Nonequilibrium Effects Research Articles

Role of bulk viscosity on the flow physics past a rotating cylinder

The present study investigates the impact of bulk viscosity on the complex flow dynamics past a rotating cylinder, with particular emphasis on compressible and non-equilibrium effects that emerge in nitrogen (N2) and carbon dioxide (CO2). By solving unsteady conservation laws obtained from the Boltzmann–Curtiss transport equation, the research focuses on key flow features such as vortex shedding, vorticity generation, enstrophy, kinetic energy dissipation, and the degree of thermal non-equilibrium. Numerical simulations are performed at a Mach number of 0.6 using the dbnsTurbFoam solver with unstructured meshes, and the computational model is verified using available data for flow past a rotating cylinder. The results reveal that bulk viscosity significantly affects vortex shedding, particularly suppressing vortex formation and reducing flow instability. In CO2, high bulk viscosity nearly eliminates vortex shedding, leading to a laminar wake, while in N2, vortex shedding is dampened but persists. Enstrophy and vorticity production through stretching and baroclinic effects are also reduced in both gases as bulk viscosity increases, with CO2 showing more dramatic reductions due to its higher inherent viscosity. The study further indicates that bulk viscosity enhances kinetic energy dissipation in both gases, with N2 exhibiting sharper dissipation than CO2. Additionally, the role of rotational speed is explored, showing that higher rotational speeds amplify vorticity production and energy dissipation. While high-speed rotation induces more turbulence and instability in N2, it stabilizes the flow in CO2, leading to a more organized wake. The findings demonstrate that bulk viscosity and rotational speed are crucial in controlling flow stability and energy dissipation, with significant variations depending on the gas properties.

Influence of vibrational and chemical non-equilibrium on the velocity-gradient and the pressure-Hessian fields in compressible turbulence

The influence of vibrational and chemical non-equilibrium on the dynamics of velocity gradients and pressure-Hessian tensors is investigated in this study. Such non-equilibrium flows typically occur in high-speed compressible turbulence at elevated temperatures, as observed in reentry vehicles or hypersonic flights. In the first part of the study, we derive the exact evolution equations for the velocity gradients and pressure-Hessian tensors under vibrational and chemical non-equilibrium conditions. We identify the inertial, vibrational, and chemical mechanisms in this evolution equation. In the second part of this study, we focus on assessing the relative importance of these mechanisms across different simulation cases. In this work, we present direct numerical simulations of isotropic decaying turbulence, which consider both vibrational and chemical non-equilibrium effects. It is found that in the presence of chemical and vibrational non-equilibrium, (i) vibrational relaxation processes are expedited in terms of mean and fluctuating flow fields. (ii) Vortical fluctuations increase while dilatational fluctuations are suppressed. (iii) The relative strength of the pressure-Hessian tensor compared to the velocity gradient tensor is reduced. (iv) The explicit effects of vibrational mechanisms on the pressure-Hessian evolution equation are diminished, whereas chemical mechanisms substantially influence the dynamics compared to inertial mechanisms throughout the turbulence decay process. This study highlights the need for robust turbulence closure models for chemical mechanisms to accurately capture the effects of thermal non-equilibrium on the dynamics of velocity gradients in compressible flows.

Mixed convection flow in a porous channel under the effects of exothermic chemical reactions and local thermal non-equilibrium

The effect of a first-order exothermic chemical reaction on the mixed flow in a vertical channel filled with a porous medium has been studied using the local thermal non-equilibrium model. A mathematical model including the exponen-tial term for heat generation in the fluid phase and interphase heat transfer terms in both the fluid and solid phases was considered. The above dimensionless model has been solved numerically using the Matlab routine bvp4c for the energy equations together with an in-house developed code for the momentum equation. The existence of dual solutions is reported for certain values of the Frank-Kamenetskii number. The obtained profiles of temperature and velocity have been plotted. In addition, the ranges of existence of the dual solutions are reported.

Study of the N2 vibrational relaxation behaviors via the CO rovibrational thermometry.

