arXiv subjects

Nigel Goldenfeld

Publications and source records attributed to Nigel Goldenfeld.

At least 19 recordsLinked to original sources

arXiv2026

Broken Ergodicity and the Violation of the Fluctuation-Dissipation Theorem Lead to Generalization Beyond Overfitting in Machine Learning

The remarkable ability of modern neural networks to generalize improves with increasing network capacity, even when the number of model parameters or effective degrees of freedom exceeds the number of training data points. This phenomenon is all the more surprising given that generalization error diverges when the number of model parameters approaches a critical value from below. Here we use dynamical mean field theory to show that this so-called "double descent" behavior is the outcome of a phase transition in the stochastic field theory describing the training process. We calculate the critical exponents and scaling function of the double descent phase transition, and show that it is marked by a breakdown of the fluctuation-dissipation theorem associated with broken ergodicity. The corresponding response function has the same functional form as the simple London model of the superconducting transition, with the rigidity of the wave function corresponding to the neural network's ability to generalize accurately.

cond-mat.dis-nn↗

arXiv2025

Beyond chaos: fluctuations, anomalies and spontaneous stochasticity in fluid turbulence

In this perspective, we consider the development of statistical hydrodynamics, focusing on the way in which the intrinsic stochasticity of turbulent phenomena was identified and is being explored. A major purpose of our discussion is to bring out the role of anomalies in turbulent phenomena, in ways that are not usually done, and to emphasize how the description of turbulent phenomena requires delicate considerations of asymptotic limits. The scope of our narrative includes selected historical aspects that are not usually emphasized, primarily due to G.I. Taylor, as well as discussions of certain aspects of the laminar-turbulent transition, the behaviour of turbulent drag at intermediate Reynolds numbers, and the statistics of fully-developed turbulence that exhibit spontaneous stochasticity.

physics.flu-dyn↗

arXiv2025

Power-laws in phylogenetic trees and the preferential coalescent

Phylogenetic trees capture evolutionary relationships among species and reflect the forces that shaped them. While many studies rely on branch length information, the topology of phylogenetic trees (particularly their degree of imbalance) offers a robust framework for inferring evolutionary dynamics when timing data is uncertain. Classical metrics, such as the Colless and Sackin indices, quantify tree imbalance and have been extensively used to characterize phylogenies. Empirical phylogenies typically show intermediate imbalance, falling between perfectly balanced and highly skewed trees. This regime is marked by a power-law relationship between subtree sizes and their cumulative sizes, governed by a characteristic exponent. Although a recent niche-size model replicates this scaling, its mathematical origin and the exponent's value remain unclear. We present a generative model inspired by Kingman's coalescent that incorporates niche-like dynamics through preferential node coalescence. This process maps to Smoluchowski's coagulation kinetics and is described by a generalized Smoluchowski equation. Our model produces imbalanced trees with power-law exponents matching empirical and numerical observations, revealing the mathematical basis of observed scaling laws and offering new tools to interpret tree imbalance in evolutionary contexts.

q-bio.PE↗

arXiv2025

Nonlinear Diffusion and Decay of an Expanding Turbulent Blob

Turbulence, left unforced, decays and invades the surrounding quiescent fluid. Though ubiquitous, this simple phenomenon has proven hard to capture within a simple and general framework. Experiments in conventional turbulent flow chambers are inevitably complicated by proximity to boundaries and mean flow, obscuring the fundamental aspects of the relaxation to the quiescent fluid state. Here, we circumvent these issues by creating a spatially-localized blob of turbulent fluid using eight converging vortex generators focused towards the center of a tank of water, and observe its decay and spread over decades in time, using particle image velocimetry with a logarithmic sampling rate. The blob initially expands and decays until it reaches the walls of the tank and eventually transitions to a second regime of approximately spatially uniform decay. We interpret these dynamics within the framework of a nonlinear diffusion equation, which predicts that the ideal quiescent-turbulent fluid boundary is sharp and propagates non-diffusively, driven by turbulent eddies while decaying with characteristic scaling laws. We find direct evidence for this model within the expansion phase of our turbulent blob and use it to account for the detailed behavior we observe, in contrast to earlier studies. Our work provides a detailed spatially-resolved narrative for the behavior of turbulence once the forcing is removed, and demonstrates unexpectedly that the turbulent cascade leaves an indelible footprint far into the decay process.

