Nontrivial Scaling Research Articles

Molecular dynamics study of shear-induced long-range correlations in simple fluids

We investigate long-range correlations (LRCs) induced by shear flow using the molecular dynamics (MD) simulation. We observe the LRCs by comparing the MD results with the linearized fluctuating hydrodynamics (LFH). We find that the MD result has large finite-size effects, and it prevents the occurrence of LRCs in small systems. We examine the finite-size effects using a sufficiently large system consisting of more than ten million particles, and verify the existence of shear-induced LRCs without ambiguity. Furthermore, we show that MD result is quantitatively consistent with the LFH solution for the large system. As we reduce the system size $L$ or increase the shear rate $\dot{\gamma}$, the hydrodynamic description gradually breaks down in the long-wavelength region. We define a characteristic wavenumber $k^{\rm vio}$ associated with the breakdown and find the nontrivial scaling relations $k^{\rm vio} \propto L^{-\omega}$ and $k^{\rm vio} \propto \dot{\gamma}$, where $\omega$ is an exponent depending on $\dot{\gamma}$. These relations enable us to estimate the finite-size effects in a larger-size simulation from a smaller system.

Scaling of the entropy production rate in a φ^{4} model of active matter.

In active φ^{4} field theories the nonequilibrium terms play an important role in describing active phase separation; however, they are irrelevant, in the renormalization group sense, at the critical point. Their irrelevance makes the critical exponents the same as those of the Ising universality class. Despite their irrelevance, they contribute to a nontrivial scaling of the entropy production rate at criticality. We consider the nonequilibrium dynamics of a nonconserved scalar field φ (Model A) driven out-of-equilibrium by a persistent noise that is correlated on a finite timescale τ, as in the case of active baths. We perform the computation of the density of entropy production rate σ and we study its scaling near the critical point. We find that similar to the case of active Model A, and although the nonlinearities responsible for nonvanishing entropy production rates in the two models are quite different, the irrelevant parameter τ makes the critical dynamics irreversible.

Renormalization group study of the dynamics of active membranes: Universality classes and scaling laws.

Motivated by experimental observations of patterning at the leading edge of motile eukaryotic cells, we introduce a general model for the dynamics of nearly-flat fluid membranes driven from within by an ensemble of activators. We include, in particular, a kinematic coupling between activator density and membrane slope which generically arises whenever the membrane has a nonvanishing normal speed. We unveil the phase diagram of the model by means of a perturbative field-theoretical renormalization group analysis. Due to the aforementioned kinematic coupling the natural early-time dynamical scaling is acoustic, that is the dynamical critical exponent is 1. However, as soon as the the normal velocity of the membrane is tuned to zero, the system crosses over to diffusive dynamic scaling in mean field. Distinct critical points can be reached depending on how the limit of vanishing velocity is realized: in each of them corrections to scaling due to nonlinear coupling terms must be taken into account. The detailed analysis of these critical points reveals novel scaling regimes which can be accessed with perturbative methods, together with signs of strong coupling behavior, which establishes a promising ground for further nonperturbative calculations. Our results unify several previous studies on the dynamics of active membrane, while also identifying nontrivial scaling regimes which cannot be captured by passive theories of fluctuating interfaces and are relevant for the physics of living membranes.

Critical peeling of tethered nanoribbons.

The peeling of an immobile adsorbed membrane is a well-known problem in engineering and macroscopic tribology. In the classic setup, picking up at one extreme and pulling off result in a peeling force that is a decreasing function of the pickup angle. As one end is lifted, the detachment front retracts to meet the immobile tail. At the nanoscale, interesting situations arise with the peeling of graphene nanoribbons (GNRs) on gold, as realized, e.g., by atomic force microscopy. The nanosized system shows a constant-force steady peeling regime, where the tip lifting h produces no retraction of the ribbon detachment point, and just an advancement h of the free tail end. This is opposite to the classic case, where the detachment point retracts and the tail end stands still. Here we characterise, by analytical modeling and numerical simulations, a third, experimentally relevant setup where the nanoribbon, albeit structurally lubric, does not have a freely moving tail end, which is instead elastically tethered. Surprisingly, novel nontrivial scaling exponents appear that regulate the peeling evolution. As the detachment front retracts and the tethered tail is stretched, power laws of h characterize the shrinking of the adhered length, the growth of peeling force and the peeling angle. These exponents precede the final total detachment as a critical point, where the entire ribbon eventually hangs suspended between the tip and the tethering spring. These analytical predictions are confirmed by realistic MD simulations, retaining the full atomistic description, also confirming their survival at finite experimental temperatures.

