Coarse-graining Length Research Articles

Witnessing quantum chaos using observational entropy.

We study observation entropy (OE) for the quantum kicked top model, whose classical counterpart possesses different phases: regular, mixed, or chaotic, depending on the strength of the kicking parameter. We show that OE grows logarithmically with coarse-graining length beyond a critical value in the regular phase, while OE growth is much faster in the chaotic regime. In the dynamics, we demonstrate that the short-time growth rate of OE acts as a measure of the chaoticity in the system, and we compare our results with out-of-time-ordered correlators (OTOC). Moreover, we show that in the deep quantum regime, the results obtained from OE are much more robust compared to OTOC results. Finally, we also investigate the long-time behavior of OE to distinguish between saddle-point scrambling and true chaos, where the former shows large persistent fluctuations compared to the latter.

Clusters in the critical branching Brownian motion

Brownian particles that are replicated and annihilated at equal rate have strongly correlated positions, forming a few compact clusters separated by large gaps. We characterize the distribution of the particles at a given time, using a definition of clusters in terms of a coarse-graining length recently introduced by some of us. We show that, in a non-extinct realization, the average number of clusters grows as where is the Hausdorff dimension of the boundary of the super-Brownian motion (SBM), found by Mueller, Mytnik, and Perkins. We also compute the distribution of gaps between consecutive particles. We find two regimes separated by the characteristic length scale where D is the diffusion constant and β the branching rate. The average number of gaps greater than g decays as for and for . Finally, conditioned on the number of particles n, the above distributions are valid for ; the average number of gaps greater than is much less than one, and decays as , in agreement with the universal gap distribution predicted by Ramola, Majumdar, and Schehr. Our results interpolate between a dense SBM regime and a large-gap regime, unifying two previously independent approaches.

Microscopic Theory of Fluctuating Hydrodynamics in Nonlinear Lattices.

The theory of fluctuating hydrodynamics has been an important tool for analyzing macroscopic behavior in nonlinear lattices. However, despite its practical success, its microscopic derivation is still incomplete. In this work, we provide the microscopic derivation of fluctuating hydrodynamics, using the coarse-graining and projection technique; the equivalence of ensembles turns out to be critical. The Green-Kubo (GK)-like formula for the bare transport coefficients are presented in a numerically computable form. Our numerical simulations show that the bare transport coefficients exist for a sufficiently large but finite coarse-graining length in the infinite lattice within the framework of the GK-like formula. This demonstrates that the bare transport coefficients uniquely exist for each physical system.

Quantum relaxation in a system of harmonic oscillators with time-dependent coupling.

In the context of the de Broglie-Bohm pilot-wave theory, numerical simulations for simple systems have shown that states that are initially out of quantum equilibrium-thus violating the Born rule-usually relax over time to the expected |ψ|2 distribution on a coarse-grained level. We analyse the relaxation of non-equilibrium initial distributions for a system of coupled one-dimensional harmonic oscillators in which the coupling depends explicitly on time through numerical simulations, focusing on the influence of different parameters such as the number of modes, the coarse-graining length and the coupling constant. We show that in general the system studied here tends to equilibrium, but the relaxation can be retarded depending on the values of the parameters, particularly to the one related to the strength of the interaction. Possible implications on the detection of relic non-equilibrium systems are discussed.

Non-monotonic temperature evolution of nonlocal structure–dynamics correlation in CuZr glass-forming liquids* *Project supported by the National Natural Science Foundation of China (Grant Nos. 52031016 and 51631003).

The structure–dynamics correlations in a nonlocal manner were investigated in CuZr metallic glass-forming liquids via classical molecular dynamics simulations. A spatial coarse-graining approach was employed to incorporate the nonlocal structural information of given structural order parameters in the structure–dynamics relationship. It is found that the correlation between structure order parameters and dynamics increases with increasing coarse-graining length and has a characteristic length scale. Moreover, the characteristic correlation length exhibits a non-monotonic temperature evolution as temperature approaches glass transition temperature, which is not sensitive to the considered structure order parameters. Our results unveil a striking change in the structure–dynamics correlation, which involves no fitting theoretical interpretation. These findings provide new insight into the structure–dynamics correlation in glass transition.