physics.flu-dyn↗

arXiv2024

Noise-driven odd elastic waves in living chiral active matter

Chiral active matter is predicted to exhibit odd elasticity, with nontraditional elastic response arising from a combination of chirality, being out of equilibrium, and the presence of nonreciprocal interactions. One of the resulting phenomena is the possible occurrence of odd elastic waves in overdamped systems, although its experimental realization still remains elusive. Here we show that in overdamped active systems, noise is required to generate persistent elastic waves. In the chiral crystalline phase of active matter, such as that found recently in populations of swimming starfish embryos, the noise arises from the self-driving of active particles and their mutual collisions, a key factor that has been missing in previous studies. We identify the criterion for the occurrence of noise-driven odd elastic waves and construct the corresponding phase diagram, which is also applicable to general chiral active crystals. Our results can be used to predict the experimental conditions for achieving a transition to self-sustained elastic waves in overdamped active systems.

cond-mat.soft↗

arXiv2024

Self-consistent expansion and field-theoretic renormalization group for a singular nonlinear diffusion equation with anomalous scaling

The method of self-consistent expansions is a powerful tool for handling strong coupling problems that might otherwise be beyond the reach of perturbation theory, providing surprisingly accurate approximations even at low order. First applied in its embryonic form to fully-developed turbulence, it has subsequently been successfully applied to a variety of problems that include polymer statistics, interface dynamics and high order perturbation theory for the anharmonic oscillator. Here we show that the self-consistent expansion can be applied to singular perturbation problems arising in the theory of partial differential equations. We demonstrate its application to Barenblatt's nonlinear diffusion equation for porous media filtration, where the long-time asymptotics exhibits anomalous dimensions that can be systematically calculated using the perturbative renormalization group. We find that even the first order self-consistent expansion improves the approximation of the anomalous dimension obtained by the first-order perturbative renormalization group, especially in the strong coupling regime. We also develop a field-theoretic framework for deterministic partial differential equations to facilitate the application of self-consistent expansions to other dynamic systems, and illustrate its application using the example of Barenblatt's equation. The scope of our results on the combination of renormalization group and self-consistent expansions is limited to partial differential equations whose long-time asymptotics is controlled by incomplete similarity. However, our work suggests that these methods could be applied to a broader suite of singular perturbation problems such as boundary layer theory, multiple scales analysis and matched asymptotic expansions, for which excellent approximations using renormalization group methods alone are already available.

cond-mat.stat-mech↗

arXiv2024

Spontaneous stochasticity amplifies even thermal noise to the largest scales of turbulence in a few eddy turnover times

How predictable are turbulent flows? Here we use theoretical estimates and shell model simulations to argue that Eulerian spontaneous stochasticity, a manifestation of the non-uniqueness of the solutions to the Euler equation that is conjectured to occur in Navier-Stokes turbulence at high Reynolds numbers, leads to universal statistics at finite times, not just at infinite time as for standard chaos. These universal statistics are predictable, even though individual flow realizations are not. Any small-scale noise vanishing slowly enough with increasing Reynolds number can trigger spontaneous stochasticity and here we show that thermal noise alone, in the absence of any larger disturbances, would suffice. If confirmed for Navier-Stokes turbulence, our findings would imply that intrinsic stochasticity of turbulent fluid motions at all scales can be triggered even by unavoidable molecular noise, with implications for modeling in engineering, climate, astrophysics and cosmology.

physics.flu-dyn↗

arXiv2023

There's Plenty of Room in the Middle: The Unsung Revolution of the Renormalization Group