Numerical evidence of persisting surface roughness when deposition stops

Do evolving surfaces become flat or not with time evolving when material deposition stops? As one qualitative exploration of this interesting issue, modified stochastic models for persisting roughness have been proposed by Schwartz and Edwards (2004 Phys. Rev. E 70 061602). In this work, we perform numerical simulations on the modified versions of Edwards–Wilkinson (EW) and Kardar–Parisi–Zhang (KPZ) systems when the angle of repose is introduced. Our results show that the evolving surface always presents persisting roughness during the flattening process, and sand dune-like morphology could gradually appear, even when the angle of repose is very small. Nontrivial scaling properties and differences of evolving surfaces between the modified EW and KPZ systems are also discussed.

Control of noise-induced coherent oscillations in three-neuron motifs

The phenomenon of self-induced stochastic resonance (SISR) requires a nontrivial scaling limit between the deterministic and the stochastic timescales of an excitable system, leading to the emergence of coherent oscillations which are absent without noise. In this paper, we numerically investigate SISR and its control in single neurons and three-neuron motifs made up of the Morris–Lecar model. In single neurons, we compare the effects of electrical and chemical autapses on the degree of coherence of the oscillations due to SISR. In the motifs, we compare the effects of altering the synaptic time-delayed couplings and the topologies on the degree of SISR. Finally, we provide two enhancement strategies for a particularly poor degree of SISR in motifs with chemical synapses: (1) we show that a poor SISR can be significantly enhanced by attaching an electrical or an excitatory chemical autapse on one of the neurons, and (2) we show that by multiplexing the motif with a poor SISR to another motif (with a high SISR in isolation), the degree of SISR in the former motif can be significantly enhanced. We show that the efficiency of these enhancement strategies depends on the topology of the motifs and the nature of synaptic time-delayed couplings mediating the multiplexing connections.

On the limits of quasi-static theory in plasmonic nanostructures

The approximated analytical approach of quasi-static theory (QST) is widely used in modelling the optical response of plasmonic nanoparticles. It is well known that its accuracy is remarkable provided that the particle is much smaller than the wavelength of the interacting radiation and that the field induced inside the structure is approximately uniform. Here, we investigate the limits of QST range of validity for gold nanostructures freestanding in air. First, we compare QST predictions of scattering spectra of nanospheres and cylindrical nanowires of various sizes with the exact results provided by Mie scattering theory. We observe a non-monotonic behaviour of the error of QST as a function of the characteristic length of the nanostructures, revealing a non-trivial scaling of its accuracy with the scatterer size. Second, we study nanowires with elliptical section upon different excitation conditions by performing finite element numerical analysis. Comparing simulation results with QST estimates of the extinction cross-section, we find that QST accuracy is strongly dependent on the excitation conditions, yielding good results even if the field is highly inhomogeneous inside the structure.

Scaling concepts in ‘omics: Nuclear lamin-B scales with tumor growth and often predicts poor prognosis, unlike fibrosis

Physicochemical principles such as stoichiometry and fractal assembly can give rise to characteristic scaling between components that potentially include coexpressed transcripts. For key structural factors within the nucleus and extracellular matrix, we discover specific gene-gene scaling exponents across many of the 32 tumor types in The Cancer Genome Atlas, and we demonstrate utility in predicting patient survival as well as scaling-informed machine learning (SIML). All tumors with adjacent tissue data show cancer-elevated proliferation genes, with some genes scaling with the nuclear filament LMNB1, including the transcription factor FOXM1 that we show directly regulates LMNB1 SIML shows that such regulated cancers cluster together with longer overall survival than dysregulated cancers, but high LMNB1 and FOXM1 in half of regulated cancers surprisingly predict poor survival, including for liver cancer. COL1A1 is also studied because it too increases in tumors, and a pan-cancer set of fibrosis genes shows substoichiometric scaling with COL1A1 but predicts patient outcome only for liver cancer-unexpectedly being prosurvival. Single-cell RNA-seq data show nontrivial scaling consistent with power laws from bulk RNA and protein analyses, and SIML segregates synthetic from contractile cancer fibroblasts. Our scaling approach thus yields fundamentals-based power laws relatable to survival, gene function, and experiments.

