Small Dissipation Research Articles

A novel capacitive sensor interface based on a simple capacitance-to-phase converter

Abstract A novel room temperature capacitive sensor interface circuit is proposed and successfully tested, which uses a modified All-Pass filter architecture combined with a simple series resonant tank circuit with a moderate Q-factor. It is fashioned from a discrete inductor with small dissipation resonating with a grounded capacitor acting as the sensing element to obtain a resolution of ∆C ~ 2 zF in a capacitance range of 10 – 30 pF. The circuit converts the change in capacitance to the change in the phase of a carrier signal in a frequency range with a central frequency set up by the tank circuit’s resonant frequency and is configured to act as a close approximation of the ideal All-Pass filter. This cancels out the effects of amplitude modulation when the carrier signal is imperfectly tuned to the resonance. The proposed capacitive sensor interface has been specifically developed for use as a front-end constituent in ultra-precision mechanical displacement measurement systems, such as accelerometers, seismometers, gravimeters and gravity gradiometers, where moving plate grounded air gap capacitors are frequently used. Some other applications of the proposed circuit are possible including the measurement of the electric field, where the sensing capacitor depends on the applied electric field, and cost effective capacitive gas sensors. In addition, the circuit can be easily adapted to function with very small capacitance values (1 - 2 pF) as is typical in MEMS-based transducers.

Dissipative quintessential cosmic inflation

In this paper we construct a dissipative quintessential cosmic inflation. For this purpose, we add a multiplicative dissipative term in the standard quintessence field Lagrangian. We consider the specific form of dissipation as the time integral including the Hubble parameter and an arbitrary function that describes the dissipative properties of the quintessential scalar field. Inflation parameters and observables are calculated under slow-roll approximations and a detailed calculation of the cosmological perturbations is performed in this setup. We consider different forms of potentials and calculate the scalar spectral index and tensor-to-scalar ratio for a constant as well as variable dissipation function. To check the reliability of this model, a numerical analysis on the model parameters space is done in confrontation with recent observational data. By comparing the results with observational joint datasets at 68% and 95% confidence levels, we obtain some constraints on the model parameters space, specially the dissipation factor with e-folds numbers N=55 and N=60. As some specific results, we show that the power-law potential with a constant dissipation factor and N=60 is mildly consistent with observational data in some restricted domains of the model parameter space with very small and negative dissipation factor and a negligible tensor-to-scalar ratio. But this case with N=55 is consistent with observation considerably. For power-law potential and variable dissipation factor as Q=αϕn, the consistency with observation is also considerable with a reliable tensor-to-scalar ratio. The quadratic and quartic potentials with variable dissipation function as Q=αϕn are consistent with Planck2018 TT, TE, EE+lowE+lensing data at the 68% and 95% levels of confidence for some intervals of the parameter n.

Statistics of kinetic and thermal energy dissipation rates in two-dimensional thermal vibrational convection

We investigate the statistical properties of kinetic ϵu and thermal ϵθ energy dissipation rates in two-dimensional (2D) thermal vibrational convection (TVC). Direct numerical simulations were conducted in a unit aspect ratio box across a dimensionless angular frequency range of 103≤ω≤107 for amplitudes 0.001≤a≤0.1, with a fixed Prandtl number of 4.38. Our findings indicate ϵu is primarily associated with the characteristics of the vibration force, while ϵθ is more related to the large-scale columnar structures. Both energy dissipation rates exhibit a power-law relationship with the oscillational Reynolds number Reos. ϵu exhibits a scaling relation as ⟨ϵu⟩V,t∼a−1Reos0.93±0.01, while ϵθ exhibits two distinct scaling behaviors, i.e., ⟨ϵθ⟩V,t∼a−1Reos1.97±0.04 for Reos<Reos,cr and ⟨ϵθ⟩V,t∼a−1Reos1.31±0.02 for Reos>Reos,cr, where the fitted critical oscillational Reynolds number is approximately Reos,cr≈80. The different scaling of ⟨ϵθ⟩V,t is determined by the competition between the thermal boundary layer and the oscillating boundary layer. Moreover, the probability density functions (PDFs) of both dissipation rates deviate significantly from the lognormal distribution and exhibit a bimodal shape. By partitioning the contributions from the boundary layer and bulk regions, it is shown that the bulk contributes to the small and moderate dissipation rates, whereas the high dissipation rates are predominantly contributed by the boundary layer. As Reos increases, the heavy tail of the PDFs becomes more pronounced, revealing an enhanced level of small-scale intermittency. This small-scale intermittency is mainly caused by the influence of BL due to vibration. Our study provides insight into the small-scale characteristics of 2D TVC, highlighting the non-trivial scaling laws and intermittent behavior of energy dissipation rates with respect to vibration intensity.