Hidden Order Beyond Hyperuniformity in Critical Absorbing States.

Disordered hyperuniformity is a description of hidden correlations in point distributions revealed by an anomalous suppression in fluctuations of local density at various coarse-graining length scales. In the absorbing phase of models exhibiting an active-absorbing state transition, this suppression extends up to a hyperuniform length scale that diverges at the critical point. Here, we demonstrate the existence of additional many-body correlations beyond hyperuniformity. These correlations are hidden in the higher moments of the probability distribution of the local density and extend up to a longer length scale with a faster divergence than the hyperuniform length on approaching the critical point. Our results suggest that a hidden order beyond hyperuniformity may generically be present in complex disordered systems.

Determining the dependence of the power supply to the ocean on the length and time scales of the dynamics between the meso-scale and the synoptic scale, from satellite data

The input of mechanical power to the ocean due to the surface wind stress, in regions which correspond to different regimes of ocean dynamics, is considered using data from satellites observations. Its dependence on the coarse-graining range of the atmospheric and oceanic velocity in space from 0.5 to 10∘ and time from 6 h to 40 days is determined. In the area of the Gulf Stream and the Kuroshio extensions, the dependence of the power input on space-time coarse-graining varies over tenfold for the coarse-graining considered. It decreases over twofold for the Gulf Stream extension and threefold for the Kuroshio extension, when the coarse-graining length scale passes from a few degrees to 0.5∘ at a temporal coarse-graining scale of a few days. It increases over threefold in the Gulf Stream and the Kuroshio extensions when the coarse-graining passes from several days to 6 h at a spatial coarse-graining of a few degrees. The power input is found to increase monotonically with shorter coarse-graining in time. Its variation with coarse-graining in space has no definite sign. Results show that including the dynamics at scales below a few degrees reduces considerably the power input by air-sea interaction in regions of strongly nonlinear ocean currents. When the ocean velocities are not considered in the shear calculation, the power input is considerably (up to threefold) increased. The dependence of the power input on coarse-graining in space and time is close to being multiplicatively separable in all regions and for most of the coarse-graining domain considered.

Towards systematic grid selection in LES: Identifying the optimal spatial resolution by minimizing the solution sensitivity

A method for finding a nearly optimal computational grid or, more generally, filter-width distribution for a large eddy simulation is proposed and assessed. The core idea is that the optimal resolution (or coarse-graining length scale, or filter-width, or grid spacing) for LES is the coarsest resolution for which the LES solution is sufficiently accurate and exhibits minimal sensitivity to the resolution. This idea is formulated based on an error indicator that measures the dependence of the solution on the grid/filter as a residual term in the governing equation and a criterion that determines how that error indicator should vary in space and direction to minimize the overall sensitivity of the solution. The final definition of the error indicator becomes very similar to the divergence of the error in the Germano identity, with its derivation offering an alternative explanation for the success of the dynamic procedure. Furthermore, the solution to the optimization problem of grid/filter-width adaptation is that the cell-integrated error indicator should be equi-distributed; a corollary is that one cannot link the accuracy in LES to quantities that are not cell-integrated, including the common belief that LES is accurate whenever 80-90% of the energy is resolved. The full method is tested on wall-resolved LES of turbulent channel flow and the flow over a backward-facing step, with final length scale fields (or filter-width fields, or grids) that are close to what is considered “best practice” in the LES literature.

Stress Transmission and Failure in Disordered Porous Media.