This paper performed a comprehensive study of the thermal nonequilibrium effects of CO/Ar mixtures with various degrees of N2 additions and probed the N2 relaxation behaviors via the CO rovibrational thermometry. The rovibrational temperature time histories of shock-heated CO/N2/Ar mixtures were measured via a laser-absorption technique, and the corresponding vibrational relaxation data were summarized at 1890-3490K. The measured results were compared with predictions from the Schwartz-Slawsky-Herzfeld (SSH) formula and the state-to-state (StS) approach (treating CO and N2 as pseudo-species). The vibrational state-specific inelastic rate coefficients for N2-N2 collisions were supplemented using the mixed quantum-classical calculations. The StS predictions, informed by experimentally measured pressures, showed good agreement with experimental data. Additionally, the impact of coupling between flow dynamics and StS kinetics behind reflected shock waves was evaluated using two different one-dimensional approaches, which provide limiting bounds (accounting for unsteady flow and end wall effects) in post-reflected shock flow conditions. Moreover, the vibrational relaxation data of the N2-N2 system were modified via sensitivity analysis to improve the performance of the SSH formula. Further analysis highlighted that the vibration-vibration-translation path provides an efficient way for vibrational energy transfer between CO and N2, resulting in almost the same vibrational temperature time histories for CO and N2. Therefore, the N2 relaxation behaviors can be characterized by the CO rovibrational thermometry, considering N2 is infrared inactive. Finally, the heat sink effects and the reflected-shock-bifurcation phenomena were highlighted.

Influence of band occupation on electron-phonon coupling in gold

Electron-phonon coupling is a fundamental process that governs the energy relaxation dynamics of solids excited by ultrafast laser pulses. It has been found to strongly depend on the electron temperature as well as on nonequilibrium effects. Recently, the effect of occupational nonequilibrium in noble metals, which outlasts the fully kinetic stage, has come into increased focus. In this study, we investigate the influence of nonequilibrium density distributions in gold on the electron-phonon coupling. We find a large effect on the coupling parameter which describes the energy exchange between electrons and phonons. Our results challenge the conventional view that electron temperature alone is a sufficient predictor of electron-phonon coupling.

Monomer size effect in inelastic collisional dynamics of non-equilibrium soot nucleation.

Molecular dynamics (MD) simulations of the collisional dynamics of the coronene-acepyrene and coronene radical-acepyrene pairs have been carried out to investigate the size effect of monomers of polycyclic aromatic hydrocarbons (PAH) on their non-equilibrium dimerization. The results compared to the previous MD simulations of the smaller pyrene-acepyrene and pyrenyl-acepyrene systems corroborate the non-equilibrium hypothesis of crosslinking PAH dimerization enhanced by physical interaction between the monomers. The phenomenon of inelastic collisional dynamics responsible for non-equilibrium van der Waals dimerization, which fosters a covalent bond formation between the monomers, amplifies with increasing PAH size. The increase in the size of the colliding monomers enhances the non-equilibrium effects as the growing pool of low-frequency modes provides a larger sink for the energy of the colliding PAH monomers. Based on the direct count of the crosslinking reaction events observed in the MD simulations, the forward rate constant for the coronene radical-acepyrene association is estimated at ∼10-11cm3molecule-1s-1, showing a 15-fold increase with respect to the value from the statistical Rice-Ramsperger-Kassel-Marcus calculations. A comparison with the eightfold increase reported previously for the pyrenyl-acepyrene system shows that the statistical (equilibrium-based) calculations increasingly underestimate the reaction rate with the increasing size of the interacting PAHs from pyrene to coronene. The total increase of the MD-assessed rate constant for the coronene radical-acepyrene dimerization reaction as compared to pyrenyl-acepyrene is a factor of 2.4, with the overall collision efficiency to produce dimerized products growing by 30%.

Numerical study of thermal nonequilibrium effects on Richtmyer-Meshkov flow driven by a heavy forward-triangular bubble

Numerical study of thermal nonequilibrium effects on Richtmyer-Meshkov flow driven by a heavy forward-triangular bubble

Modeling of vibrational nonequilibrium effects on pressure Hessian tensor using physics-assisted deep neural networks

