latest time, a convergence of magnetic field lines to the location of maximum current is probably leading locally to a strong bending and directional variability of such lines. A novel analytical method, based on sharp analysis inequalities, is used to assess the validity of the finite-time singularity scenario. This method allows one to rule out spurious singularities by evaluating the rate at which the logarithmic decrement of the analyticity-strip method goes to zero. The result is that the finite-time singularity scenario cannot be ruled out, and the singularity time could be somewhere between t = 2.33 and t = 2.70. More robust conclusions will require higher resolution runs and grid-point interpolation measurements of maximum current and vorticity.
DOI
14
Quantum walks as massless Dirac fermions in curved space-time - Di Molfetta, Giuseppe and Brachet, M. and Debbasch, Fabrice
PHYSICAL REVIEW A 88 (2013) 
LPS


Abstract : A particular family of time- and space-dependent discrete-time quantum walks (QWs) is considered in one-dimensional physical space. The continuous limit of these walks is defined through a procedure discussed here and computed in full detail. In this limit, the walks coincide with the propagation of a massless Dirac fermion in an arbitrary gravitational field. A QW mimicking the radial propagation of a fermion outside and inside the event horizon of a Schwarzschild black hole is explicitly constructed and simulated numerically. Thus, the family of QWs considered in our manuscript provides an analog system to study experimentally coherent quantum propagation in curved spacetime.
DOI
15
Turbulence in the two-dimensional Fourier-truncated Gross-Pitaevskii equation - Shukla, Vishwanath and Brachet, Marc and Pandit, Rahul
NEW JOURNAL OF PHYSICS 15 (2013) 
LPS


Abstract : We undertake a systematic, direct numerical simulation of the twodimensional, Fourier-truncated, Gross-Pitaevskii equation to study the turbulent evolutions of its solutions for a variety of initial conditions and a wide range of parameters. We find that the time evolution of this system can be classified into four regimes with qualitatively different statistical properties. Firstly, there are transients that depend on the initial conditions. In the second regime, powerlaw scaling regions, in the energy and the occupation-number spectra, appear and start to develop; the exponents of these power laws and the extents of the scaling regions change with time and depend on the initial condition. In the third regime, the spectra drop rapidly for modes with wave numbers k > kc and partial thermalization takes place for modes with k < kc; the self-truncation wave number kc(t) depends on the initial conditions and it grows either as a power of t or as log t. Finally, in the fourth regime, complete thermalization is achieved and, if we account for finite-size effects carefully, correlation functions and spectra are consistent with their nontrivial Berezinskii-Kosterlitz-Thouless forms. Our work is a natural generalization of recent studies of thermalization in the Euler and other hydrodynamical equations; it combines ideas from fluid dynamics and turbulence, on the one hand, and equilibrium and nonequilibrium statistical mechanics on the other.
 
2012
DOI
16
Long-time properties of magnetohydrodynamic turbulence and the role of symmetries - Stawarz, Julia E. and Pouquet, Annick and Brachet, Marc-Etienne
PHYSICAL REVIEW E 86 (2012) 
LPS


Abstract : Using direct numerical simulations with grids of up to 5123 points, we investigate long-time properties of three-dimensional magnetohydrodynamic turbulence in the absence of forcing and examine in particular the roles played by the quadratic invariants of the system and the symmetries of the initial configurations. We observe that when sufficient accuracy is used, initial conditions with a high degree of symmetries, as in the absence of helicity, do not travel through parameter space over time, whereas by perturbing these solutions either explicitly or implicitly using, for example, single precision for long times, the flows depart from their original behavior and can either become strongly helical or have a strong alignment between the velocity and the magnetic field. When the symmetries are broken, the flows evolve towards different end states, as already predicted by statistical arguments for nondissipative systems with the addition of an energy minimization principle. Increasing the Reynolds number by an order of magnitude when using grids of 643-5123 points does not alter these conclusions. Furthermore, the alignment properties of these flows, between velocity, vorticity, magnetic potential, induction, and current, correspond to the dominance of two main regimes, one helically dominated and one in quasiequipartition of kinetic and magnetic energies. We also contrast the scaling of the ratio of magnetic energy to kinetic energy as a function of wave number to the ratio of eddy turnover time to Alfven time as a function of wave number. We find that the former ratio is constant with an approximate equipartition for scales smaller than the largest scale of the flow, whereas the ratio of time scales increases with increasing wave number.
DOI
17
Gross-Pitaevskii description of superfluid dynamics at finite temperature: A short review of recent results - Brachet, Marc
COMPTES RENDUS PHYSIQUE 13954-965 (2012) 
LPS