The remarkable technical contributions of Michael E. Fisher to statistical physics and the development of the renormalization group are widely known and deeply influential. But less well-known is his early and profound appreciation of the way in which renormalization group created a revolution in our understanding of how physics -- in fact, all science -- is practiced, and the concomitant adjustment that needs to be made to our conception of the purpose and philosophy of science. In this essay, I attempt to redress this imbalance, with examples from Fisher's writings and my own work. It is my hope that this tribute will help remove some of the confusion that surrounds the scientific usage of minimal models and renormalization group concepts, as well as their limitations, in the ongoing effort to understand emergence in complex systems. This paper will be published in "50 years of the renormalization group", dedicated to the memory of Michael E. Fisher, edited by Amnon Aharony, Ora Entin-Wohlman, David Huse and Leo Radzihovsky, World Scientific (in press).

cond-mat.stat-mech↗

arXiv2021

Frequency of surveillance testing necessary to reduce transmission of SARS-CoV-2

We estimate the reduction in transmission of SARS-CoV-2 achievable by surveillance testing of a susceptible population at different frequencies, comparing the cases of both the original Wuhan strain and the Delta variant. We estimate the viral dynamics using viral copy number at first detection combined with considerations arising from aerosol transmission. We take into account the recent findings that infected vaccinated adults may have live viral loads at the same level as infected unvaccinated adults. Our estimates suggest that twice weekly testing, which was adequate for the original strains of SARS-CoV-2 will be insufficient on its own to contain the spread of the Delta variant of concern. We exclude consideration of contact tracing since the rapidity of the onset of viral titre in the case of the Delta variant suggests that unless contact tracing and quarantine are performed very rapidly (ie. much less than a day), these mitigations will be of minimal impact in reducing transmission. These crude estimates do not take into account heterogeneity of susceptibility, social activity, and compliance, nor do they include the additional reduction in transmission that could be achieved by masking and social distancing. In the setting of a large public university, these considerations suggest that risk-targeted testing of vaccinated students, staff and faculty combined with surveillance testing of all unvaccinated individuals is the most efficient way to reduce transmission of COVID-19.

physics.soc-ph↗

arXiv2021

Dissipation-Range Fluid Turbulence and Thermal Noise

We revisit the issue of whether thermal fluctuations are relevant for incompressible fluid turbulence, and estimate the scale at which they become important. As anticipated by Betchov in a prescient series of works more than six decades ago, this scale is about equal to the Kolmogorov length, even though that is several orders of magnitude above the mean free path. This result implies that the deterministic version of the incompressible Navier-Stokes equation is inadequate to describe the dissipation range of turbulence in molecular fluids. Within this range, the fluctuating hydrodynamics equation of Landau and Lifschitz is more appropriate. In particular, our analysis implies that both the exponentially decaying energy spectrum and the far-dissipation range intermittency predicted by Kraichnan for deterministic Navier-Stokes will be generally replaced by Gaussian thermal equipartition at scales just below the Kolmogorov length. Stochastic shell model simulations at high Reynolds numbers verify our theoretical predictions and reveal furthermore that inertial-range intermittency can propagate deep into the dissipation range, leading to large fluctuations in the equipartition length scale. We explain the failure of previous scaling arguments for the validity of deterministic Navier-Stokes equations at any Reynolds number and we provide a mathematical interpretation and physical justification of the fluctuating Navier-Stokes equation as an ``effective field-theory'' valid below some high-wavenumber cutoff $\Lambda$, rather than as a continuum stochastic partial differential equation. At Reynolds number around a million the strongest turbulent excitations observed in our simulation penetrate down to a length-scale of microns. However, for longer observation times or higher Reynolds numbers, more extreme turbulent events could lead to a local breakdown of fluctuating hydrodynamics.

physics.flu-dyn↗

arXiv2021

Thermal noise competes with turbulent fluctuations below millimeter scales

Turbulent flows frequently accompany physical, chemical and biological processes, such as mixing, two-phase flow, combustion and even foraging by bacteria and plankton larvae, all of which are in principle subject to thermal fluctuations already on scales of several microns. Nevertheless the large separation between the millimeter scale at which turbulent fluctuations begin to be strongly damped and the mean free path of the fluid has been generally assumed to imply that thermal fluctuations are irrelevant to the turbulent dissipation range. Here we use statistical mechanical estimates to show that thermal fluctuations are not negligible compared to turbulent eddies in the dissipation range. Simulation of the Sabra shell model shows that intermittent bursts of turbulence lead to a fluctuating length scale below which thermal fluctuations are important: over three decades of length, from sub-millimeter scales down to the mean free path, thermal fluctuations coexist with hydrodynamics. Our results imply that thermal fluctuations cannot be neglected when modeling turbulent phenomena in the far dissipation range.