By means of extensive lattice-element simulations, we investigate stress transmission and its relation with failure properties in increasingly disordered porous systems. We observe a non-Gaussian broadening of stress probability density functions under tensile loading with increasing porosity and disorder, revealing a gradual transition from a state governed by single-pore stress concentration to a state controlled by multipore interactions and metric disorder. This effect is captured by the excess kurtosis of stress distributions and shown to be nicely correlated with the second moment of local porosity fluctuations, which appears thus as a (dis)order parameter for the system. By generating statistical ensembles of porous textures with varying porosity and disorder, we derive a general expression for the fracture stress as a decreasing function of porosity and disorder. Focusing on critical sites where the local stress is above the global fracture threshold, we also analyze the transition to failure in terms of a coarse-graining length. These findings provide a general framework which can also be more generally applied to multiphase and structural heterogeneous materials.

Coarse graining atomistic simulations of plastically deforming amorphous solids.

The primary mode of failure in disordered solids results from the formation and persistence of highly localized regions of large plastic strains known as shear bands. Continuum-level field theories capable of predicting this mechanical response rely upon an accurate representation of the initial and evolving states of the amorphous structure. We perform molecular dynamics simulations of a metallic glass and propose a methodology for coarse graining discrete, atomistic quantities, such as the potential energies of the elemental constituents. A strain criterion is established and used to distinguish the coarse-grained degrees-of-freedom inside the emerging shear band from those of the surrounding material. A signal-to-noise ratio provides a means of evaluating the strength of the signal of the shear band as a function of the coarse graining. Finally, we investigate the effect of different coarse graining length scales by comparing a two-dimensional, numerical implementation of the effective-temperature description in the shear transformation zone (STZ) theory with direct molecular dynamics simulations. These comparisons indicate the coarse graining length scale has a lower bound, above which there is a high level of agreement between the atomistics and the STZ theory, and below which the concept of effective temperature breaks down.

Coarse-grained incompressible magnetohydrodynamics: analyzing the turbulent cascades

We formulate a coarse-graining approach to the dynamics of magnetohydrodynamic (MHD) fluids at a continuum of length-scales ℓ. In this methodology, effective equations are derived for the observable velocity and magnetic fields spatially-averaged at an arbitrary scale of resolution. The microscopic equations for the ‘bare’ velocity and magnetic fields are ‘renormalized’ by coarse-graining to yield macroscopic effective equations that contain both a subscale stress and a subscale electromotive force (EMF) generated by nonlinear interaction of eliminated fields and plasma motions. Particular attention is given to the effects of these subscale terms on the balances of the quadratic invariants of ideal incompressible MHD—energy, cross-helicity and magnetic helicity. At large coarse-graining length-scales, the direct dissipation of the invariants by microscopic mechanisms (such as molecular viscosity and Spitzer resistivity) is shown to be negligible. The balance at large scales is dominated instead by the subscale nonlinear terms, which can transfer invariants across scales, and are interpreted in terms of work concepts for energy and in terms of topological flux-linkage for the two helicities. An important application of this approach is to MHD turbulence, where the coarse-graining length ℓ lies in the inertial cascade range. We show that in the case of sufficiently rough velocity and/or magnetic fields, the nonlinear inter-scale transfer need not vanish and can persist to arbitrarily small scales. Although closed expressions are not available for subscale stress and subscale EMF, we derive rigorous upper bounds on the effective dissipation they produce in terms of scaling exponents of the velocity and magnetic fields. These bounds provide exact constraints on phenomenological theories of MHD turbulence in order to allow the nonlinear cascade of energy and cross-helicity. On the other hand, we prove a very strong version of the Woltjer-Taylor conjecture on conservation of magnetic helicity. Our bounds show that forward cascade of magnetic helicity to asymptotically small scales is impossible unless 3rd-order moments of either velocity or magnetic field become infinite.

Emergence of linear elasticity from the atomistic description of matter.

We investigate the emergence of the continuum elastic limit from the atomistic description of matter at zero temperature considering how locally defined elastic quantities depend on the coarse graining length scale. Results obtained numerically investigating different model systems are rationalized in a unifying picture according to which the continuum elastic limit emerges through a process determined by two system properties, the degree of disorder, and a length scale associated to the transverse low-frequency vibrational modes. The degree of disorder controls the emergence of long-range local shear stress and shear strain correlations, while the length scale influences the amplitude of the fluctuations of the local elastic constants close to the jamming transition.