This study focuses on modeling the effect of vibrational nonequilibrium on the pressure Hessian tensor. The pressure Hessian tensor is one of the unclosed processes involved in the velocity gradient evolution equation. Accessing the velocity gradient tensor following a fluid particle is needed to understand the physics of various nonlinear turbulent processes. Our modeling strategy employs a combination of two different deep neural networks (DNNs) to model the magnitude and the directional aspects of the vibrational nonequilibrium tensor separately. Both the DNNs are optimized using two appropriate physics-assisted custom loss functions, comparing the desired features of the DNN predictions against the exact behavior observed in the direct numerical simulation (DNS) database. A detailed investigation of four different DNS databases of vibrationally excited decaying compressible turbulence with different initial Reynolds numbers, Mach numbers, and vibrational Damköhler numbers is performed to identify the appropriate normalized forms of input and output quantities for the two neural networks. The training of both the DNNs is done using the DNS data of merely one simulation. The trained models are then subjected to extensive evaluation against different DNS databases. Indeed, the new model captures many DNS features quite well. Such an extensive evaluation of the new model proves the generalizability of the model, at least in the range of parameters involved in our study.

Computational Study on Flow Characteristics of Shocked Light Backward-Triangular Bubbles in Polyatomic Gas

This study computationally examined the Richtmyer–Meshkov instability (RMI) evolution in a helium backward-triangular bubble immersed in monatomic argon, diatomic nitrogen, and polyatomic methane under planar shock wave interactions. Using high-fidelity numerical simulations based on the compressible Navier–Fourier equations based on the Boltzmann–Curtiss kinetic framework and simulated via a modal discontinuous Galerkin scheme, we analyze the complex interplay of shock-bubble dynamics. Key findings reveal distinct thermal non-equilibrium effects, vorticity generation, enstrophy evolution, kinetic energy dissipation, and interface deformation across gases. Methane, with its molecular complexity and higher viscosity, exhibits the highest levels of vorticity production, enstrophy, and kinetic energy, leading to pronounced Kelvin–Helmholtz instabilities and enhanced mixing. Conversely, argon, due to its simpler atomic structure, shows weaker deformation and mixing. Thermal non-equilibrium effects, quantified by the Rayleigh–Onsager dissipation function, are most significant in methane, indicating delayed energy relaxation and intense turbulence. This study highlights the pivotal role of molecular properties, specific heat ratio, and bulk viscosity in shaping RMI dynamics in polyatomic gases, offering insights on uses such as high-speed aerodynamics, inertial confinement fusion, and supersonic mixing.

Activity-driven polymer knotting for macromolecular topology engineering.

Macromolecules can gain special properties by adopting knotted conformations, but engineering knotted macromolecules is a challenging task. Here, we unexpectedly find that knots can be efficiently generated in active polymer systems. When one end of an actively reptative polymer is anchored, it undergoes continual self-knotting as a result of intermittent giant conformation fluctuations and the outward reptative motion. Once a knot is formed, it migrates to the anchoring point due to a nonequilibrium ratchet effect. Moreover, when the active polymer is grafted on a passive polymer, it can function as a self-propelling soft needle to either transfer its own knots or directly braid knots on the passive polymer. We further show that these active needles can create intermolecular bridging knots between two passive polymers. Our finding highlights the nonequilibrium effects in modifying the dynamic pathways of polymer systems, which have potential applications in macromolecular topology engineering.

Physics-enhanced data-driven turbulence model for flow around submerged bodies

Data-driven turbulence modeling is regarded as an effective approach to enhance the predictive performance of Reynolds-Averaged Navier−Stokes (RANS) models. However, the stability and accuracy of such methods still require further improvement. Therefore, we propose a novel modified model based on a combined neural network. Within this framework, we construct a fully connected neural network to predict the eddy viscosity coefficient in the RANS equations, thereby achieving an implicit solution for the linear part of the Reynolds stress. Additionally, we introduce a tensor basis neural network to predict the higher-order eddy viscosity relationships between unclosed quantities and analytical quantities. Furthermore, the model incorporates pressure gradients and turbulent kinetic energy gradients as inputs, fully considering the influence of pressure gradients and strong non-equilibrium effects in turbulence, thereby enhancing the physical foundation of the model. To evaluate the performance of the constructed model in different-dimensional flow fields with higher Reynolds numbers, we conduct validation analyzes on a two-dimensional NACA0012 model and a three-dimensional SUBOFF model without appendages. The results indicate that the modified model consistently outperforms the RANS model, particularly in predicting the velocity field near the leading edge of the NACA0012 hydrofoil. The predictions of the modified model are closer to those of large eddy simulation (LES) results. Specifically, the root mean square error (RMSE) of the interpolated and extrapolated modified models is reduced by 49.2% and 33.3%, respectively, compared to the RMSE of the RANS model. When applied to three-dimensional isolated SUBOFF flows, the modified model’s predictions for the average velocity field and pressure coefficients at both ends of the SUBOFF are in high agreement with LES results, but the prediction accuracy in the middle section is slightly insufficient. Nevertheless, the prediction errors of the interpolated and extrapolated modified models are reduced by 69.5% and 63.6%, respectively, compared to the baseline RANS model, which fully demonstrates the high accuracy of the modified model in predicting the overall pressure distribution of the SUBOFF. The methods developed in this study provide valuable insights into high-precision intelligent modeling for two-dimensional and three-dimensional flows.