physics.flu-dyn↗

arXiv2021

DNA Supercoiling Drives a Transition between Collective Modes of Gene Synthesis

Recent experiments showed that multiple copies of the molecular machine RNA polymerase (RNAP) can efficiently synthesize mRNA collectively in the active state of the promoter. However, environmentally-induced promoter repression results in long-distance antagonistic interactions that drastically reduce the speed of RNAPs and cause a quick arrest of mRNA synthesis. The mechanism underlying this transition between cooperative and antagonistic dynamics remains poorly understood. In this Letter, we introduce a continuum deterministic model for the translocation of RNAPs, where the speed of an RNAP is coupled to the local DNA supercoiling as well as the density of RNAPs on the gene. We assume that torsional stress experienced by individual RNAPs is exacerbated by high RNAP density on the gene and that transcription factors act as physical barriers to the diffusion of DNA supercoils. We show that this minimal model exhibits two transcription modes mediated by the torsional stress: a fluid mode when the promoter is active and a torsionally stressed mode when the promoter is repressed, in quantitative agreement with experimentally observed dynamics of co-transcribing RNAPs. Our work provides an important step towards understanding the collective dynamics of molecular machines involved in gene expression.

q-bio.SC↗

arXiv2021

Stochastic social behavior coupled to COVID-19 dynamics leads to waves, plateaus and an endemic state

It is well recognized that population heterogeneity plays an important role in the spread of epidemics. While individual variations in social activity are often assumed to be persistent, i.e. constant in time, here we discuss the consequences of dynamic heterogeneity. By integrating the stochastic dynamics of social activity into traditional epidemiological models we demonstrate the emergence of a new long timescale governing the epidemic in broad agreement with empirical data. Our model captures multiple features of real-life epidemics such as COVID-19, including prolonged plateaus and multiple waves, which are transiently suppressed due to the dynamic nature of social activity. The existence of the long timescale due to the interplay between epidemic and social dynamics provides a unifying picture of how a fast-paced epidemic typically will transition to the endemic state.

q-bio.PE↗

arXiv2020

A Field-Theoretic Model for Chemotaxis in Run and Tumble Particles

In this paper we develop a field-theoretic description for run and tumble chemotaxis, based on a density functional description of crystalline materials modified to capture orientational ordering. We show that this framework, with its in-built multi-particle interactions, soft-core repulsion and elasticity is ideal for describing continuum collective phases with particle resolution, but on diffusive timescales. We show that our model exhibits particle aggregation in an externally imposed constant attractant field, as is observed for phototactic or thermotactic agents. We also show that this model captures particle aggregation through self-chemotaxis, an important mechanism that aids quorum dependent cellular interactions.

cond-mat.soft↗

arXiv2020

Time-dependent heterogeneity leads to transient suppression of the COVID-19 epidemic, not herd immunity

Epidemics generally spread through a succession of waves that reflect factors on multiple timescales. On short timescales, super-spreading events lead to burstiness and overdispersion, while long-term persistent heterogeneity in susceptibility is expected to lead to a reduction in the infection peak and the herd immunity threshold (HIT). Here, we develop a general approach to encompass both timescales, including time variations in individual social activity, and demonstrate how to incorporate them phenomenologically into a wide class of epidemiological models through parameterization. We derive a non-linear dependence of the effective reproduction number Re on the susceptible population fraction S. We show that a state of transient collective immunity (TCI) emerges well below the HIT during early, high-paced stages of the epidemic. However, this is a fragile state that wanes over time due to changing levels of social activity, and so the infection peak is not an indication of herd immunity: subsequent waves can and will emerge due to behavioral changes in the population, driven (e.g.) by seasonal factors. Transient and long-term levels of heterogeneity are estimated by using empirical data from the COVID-19 epidemic as well as from real-life face-to-face contact networks. These results suggest that the hardest-hit areas, such as NYC, have achieved TCI following the first wave of the epidemic, but likely remain below the long-term HIT. Thus, in contrast to some previous claims, these regions can still experience subsequent waves.