Sound velocity fluctuations in confined granular materials: Coarse-graining lengths and elastic heterogeneities

We measure the consequences of elastic heterogeneities in confined granular layers using long-wavelength sound velocity determination. By progressively decreasing the coarse-graining length w, which is determined here by the sample size L, we measure the standard deviation of the longitudinal sound velocity and the packing density ϕ, normalized by their ensemble-averaged values. We find that the relative fluctuations in VL and ϕ increase when w is decreased. Importantly, we observe that decreasing the confining pressure P or using nonspherical particles leads to an important increase of the fluctuations in . We conduct simulations of sound propagation in 2D hexagonal packings with contact-stiffness disorder to mimic the inhomogeneous contact networks. The sound velocity fluctuations of coherent longitudinal waves increase either with decreasing the sample size or with increasing the elastic disorder related to confining pressure, in consistency with the experiments. Our experimental observations thus support the scenario of a pressure-dependent mesoscopic length (at ), below which the continuum elasticity breaks down, likely due to the large spatial fluctuation of the shear modulus .

Nonlinear and heterogeneous elasticity of multiply-crosslinked biopolymer networks

We simulate randomly crosslinked networks of biopolymers, characterizing linear and nonlinear elasticity under different loading conditions (uniaxial extension, simple shear, and pure shear). Under uniaxial extension, and upon entering the nonlinear regime, the network switches from a dilatant to contractile response. Analogously, under isochoric conditions (pure shear), the normal stresses change their sign. Both effects are readily explained with a generic weakly nonlinear elasticity theory. The elastic moduli display an intermediate super-stiffening regime, where moduli increase much stronger with applied stress σ than predicted by the force-extension relation of a single wormlike-chain (). We interpret this super-stiffening regime in terms of the reorientation of filaments with the maximum tensile direction of the deformation field. A simple model for the reorientation response gives an exponential stiffening, , in qualitative agreement with our data. The heterogeneous, anisotropic structure of the network is reflected in correspondingly heterogeneous and anisotropic elastic properties. We provide a coarse-graining scheme to quantify the local anisotropy, the fluctuations of the elastic moduli, and the local stresses as a function of coarse-graining length. Heterogeneities of the elastic moduli are strongly correlated with the local density and increase with applied strain.

Geometrical Detection of Weak Non-Gaussianity upon Coarse-Graining

Measures of the non-Gaussianity of a random field depend on how accurately one is able to measure the field. If a signal measured at a certain point is to be averaged with its surroundings, or coarse-grained, the magnitude of its non-Gaussian component can vary. In this article, we investigate the variation of the "apparent" non-Gaussianity, as a function of the coarse-graining length, when we measure non-Gaussianity using the statistics of extrema in the field. We derive how the relative difference between maxima and minima -- which is a geometrical measure of the field's non-Gaussianity -- behaves as the field is coarse-grained over increasingly larger length scales. Measuring this function can give extra information about the non-Gaussian statistics and facilitate its detection.

Coarse-grained local and objective continuum description of three-dimensional granular flows down an inclined surface

Dry, frictional, steady-state granular flows down an inclined, rough surface are studied with discrete particle simulations. From this exemplary flow situation, macroscopic fields, consistent with the conservation laws of continuum theory, are obtained from microscopic data by time-averaging and spatial smoothing (coarse-graining). Two distinct coarse-graining length scale ranges are identified, where the fields are almost independent of the smoothing length w. The smaller, sub-particle length scale, w ≪ d, resolves layers in the flow near the base boundary that cause oscillations in the macroscopic fields. The larger, particle length scale, w ≈ d, leads to smooth stress and density fields, but the kinetic stress becomes scale-dependent; however, this scale-dependence can be quantified and removed. The macroscopic fields involve density, velocity, granular temperature, as well as strain-rate, stress, and fabric (structure) tensors. Due to the plane strain flow, each tensor can be expressed in an inherently anisotropic form with only four objective, coordinate frame invariant variables. For example, the stress is decomposed as: (i) the isotropic pressure, (ii) the “anisotropy” of the deviatoric stress, i.e., the ratio of deviatoric stress (norm) and pressure, (iii) the anisotropic stress distribution between the principal directions, and (iv) the orientation of its eigensystem. The strain rate tensor sets the reference system, and each objective stress (and fabric) variable can then be related, via discrete particle simulations, to the inertial number, I. This represents the plane strain special case of a general, local, and objective constitutive model. The resulting model is compared to existing theories and clearly displays small, but significant deviations from more simplified theories in all variables – on both the different length scales.