Epidemic spreading and herd immunity in a driven non-equilibrium system of strongly-interacting atoms

It is increasingly important to understand the spatial dynamics of epidemics. While there are numerous mathematical models of epidemics, there is a scarcity of physical systems with sufficiently well-controlled parameters to allow quantitative model testing. It is also challenging to replicate the macro non-equilibrium effects of complex models in microscopic systems. In this work, we demonstrate experimentally a physics analog of epidemic spreading using optically-driven non-equilibrium phase transitions in strongly interacting Rydberg atoms. Using multiple laser beams we can impose any desired spatial structure. The observed spatially localized phase transitions simulate the outbreak of an infectious disease in multiple locations, and the splitting of the outbreak in subregions, as well as the dynamics towards “herd immunity” and “endemic state” in different regimes. The reported results indicate that Rydberg systems are versatile enough to model complex spatial-temporal dynamics.

Optimisation of a converging-diverging nozzle for the wet-to-dry expansion of the siloxane MM

Wet-to-dry expansion within the nozzle guide vane of an ORC turbine has been proposed as a means to improve the power output of ORC systems for waste-heat recovery (<250 °C). However, given the rapid fluid acceleration in the stator, the phases can develop significant velocity and temperature disparity due to density difference and finite rate of interphase heat transfer. Since these factors can significantly affect the phase-change process, wet-to-dry nozzle design techniques must account for non-equilibrium effects. The first part of this paper aims to further verify a previously developed quasi-1D inviscid non-equilibrium nozzle design tool by comparing it to non-equilibrium CFD simulations, which, unlike the design model, account for lateral flow variations, viscous and turbulence effects, along with secondary momentum forces. Within the CFD model, the interphase mass, momentum, and energy exchange models have been updated using correlations better tailored to evaporating droplet flows and a corrected drag equation. Moreover, the definition of the vapour mass fraction has been modified, while a simplified droplet breakup model has been used to predict the droplet size. The results from the CFD simulations indicate that the outlet vapour mass fraction is approximately 10 to 15% lower than that predicted by the quasi-1D tool. However, the overall flow behaviour and phase-change pattern were in satisfactory agreement, justifying the use of the design tool for 1D optimisation. As such, the quasi-1D tool is coupled to a gradient-based optimiser to optimise the nozzle pressure profile and enhance evaporation of siloxane MM for expansions with an inlet pressure ranging from 450 to 650 kPa, and inlet vapour quality of 0.3. CFD simulations of the optimised geometries indicate an increase of 3.3 to 5.7% in the outlet vapour mass fraction, which was raised from 84.9, 87.7 and 90.5% to 88.2, 93.4 and 95.7% for 450, 550 and 650 kPa inlet pressures respectively. However, a more abrupt expansion in the optimised nozzles resulted in the development of a shock and led to deterioration in nozzle efficiency compared to the baseline nozzles. Finally, a CFD-based shape optimisation was conducted, which demonstrated that it may be difficult to further enhance the vapourisation rate. However, the optimised geometry did mitigate the effect of the oblique shock that appears in the diverging section of the nozzle, raising the expansion efficiency by around 3%.