Microwave-multiplexed qubit controller using adiabatic superconductor logic

Cryogenic qubit controllers (QCs) are the key to build large-scale superconducting quantum processors. However, developing scalable QCs is challenging because the cooling power of a dilution refrigerator is too small (~10 μW at ~10 mK) to operate conventional logic families, such as complementary metal-oxide-semiconductor logic and superconducting single-flux-quantum logic, near qubits. Here we report on a scalable QC using an ultra-low-power superconductor logic family, namely adiabatic quantum-flux-parametron (AQFP) logic. The AQFP-based QC, referred to as the AQFP-multiplexed QC (AQFP-mux QC), produces multi-tone microwave signals for qubit control with an extremely small power dissipation of 81.8 pW per qubit. Furthermore, the AQFP-mux QC adopts microwave multiplexing to reduce the number of coaxial cables for operating the entire system. As a proof of concept, we demonstrate an AQFP-mux QC chip that produces microwave signals at two output ports through microwave multiplexing and demultiplexing. Experimental results show an output power of approximately −80 dBm and on/off ratio of ~40 dB at each output port. Basic mixing operation is also demonstrated by observing sideband signals.

Seismic vulnerability analysis of a high‐rise steel frame structure equipped with novel displacement‐amplified viscoelastic dampers

SummaryThe energy dissipation capacity of conventional viscoelastic dampers (VEDs) cannot be fully exerted due to the relatively small inter‐story displacement of building structures. Thus, a novel VED with a displacement amplification mechanism based on the lever principle is developed in this study to achieve small displacement but high energy dissipation ability. First, the structural layout of an amplified viscoelastic damper (AVED) system is briefly introduced, and then the displacement amplification effect is described in detail. According to the working principle of the AVED and the relationship of force and geometric displacement in an actual frame structure, the restoring force theoretical formula of the corresponding amplified damper is derived. Based on the restoring force of the AVED and in combination with the secondary development function of the ABAQUS unit, a VUEL subroutine suitable for the explicit algorithm is programmed with the FORTRAN language. Then, the correctness of the subroutine is verified through a comparative analysis of the ABAQUS simulation and theoretically calculated results. Consequently, the vulnerability analysis of a high‐rise steel frame structure is conducted by using the incremental dynamic analysis (IDA) method, and the influence of the AVED on the seismic performance of the steel frame structure is quantitatively analyzed from the perspective of probability statistics. The analysis results show that compared with the VED structure, the failure probability of the AVED‐2 and AVED‐3 structures reaching the ultimate failure state at all levels is reduced by approximately 43% and 57%, respectively, and the collapse margin ratio (CMR) of structures under large earthquakes is increased by approximately 35% and 68%, respectively. This indicates that the AVED structures with displacement‐amplified dampers have a better effect on the seismic stability of tall steel frame structures under random seismic excitations. This study aims to provide comprehensive information for the seismic‐resistant design of tall buildings in earthquake‐prone regions.

Monomial warm inflation revisited

Abstract We revisit the idea that the inflaton may have dissipated part of its energy into a thermal bath during inflation, considering monomial inflationary potentials and three different forms of dissipation rate. Using a numerical Fokker-Planck approach to describe the stochastic dynamics of inflationary fluctuations, we confront this scenario with current bounds on the spectrum of curvature fluctuations and primordial gravitational waves. We also obtain purely analytical approximations that improve over previously used ones in the small dissipation regime for the amplitude of the spectrum and its tilt. We show that only our numerical Fokker-Planck method is accurate, fast and precise enough to test these models against current data. We advocate its use in future studies of warm inflation. We also apply the stochastic inflation formalism to this scenario, finding that the resulting spectrum is the same as the one obtained with standard perturbation theory. We discuss the origin and convenience of using a commonly implemented large thermal correction to the primordial spectrum and the implications of such a term for a specific scenario. Improved bounds on the scalar spectral index will further constrain warm inflation in the near future.