A Note on the Role of Seasonal Expansions and Contractions of the Flowing Fluvial Network on Metapopulation Persistence

AbstractDoes a dynamic drainage density have a role on species persistence in the river basin? The general viability of a focus species under time‐varying hydrologic connectivity and habitat quality is a topic gaining traction in view of recent advances in our understanding of flowing fluvial network dynamics and of ecological interactions occurring on directed trees. Here, we combine metapopulation dynamics and scaling theory to investigate how the structure of river networks and time‐changing hydrological and geomorphological attributes control local metapopulation survival. This is done by introducing seasonal fluctuations of the drainage density subsuming overall time‐changing connectivity and distributed changes in habitat quality of the fluvial domain. Suitable replicas of channel networks within an assigned domain are used to compute the statistics of evolving metapopulation capacities, properties of a landscape matrix measuring the viability of the focus species. To obtain consistent replicas of the substrate for ecological interactions, we employ constructs whose suitability for the task has long been established. We find that the river network structure blends the fluctuations into a nontrivial scaling of the metapopulation capacity with the sum of total active contributing sites at any point of the flowing network. The latter is proportional to the mean distance to the outlet of the flowing dendrite and to the tree diameter—a measure of the overall connectivity of the active stream links. Scaling emerges as a robust ensemble property that enables the linkage of ecological patterns across a river network to clearly identified hydrological and geomorphological factors.

The distance exponent for Liouville first passage percolation is positive

Discrete Liouville first passage percolation (LFPP) with parameter \(\xi > 0\) is the random metric on a sub-graph of \(\mathbb Z^2\) obtained by assigning each vertex z a weight of \(e^{\xi h(z)}\), where h is the discrete Gaussian free field. We show that the distance exponent for discrete LFPP is strictly positive for all \(\xi > 0\). More precisely, the discrete LFPP distance between the inner and outer boundaries of a discrete annulus of size \(2^n\) is typically at least \(2^{\alpha n}\) for an exponent \(\alpha > 0\) depending on \(\xi \). This is a crucial input in the proof that LFPP admits non-trivial subsequential scaling limits for all \(\xi > 0\) and also has theoretical implications for the study of distances in Liouville quantum gravity.

Comment on "Turbulent compressible fluid: Renormalization group analysis, scaling regimes, and anomalous scaling of advected scalar fields".

Recently, asymptotic scaling behavior of the compressible randomly forced Navier-Stokes equation has been analyzed with the use of field-theoretic renormalization group near four dimensions [Phys. Rev. E 95, 033120 (2017)PREHBM2470-004510.1103/PhysRevE.95.033120]. Two infrared stable nontrivial asymptotic scaling patterns have been found and their parameters determined in the one-loop approximation. Here, it is pointed out that the asymptotic scaling behavior predicted in this way may not be realized in physical fluid systems. This is a consequence of restrictions on viscosities imposed by the energy balance equation which the one-loop fixed-point value of the relative viscosity fails to meet.

Distribution of the time of the maximum for stationary processes

We consider a one-dimensional stationary stochastic process of duration T. We study the probability density function (PDF) of the time at which reaches its global maximum. By using a path integral method, we compute for a number of equilibrium and nonequilibrium stationary processes, including the Ornstein-Uhlenbeck process, Brownian motion with stochastic resetting and a single confined run-and-tumble particle. For a large class of equilibrium stationary processes that correspond to diffusion in a confining potential, we show that the scaled distribution , for large T, has a universal form (independent of the details of the potential). This universal distribution is uniform in the “bulk”, i.e., for and has a nontrivial edge scaling behavior for (and when ), that we compute exactly. Moreover, we show that for any equilibrium process the PDF is symmetric around , i.e., . This symmetry provides a simple method to decide whether a given stationary time series is at equilibrium or not.