Abstract : The Gross-Pitaevskii equation (GPE) describes the dynamics of superflows and Bose-Einstein Condensates (BEC) at very low temperature. When a truncation of Fourier modes is performed, the resulting truncated GPE (TGPE) can also describe the correct thermal behavior of a Bose gas, as long as all relevant modes are highly occupied [M.J. Davis, S.A. Morgan, K. Burnett, Simulations of Bose fields at finite temperature, Phys. Rev. Lett. 87 (16) (2001) 160402]. We review some of our group's recent GPE- and TGPE-based numerical studies of superfluid dynamics and BEC stability. The relations with experiments are discussed. (C) 2012 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.
DOI
18
Interplay between the Beale-Kato-Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem - Bustamante, Miguel D. and Brachet, Marc
PHYSICAL REVIEW E 86 (2012) 
LPS


Abstract : Numerical simulations of the incompressible Euler equations are performed using the Taylor-Green vortex initial conditions and resolutions up to 4096(3). The results are analyzed in terms of the classical analyticity-strip method and Beale, Kato, and Majda (BKM) theorem. A well-resolved acceleration of the time decay of the width of the analyticity strip delta(t) is observed at the highest resolution for 3.7 < t < 3.85 while preliminary three-dimensional visualizations show the collision of vortex sheets. The BKM criterion on the power-law growth of the supremum of the vorticity, applied on the same time interval, is not inconsistent with the occurrence of a singularity around t similar or equal to 4. These findings lead us to investigate how fast the analyticity-strip width needs to decrease to zero in order to sustain a finite-time singularity consistent with the BKM theorem. A simple bound of the supremum norm of vorticity in terms of the energy spectrum is introduced and used to combine the BKM theorem with the analyticity-strip method. It is shown that a finite-time blowup can exist only if delta(t) vanishes sufficiently fast at the singularity time. In particular, if a power law is assumed for delta(t) then its exponent must be greater than some critical value, thus providing a new test that is applied to our 4096(3) Taylor-Green numerical simulation. Our main conclusion is that the numerical results are not inconsistent with a singularity but that higher-resolution studies are needed to extend the time interval on which a well-resolved power-law behavior of delta(t) takes place and check whether the new regime is genuine and not simply a crossover to a faster exponential decay. DOI: 10.1103/PhysRevE.86.066302
 
2011
DOI
19
Dispersive Bottleneck Delaying Thermalization of Turbulent Bose-Einstein Condensates - Krstulovic, Giorgio and Brachet, Marc
PHYSICAL REVIEW LETTERS 106 (2011) 
LPS


Abstract : A new mechanism of thermalization involving a direct energy cascade is obtained in the truncated Gross-Pitaevskii dynamics. A long transient with partial thermalization at small scales is observed before the system reaches equilibrium. Vortices are found to disappear as a prelude to final thermalization. A bottleneck that produces spontaneous effective self-truncation and delays thermalization is characterized when large dispersive effects are present at the truncation wave number. Order of magnitude estimates indicate that self-truncation takes place in turbulent Bose-Einstein condensates. This effect should also be present in classical hydrodynamics and models of turbulence.
DOI
20
Anomalous vortex-ring velocities induced by thermally excited Kelvin waves and counterflow effects in superfluids - Krstulovic, Giorgio and Brachet, Marc
PHYSICAL REVIEW B 83 (2011) 
LPS


Abstract : Dynamical counterflow effects on vortex evolution under the truncated Gross-Pitaevskii equation are investigated. Standard longitudinal mutual-friction effects are produced and a dilatation of vortex rings is obtained at large counterflows. A strong temperature-dependent anomalous slowdown of vortex rings is observed and attributed to the presence of thermally excited Kelvin waves. This generic effect of finite-temperature superfluids is estimated using energy equipartition and orders of magnitude are given for weakly interacting Bose-Einstein condensates and superfluid (4)He. The relevance of thermally excited Kelvin waves is discussed in the context of quantum turbulence.