Influence of surface acoustic wave (SAW) on nanoscale in-plane magnetic tunnel junctions

The use of voltage induced strain to switch magnetic tunnel junctions (MTJs) is a promising solution for reducing the switching energy in MRAM technologies. The MTJ is integrated with a piezoelectric layer to generate the strain. A very thin layer is needed to switch with small voltages and small energy dissipation. It is challenging to synthesize ultrathin piezoelectric layers that retain a high degree of piezoelectricity. An alternate approach is to use time-varying strain generated by a surface acoustic wave (SAW). This approach does not require a thin piezoelectric layer since the SAW is confined to the surface of the layer. In this study, we fabricated in-plane MTJs on piezoelectric LiNbO3 substrates and used IDTs to generate the SAW signal within the substrate. Our results showed that the SAW signal had a significant influence on the resistance and the tunneling magnetoresistance (TMR) ratio of the MTJs. The influence was much less significant in nanometer size MTJs than in micrometer sized ones. Most surprisingly, the SAW signal caused the tunneling magnetoresistance ratio (TMR) to drop below zero for the micrometer size MTJ, meaning that the antiparallel resistance RAP is temporarily less than the parallel resistance RP under SAW excitation. Our results provide insight into the dynamic behavior of MTJs under periodic strain and the dependence of this behavior on the device dimensions as they are scaled down to nanometer sizes.

Staggered grids for multidimensional multiscale modelling

For high accuracy and to improve simulated wave characteristics, this article extends the concept of staggered grids to novel multidimensional multiscale modelling enabling efficient computation on sparse patches. Computational schemes for wave-like systems with small dissipation are often inaccurate and unstable due to truncation errors and numerical roundoff errors. Hence simulations of wave-like systems lacking proper handling of these numerical issues often fail to represent the physical characteristics of wave phenomena. This challenge gets even more intricate for multiscale modelling, especially in multiple dimensions. But numerical schemes on staggered grids are significantly less dispersive, better model the group velocity, and preserve much of the wave characteristics. This article develops and exhaustively studies all 167040 possible 2D multiscale staggered grids. Our catalog (Divahar, 2023) interactively plots all of them. Only 120 multiscale staggered patch grids give stable and accurate multiscale schemes. Specifically, this article develops these 120 multiscale staggered grids and demonstrates their stability, accuracy, and wave-preserving characteristic for equation-free multiscale modelling of weakly damped linear waves. These characteristics of the developed multiscale staggered grids must also hold in general for multiscale modelling of many complex spatio-temporal physical phenomena such as the general computational fluid dynamics.

Ultralow Voltage Resistive Switching in Hafnium–Zirconium Oxide

AbstractUltralow SET and RESET voltage are essential for high‐density, low‐power, and small heat dissipation nonvolatile random‐access memory (NVRAM) elements. A nanoscale polycrystalline Hf0.75Zr0.25O2 (HZO) thin films on Pt/Si substrate are fabricated and investigated for suitability for bipolar resistive switching. The device illustrates monoclinic and tetragonal/orthorhombic phases with weak ferroelectricity and robust resistive switching. Small remanent polarization (≈0.1 μC cm−2) may assist in the height reduction of barrier height and ease the electron for transport. Remarkably, the Al/HZO/Pt/Si device, consisting of thin films with 10 and 5 nm thicknesses, exhibits a switching voltage below −30 mV from a low‐resistance state (LRS) to a high‐resistance state (HRS). It shows a significant ROFF/RON ratio of 106, making it suitable for low power consumption and minimal heat dissipation devices. Moreover, the utilization of an ultrathin film (5 nm) results in an improved reduction (< 0.7 V) of the operating window at the positive voltage. Direct tunneling and the Fowler–Nordheim tunneling model are performed in current–voltage (I–V) data to study the charge transportation behavior over a trapezoidal and triangular potential barrier. These results of the HZO candidate may stimulate the futuristic nonvolatile resistive random‐access memory (ReRAM) in the optoelectronic industry.

Fluctuations, correlations, and Casimir-like forces in the homogeneous cooling state of a granular gas

The fluctuating hydrodynamics by Brey et al. [Phys. Rev. E 83, 041303 (2011)] is analytically solved to get the long-time limit of the fluctuations of the number density, velocity field, and energy density around the homogeneous cooling state of a granular gas, under physical conditions where it keeps stable. Explicit expressions are given for the nonwhite contributions in the elastic limit. For small dissipation, the latter is shown to be much smaller than the inelastic contributions, in general. The fluctuation-induced Casimir-like forces on the walls of the system are calculated assuming a fluctuating pressure tensor resulting from perturbing its Navier–Stokes expression. This way, the Casimir-like forces emerge as the correlation between the longitudinal velocity and the energy density. Interestingly, the fluctuation-induced forces push/pull the system toward the square or rectangular geometry where they vanish, in good agreement with the event-driven numerical simulations.