Wigner function for noninteracting fermions in hard-wall potentials

The Wigner function $W_N({\bf x}, {\bf p})$ is a useful quantity to characterize the quantum fluctuations of an $N$-body system in its phase space. Here we study $W_N({\bf x}, {\bf p})$ for $N$ noninteracting spinless fermions in a $d$-dimensional spherical hard box of radius $R$ at temperature $T=0$. In the large $N$ limit, the local density approximation (LDA) predicts that $W_N({\bf x}, {\bf p}) \approx 1/(2 \pi \hbar)^d$ inside a finite region of the $({\bf x}, {\bf p})$ plane, namely for $|{\bf x}| < R$ and $|{\bf p}| < k_F$ where $k_F$ is the Fermi momentum, while $W_N({\bf x}, {\bf p})$ vanishes outside this region, or "droplet", on a scale determined by quantum fluctuations. In this paper we investigate systematically, in this quantum region, the structure of the Wigner function along the edge of this droplet, called the Fermi surf. In one dimension, we find that there are three distinct edge regions along the Fermi surf and we compute exactly the associated nontrivial scaling functions in each regime. We also study the momentum distribution $\hat \rho_N(p)$ and find a striking algebraic tail for very large momenta $\hat \rho_N(p) \propto 1/p^4$, well beyond $k_F$, reminiscent of a similar tail found in interacting quantum systems (discussed in the context of Tan's relation). We then generalize these results to higher $d$ and find, remarkably, that the scaling function close to the edge of the box is universal, i.e., independent of the dimension~$d$.

Scaling of temporal entanglement in proximity to integrability

Describing dynamics of quantum many-body systems is a formidable challenge due to rapid generation of quantum entanglement between remote degrees of freedom. A promising approach to tackle this challenge, which has been proposed recently, is to characterize the quantum dynamics of a many-body system and its properties as a bath via the Feynman-Vernon influence matrix (IM), which is an operator in the space of time trajectories of local degrees of freedom. Physical understanding of the general scaling of the IM's temporal entanglement and its relation to basic dynamical properties is highly incomplete to present day. In this Article, we analytically compute the exact IM for a family of integrable Floquet models - the transverse-field kicked Ising chain - finding a Bardeen-Cooper-Schrieffer-like "wavefunction" on the Schwinger-Keldysh contour with algebraically decaying correlations. We demonstrate that the IM exhibits area-law temporal entanglement scaling for all parameter values. Furthermore, the entanglement pattern of the IM reveals the system's phase diagram, exhibiting jumps across transitions between distinct Floquet phases. Near criticality, a non-trivial scaling behavior of temporal entanglement is found. The area-law temporal entanglement allows us to efficiently describe the effects of sizeable integrability-breaking perturbations for long evolution times by using matrix product state methods. This work shows that tensor network methods are efficient in describing the effect of non-interacting baths on open quantum systems, and provides a new approach to studying quantum many-body systems with weakly broken integrability.

Active Brownian motion with directional reversals

Active Brownian motion with intermittent direction reversals is common in bacteria like Myxococcus xanthus and Pseudomonas putida. We show that, for such a motion in two dimensions, the presence of the two timescales set by the rotational diffusion constant D_{R} and the reversal rate γ gives rise to four distinct dynamical regimes: (I) t≪min(γ^{-1},D_{R}^{-1}), (II) γ^{-1}≪t≪D_{R}^{-1}, (III) D_{R}^{-1}≪t≪γ^{-1}, and (IV) t≫max(γ^{-1},D_{R}^{-1}), showing distinct behaviors. We characterize these behaviors by analytically computing the position distribution and persistence exponents. The position distribution shows a crossover from a strongly nondiffusive and anisotropic behavior at short times to a diffusive isotropic behavior via an intermediate regime, II or III. In regime II, we show that, the position distribution along the direction orthogonal to the initial orientation is a function of the scaled variable z∝x_{⊥}/t with a nontrivial scaling function, f(z)=(2π^{3})^{-1/2}Γ(1/4+iz)Γ(1/4-iz). Furthermore, by computing the exact first-passage time distribution, we show that a persistence exponent α=1 emerges due to the direction reversal in this regime.

Kinetic roughening and nontrivial scaling in the Kardar–Parisi–Zhang growth with long-range temporal correlations

Long-range spatiotemporal correlations may play important roles in non-equilibrium surface growth processes. In order to explore the effects of long-range temporal correlation on dynamic scaling of growing surfaces, we carry out extensive numerical simulations of the (1 + 1)- and (2 + 1)-dimensional Kardar–Parisi–Zhang (KPZ) growth system in the presence of temporally correlated noise, and compare our results with previous theoretical predictions and numerical simulations. We find that surface morphologies are obviously affected with long-range temporal correlations, and as the temporal correlation exponent increases, the KPZ surfaces develop gradually faceted patterns in the saturated growth regimes. Our results show that the temporal correlated KPZ system displays evidently nontrivial dynamic properties when 0 < θ < 0.5, the characteristic roughness exponents satisfy α < α s, and α loc exhibits non-universal scaling within local window sizes, which differs with the existing dynamic scaling hypotheses, both in the (1 + 1)- and (2 + 1)-dimensions.