Generalized approach to global renormalization-group theory for fluids

The global renormalization-group theory (GRGT) for fluids is derived starting with the square-gradient approximation for the Helmholtz free energy functional such that any mean-field free energy density and direct correlation function can be employed. The new derivation uses Wilson's functions for representing density fluctuations, thereby relaxing the assumption of cosine variation of density fluctuations used in earlier approaches. The generality of the present approach is shown by deriving the relationships to the earlier developments. A qualitative way to infer the free parameters in the present form of GRGT is also suggested. The new theory is applied to square-well fluids of ranges 1.5 and 3.0 (in units of hard-sphere diameter) and Lennard-Jones fluids. It is shown that the present theory produces a flat isotherm in the two-phase region. Thus the theory accounts for fluctuations at all length scales and avoids the use of Maxwell's construction. An analysis of the liquid-vapor phase diagrams and the critical constants obtained for different potentials shows that, with a mean-field free energy density that is accurate away from the critical region and an appropriate coarse graining length for the mean-field theory, GRGT can provide results in good agreement with the simulation and experimental results.

Numerical simulation of 2D granular particles and its analyses by means of the micropolar fluid model

AbstractNumerical simulations of two-dimensional granular flows under uniform mean shear and external body torque were performed following the setting of the authors’ previous study [10]. Convergence of the stresses with the increase of coarse-graining length is investigated. Difference R between vorticity field and spin field is controlled by the external torque and the stresses for the region R > 0 is obtained as well as those for R < 0. The symmetry of the stresses under the change of the sign of R is discussed.

Jammed frictionless disks: Connecting local and global response

By calculating the linear response of packings of soft frictionless disks to quasistatic external perturbations, we investigate the critical scaling behavior of their elastic properties and nonaffine deformations as a function of the distance to jamming. Averaged over an ensemble of similar packings, these systems are well described by elasticity, while in single packings we determine a diverging length scale l* up to which the response of the system is dominated by the local packing disorder. This length scale, which we observe directly, diverges as 1/Deltaz , where Deltaz is the difference between contact number and its isostatic value, and appears to scale identically to the length scale which had been introduced earlier in the interpretation of the spectrum of vibrational modes. It governs the crossover from isostatic behavior at the small scale to continuum behavior at the large scale; indeed we identify this length scale with the coarse graining length needed to obtain a smooth stress field. We characterize the nonaffine displacements of the particles using the displacement angle distribution, a local measure for the amount of relative sliding, and analyze the connection between local relative displacements and the elastic moduli.

Coarse grained free energies with gradient corrections from Monte Carlo simulations in Fourier space

By performing Wang-Landau simulations in Fourier space, we study coarse grained free energies of generalized lattice spin models. The temperature dependence of the lowest order Landau-Ginzburg free energy expansion coefficients of the ${s}^{4}$ metamodel for phase transitions of the Ising universality class for a given coarse-graining length is calculated for the first time and compared to the standard assumptions made in the literature. In particular, the assumption that the so-called gradient coefficient is constant with temperature is in excellent agreement with the simulations. Our results also indicate that the coarse grained potentials follow finite size scaling. Finally, a coarse grained free energy with a temperature-dependent coarse-graining length equal to the system's correlation length is computed. The resulting potentials are compared to their mean-field counterparts.