Geothermal system lifetime with fracture heterogeneity and thermal nonequilibrium impact

Geothermal energy is a clean and sustainable alternative energy, and enhanced geothermal systems are an essential technology offering an industrial-scale method to tap geothermal energy. This technology stimulates numerous reservoir fractures, creating primary flow channels for fluids to extract heat. The important question of how the permeability of fracture swarm evolves during fluid circulation and how this evolution potentially affects the geothermal system life remains to be further answered. To address this, we develop and validate a fully coupled thermo-hydro-mechanical model based on the discrete fracture network. Results show a positive feedback mechanism between pore pressure/temperature changes and fracture permeability evolution, leading to monotone permeability-increasing in a limited number of specific fractures, indicating uneven permeability evolution of fracture swarm is crucial in shortening reservoir lifetime. Furthermore, we conduct a comparative analysis of multiple factors to determine what most significantly affects reservoir lifetime, including fracture heterogeneity, the local thermal non-equilibrium effect, and layout strategies. Results indicate the strong heterogeneity of the fracture network is dominant, and the inherent heterogeneity of hydraulic fractures is further exacerbated during geothermal production. Neglecting fractures’ mechanical response leads to an overestimated outcome. A simplified thermal-hydraulic analysis indicates a 34 % higher temperature and a 6 % higher heat extraction rate than the coupled thermo-hydro-mechanical modeling. In contrast, the impact of the local thermal non-equilibrium effect is negligible, with comparable results between equilibrium and non-equilibrium models. We further find the multi-well layout can effectively suppress the heterogeneity impact and prevent preferential flow paths. Compared to dual-well systems, the multi-well scheme has a 24 % higher temperature and a 6 % higher heat extraction rate.

Effect of non-equilibrium phase transition on the aero-thermodynamic performance of supercritical carbon dioxide compressors

Phase transition is a prominent issue for the supercritical CO2 compressors operating near the critical region. The non-equilibrium behavior of phase transition makes it difficult to accurately predict performance under these conditions. A non-equilibrium phase transition model is proposed to analyze the aero-thermodynamic performance of supercritical CO2 centrifugal compressors with different inlet conditions. The model examines phase change characteristics and compressor performance under variable operating conditions. The results indicate that this model can effectively predict the aero-thermodynamic performance of compressor near critical conditions, with an average error of less than 2% in the performance curves. The inconsistency between low-temperature and low-pressure regions, along with the increase in liquid phase volume, are typical characteristics of non-equilibrium phase transition in supercritical CO2 compressors. The non-equilibrium effect of phase transition can reduce the leakage loss at high flow rates, but also increases the likelihood of stall at low flow rates. Furthermore, the concept of “non-equilibrium degree” (NED) is introduced to quantify these effects. When NED exceeds 0.2, the isentropic efficiency of the compressor decreases by 15.1% compared to its maximum efficiency. Designing inlet conditions for supercritical CO2 compressors with NED below 0.1 is more suitable because it has a larger inlet flowrate and higher efficiency.

Mechanosensitive dose response of the bacterial flagellar motor.

The bacterial flagellar motor is both chemo- and mechanosensitive. It is sensitive to the intracellular concentration of the chemotaxis response regulator CheY-P and to external load conditions. The motor's dose-response curve, which represents the probability of the motor rotating clockwise (CW bias) as a function of CheY-P concentration, characterizes its chemical sensitivity. However, it remains unclear how this dose-response curve depends on the load conditions. Here, we measured the dose-response curves of the motor under various load conditions. Surprisingly, we found that the dose-response curve exhibited minimal changes with load at low CW biases, but shifted leftward with higher sensitivity to CheY-P concentration at high CW biases when the load increased. This observation contradicts previous model predictions that incorporated the effect of stator-rotor interaction on motor switching. Through the development of an Ising-type model for the coupled chemo- and mechanosensitivity of the flagellar switch, we revealed that the mechanism underlying the mechanosensitive dose response is the synergistic interplay between the adaptive remodeling of the motor switch complex and the nonequilibrium effect of the stator-rotor interaction.

Kinetic investigation of Kelvin–Helmholtz instability with nonequilibrium effects in a force field

The Kelvin–Helmholtz (KH) instability in a force field is simulated and investigated using a two-component discrete Boltzmann method. Both hydrodynamic and thermodynamic nonequilibrium effects in the evolution of KH instability are analyzed in two distinct states: interface roll-up and non-roll-up. It is interesting to note that there are critical thresholds for initial amplitude and Reynolds number, both of which are determined based on the vertical density gradient. Specifically, when the initial amplitude and Reynolds number exceed their respective critical thresholds, the interface undergoes roll-up. Conversely, if these parameters fall below their critical values, the interface fails to roll up. Moreover, the initial amplitude promotes the development of density gradients, mixing degree, mixing width, viscous stress tensor strength, and heat flux strength. In contrast, the Reynolds number enhances the evolution of density gradients but dampens the mixing degree, viscous stress tensor strength, and heat flux intensity. The effect of the Reynolds number on mixing width is analyzed as well.