Vafa–Witten Invariants from Exceptional Collections

Supersymmetric D-branes supported on the complex two-dimensional base S of the local Calabi–Yau threefold \(K_S\) are described by semi-stable coherent sheaves on S. Under suitable conditions, the BPS indices counting these objects (known as generalized Donaldson–Thomas invariants) coincide with the Vafa–Witten invariants of S (which encode the Betti numbers of the moduli space of semi-stable sheaves). For surfaces which admit a strong collection of exceptional sheaves, we develop a general method for computing these invariants by exploiting the isomorphism between the derived category of coherent sheaves and the derived category of representations of a suitable quiver with potential (Q, W) constructed from the exceptional collection. We spell out the dictionary between the Chern class \(\gamma \) and polarization J on S versus the dimension vector \(\vec {N}\) and stability parameters \(\vec {\zeta }\) on the quiver side. For all examples that we consider, which include all del Pezzo and Hirzebruch surfaces, we find that the BPS indices \(\Omega _\star (\gamma )\) at the attractor point (or self-stability condition) vanish, except for dimension vectors corresponding to simple representations and pure D0-branes. This opens up the possibility to compute the BPS indices in any chamber using either the flow tree or the Coulomb branch formula. In all cases we find precise agreement with independent computations of Vafa–Witten invariants based on wall-crossing and blow-up formulae. This agreement suggests that (1) generating functions of DT invariants for a large class of quivers coming from strong exceptional collections are mock modular functions of higher depth and (2) non-trivial single-centered black holes and scaling solutions do not exist quantum mechanically in such local Calabi–Yau geometries.

Simulating Hydrodynamics on Noisy Intermediate-Scale Quantum Devices with Random Circuits.

In a recent milestone experiment, Google's processor Sycamore heralded the era of "quantum supremacy" by sampling from the output of (pseudo-)random circuits. We show that such random circuits provide tailor-made building blocks for simulating quantum many-body systems on noisy intermediate-scale quantum (NISQ) devices. Specifically, we propose an algorithm consisting of a random circuit followed by a trotterized Hamiltonian time evolution to study hydrodynamics and to extract transport coefficients in the linear response regime. We numerically demonstrate the algorithm by simulating the buildup of spatiotemporal correlation functions in one- and two-dimensional quantum spin systems, where we particularly scrutinize the inevitable impact of errors present in any realistic implementation. Importantly, we find that the hydrodynamic scaling of the correlations is highly robust with respect to the size of the Trotter step, which opens the door to reach nontrivial time scales with a small number of gates. While errors within the random circuit are shown to be irrelevant, we furthermore unveil that meaningful results can be obtained for noisy time evolutions with error rates achievable on near-term hardware. Our work emphasizes the practical relevance of random circuits on NISQ devices beyond the abstract sampling task.

Long-ranged correlations in large deviations of local clustering.

In systems of diffusing particles, we investigate large deviations of a time-averaged measure of clustering around one particle. We focus on biased ensembles of trajectories, which realize large-deviation events. The bias acts on a single particle, but elicits a response that spans the whole system. We analyze this effect through the lens of macroscopic fluctuation theory, focusing on the coupling of the bias to hydrodynamic modes. This explains that the dynamical free energy has nontrivial scaling relationships with the system size, in 1 and 2 spatial dimensions. We show that the long-ranged response to a bias on one particle also has consequences when biasing two particles.

Population structure within the Hard Maple complex

Sugar maple (Acer saccharum Marsh.) is the most economically important member of the Hard Maple species complex, a group of related species that occupy a range between Canada and Mexico, which are adapted to distinct ecological niches. Sugar maple has been identified as vulnerable to rapid climate change, and sustainable solutions are needed to support its role in the production of maple sugar, as well as timber and nursery production in the northeastern United States. Genetic relationships among sugar maple and its allies are largely unknown. In the current study, genetic relationships of 278 individuals from six hard maple species were assessed using 17 multi-allelic microsatellite (SSR) loci. Genetic variance was partitioned into separate components for variation within and among populations and within and among species. Most of the divergence among populations (FPT = 0.263) reflected interspecific divergence (FST = 0.169), but provenances within species also differed at nontrivial scales (FPS = 0.113). Estimation and testing of paired interprovenance divergence showed that all population pairs were statistically divergent. Principal coordinates analysis indicated that the pattern of radiation observed among these taxa is broadly compatible with geography.