q-bio.PE↗

arXiv2020

Modeling COVID-19 dynamics in Illinois under non-pharmaceutical interventions

We present modeling of the COVID-19 epidemic in Illinois, USA, capturing the implementation of a Stay-at-Home order and scenarios for its eventual release. We use a non-Markovian age-of-infection model that is capable of handling long and variable time delays without changing its model topology. Bayesian estimation of model parameters is carried out using Markov Chain Monte Carlo (MCMC) methods. This framework allows us to treat all available input information, including both the previously published parameters of the epidemic and available local data, in a uniform manner. To accurately model deaths as well as demand on the healthcare system, we calibrate our predictions to total and in-hospital deaths as well as hospital and ICU bed occupancy by COVID-19 patients. We apply this model not only to the state as a whole but also its sub-regions in order to account for the wide disparities in population size and density. Without prior information on non-pharmaceutical interventions (NPIs), the model independently reproduces a mitigation trend closely matching mobility data reported by Google and Unacast. Forward predictions of the model provide robust estimates of the peak position and severity and also enable forecasting the regional-dependent results of releasing Stay-at-Home orders. The resulting highly constrained narrative of the epidemic is able to provide estimates of its unseen progression and inform scenarios for sustainable monitoring and control of the epidemic.

q-bio.PE↗

arXiv2020

Window of Opportunity for Mitigation to Prevent Overflow of ICU capacity in Chicago by COVID-19

We estimate the growth in demand for ICU beds in Chicago during the emerging COVID-19 epidemic, using state-of-the-art computer simulations calibrated for the SARS-CoV-2 virus. The questions we address are these: (1) Will the ICU capacity in Chicago be exceeded, and if so by how much? (2) Can strong mitigation strategies, such as lockdown or shelter in place order, prevent the overflow of capacity? (3) When should such strategies be implemented? Our answers are as follows: (1) The ICU capacity may be exceeded by a large amount, probably by a factor of ten. (2) Strong mitigation can avert this emergency situation potentially, but even that will not work if implemented too late. (3) If the strong mitigation precedes April 1st, then the growth of COVID-19 can be controlled and the ICU capacity could be adequate. The earlier the strong mitigation is implemented, the greater the probability that it will be successful. After around April 1 2020, any strong mitigation will not avert the emergency situation. In Italy, the lockdown occurred too late and the number of deaths is still doubling every 2.3 days. It is difficult to be sure about the precise dates for this window of opportunity, due to the inherent uncertainties in computer simulation. But there is high confidence in the main conclusion that it exists and will soon be closed. Our conclusion is that, being fully cognizant of the societal trade-offs, there is a rapidly closing window of opportunity to avert a worst-case scenario in Chicago, but only with strong mitigation/lockdown implemented in the next week at the latest. If this window is missed, the epidemic will get worse and then strong mitigation/lockdown will be required after all, but it will be too late.

q-bio.PE↗

arXiv2019

Evidence for photoinduced sliding of the charge-order condensate in La$_{1.875}$Ba$_{0.125}$CuO$_4$

We use femtosecond resonant soft x-ray scattering to measure the ultrafast optical melting of charge-order correlations in La$_{1.875}$Ba$_{0.125}$CuO$_4$. By analyzing both the energy-resolved and energy-integrated order parameter dynamics, we find evidence of a short-lived nonequilibrium state, whose features are compatible with a sliding charge density wave coherently set in motion by the pump. This transient state exhibits shifts in both the quasielastic line energy and its wave vector, as expected from a classical Doppler effect. The wave vector change is indeed found to directly follow the pump propagation direction. These results demonstrate the existence of sliding charge order behavior in an unconventional charge density wave system and underscore the power of ultrafast optical excitation as a tool to coherently manipulate electronic condensates.

cond-mat.str-el↗