Investigation of non-equilibrium phenomena in nitrogen RF inductively coupled plasma discharges: a state-to-state approach

Abstract This work presents a vibrational and electronic (vibronic) state-to-state (StS) model for nitrogen plasmas implemented within a multi-physics modular computational framework to study non-equilibrium effects in inductively coupled plasma (ICP) discharges. The vibronic master equations are solved in a tightly coupled fashion with the flow governing equations eliminating the need for invoking any simplifying assumptions when computing the state of the plasma, leading to a high-fidelity physical modeling. The model’s computational complexity is reduced via a maximum entropy coarse-graining approach, verified through zero-dimensional isochoric calculations. The coarse-grained StS model is employed to study the plasma discharge in the ICP facility at the von Karman Institute for Fluid Dynamics, Belgium. Results reveal pronounced discrepancies between StS predictions and those obtained based on local thermodynamic equilibrium (LTE) models, which are conventionally used in the simulation of such facilities. The analysis demonstrates a substantial departure of the internal state populations of atoms and molecules from the Boltzmann distribution. This has significant implications for energy coupling dynamics, affecting the discharge morphology. Further analysis reveals a quasi-steady-state population distribution in the plasma core, allowing for the construction of an efficient and ‘self-consistent’ macroscopic two-temperature (2T) formulation. Non-LTE simulations indicate significant disparities between the StS model and the commonly used Park 2T model, whereas the newly proposed 2T model aligns closely with StS simulations, capturing key features of non-equilibrium plasma formation. In particular, the current study highlights the importance of the vibrational-translational energy transfer term in shaping the plasma core morphology, suggesting a notable sensitivity to heavy-impact vibrational excitations and dissociative processes.

Thickened Flame Model Extension for Dual Gas Turbine Combustion: Validation Against Single Cup Atmospheric Test

Abstract The development of predictive combustion models is more and more strategic in the design definition of gas turbine (GT) combustor. The thickened flame model (TFM), despite its high computational cost, is one of the most accurate approach available in literature since it can naturally take into account the nonequilibrium effects into the flame brush (i.e., strain and heat losses) as well as preferential diffusion when hydrogen is employed. Conversely, the original formulation of this combustion model needs several adjustments to accommodate the properties of the mixture when different streams of fuels and/or oxidizers are present in the system. The present work represents a first step in the extension of this combustion model to handle multiple streams of fuels and oxidizers. More specifically, an industrial burner fed with two different fuel streams and air as oxidizer is considered. The pilot fuel line is fed with microhydrogen injections with the aim to enhance the lean blow-out margin, while the main one is with pure methane. Dedicated tests are performed at the Technology for High Temperature laboratory (University of Florence) to retrieve the main information characterizing the burner (emissions, temperature, and pressure pulsations) as well as OH* chemiluminescence for the flame shape and position at the same operating conditions. The comparison between the numerical results and the experimental data will provide highlights about the ability of the extended-TFM to capture the main features of the flame stabilization mechanisms.

Numerical Modeling of Vortex-Based Superconducting Memory Cells: Dynamics and Geometrical Optimization.

The lack of dense random-access memory is one of the main obstacles to the development of digital superconducting computers. It has been suggested that AVRAM cells, based on the storage of a single Abrikosov vortex-the smallest quantized object in superconductors-can enable drastic miniaturization to the nanometer scale. In this work, we present the numerical modeling of such cells using time-dependent Ginzburg-Landau equations. The cell represents a fluxonic quantum dot containing a small superconducting island, an asymmetric notch for the vortex entrance, a guiding track, and a vortex trap. We determine the optimal geometrical parameters for operation at zero magnetic field and the conditions for controllable vortex manipulation by short current pulses. We report ultrafast vortex motion with velocities more than an order of magnitude faster than those expected for macroscopic superconductors. This phenomenon is attributed to strong interactions with the edges of a mesoscopic island, combined with the nonlinear reduction of flux-flow viscosity due to the nonequilibrium effects in the track. Our results show that such cells can be scaled down to sizes comparable to the London penetration depth, ∼100 nm, and can enable ultrafast switching on the picosecond scale with ultralow energy per operation, ∼10-19 J.