Probabilistic description of dissipative chaotic scattering.

We investigate the extent to which the probabilistic properties of chaotic scattering systems with dissipation can be understood from the properties of the dissipation-free system. For large energies, a fully chaotic scattering leads to an exponential decay of the survival probability P(t)∼e^{-κt}, with an escape rate κ that decreases with energy. Dissipation leads to the appearance of different finite-time regimes in P(t). We show how these different regimes can be understood for small dissipations and long times from the (effective) escape rate κ (including the nonhyperbolic regime) of the conservative system, until the energy reaches a critical value at which no escape is possible. More generally, we argue that for small dissipation and long times the surviving trajectories in the dissipative system are distributed according to the conditionally invariant measure of the conservative system at the corresponding energy. Quantitative predictions of our general theory are compared with numerical simulations in the Hénon-Heiles model.

Hanging Photothermal Fabric Based on Polyaniline/Carbon Nanotubes for Efficient Solar Water Evaporation.

Solar-driven water evaporation is essential to provide sustainable and ecofriendly sources of fresh water. However, there are still great challenges in preparing materials with broadband light absorption for high photothermal efficiency as well as in designing devices with large evaporation areas and small heat dissipation areas to boost the water evaporation rate. We designed a hanging-mode solar evaporator based on the polyaniline/carbon nanotube (PANI/CNT) fabric, in which the photothermal fabric acts as the solar evaporator and the micropores on the cotton fabric act as the water transfer channels. The hanging mode provides efficient evaporation at both interfaces by greatly reducing the heat dissipation area. The hanging mode PANI/CNT fabric solar evaporator can achieve an evaporation rate of 2.81 kg·m-2·h-1 and a photothermal efficiency of 91.74% under a solar illumination of 1 kW·m-2. This high-performance evaporator is designed by regulating the photothermal material and evaporation device, which provides a novel strategy for sustainable desalination.

Effect of fiber properties and orientation on the shear rheology and Poynting effect in meat and meat analogues

The development of plant-based meat analogues is an important aspect of the transition towards using more plant proteins. Production of such products with texture (and rheology) adequate for consumer acceptance like meats is a great challenge. However, there is a current lack in quantitative methods to compare whole cut meats and plant-based analogues that also consider its fibrous structures. We tackled this problem by combining non-linear shear rheology and shear-induced normal stress measurements on whole cut meats (beef and chicken) and plant-based fibrous model systems for meat analogues (soy and pea protein isolates combined with wheat gluten). We found different response in the linear viscoelastic regime for different samples, and the non-linear viscoelastic regime provided a deeper understanding into the role of structural components. Meat samples showed rearrangements at small strains and lower energy dissipation, higher intracycle strain stiffening, and a larger normal stress response at large strains than plant-based fibrous products, due to more elastic fibers. The plant-based fibrous products behave more solid-like until a larger strain, but the fibrous structures break down at large deformations resulting in higher energy dissipation, and show a lower normal stress response than meat. We therefore explore the importance of fiber properties and orientation on the nonlinear rheology of fibrous model systems for meat analogues, and provide a robust methodology to quantitatively compare different mechanical responses of meat and plant-based fibrous meat analogues. These results thereby allow comparison and tuning mechanical responses of plant-based meat analogue products towards real meat.

The Josephson junction as a quantum engine

We treat the Cooper pairs in the superconducting electrodes of a Josephson junction (JJ) as an open system, coupled via Andreev scattering to external baths of electrons. The disequilibrium between the baths generates the direct-current bias applied to the JJ. In the weak-coupling limit we obtain a Markovian master equation that provides a simple dynamical description consistent with the main features of the JJ, including the form of the current–voltage characteristic, its hysteresis, and the appearance under periodic voltage driving of discrete Shapiro steps. For small dissipation, our model also exhibits a self-oscillation of the JJ’s electrical dipole with frequency around mean voltage V. This self-oscillation, associated with ‘hidden attractors’ of the nonlinear equations of motion, explains the observed production of monochromatic radiation with frequency Ω and its harmonics. We argue that this picture of the JJ as a quantum engine resolves open questions about the Josephson effect as an irreversible process and could open new perspectives in quantum thermodynamics and in the theory of dynamical systems.

An analytical study of capillary rise dynamics: Critical conditions and hidden oscillations

The rise of a liquid column inside a thin capillary against the action of gravity is a prototypical example of a dynamic wetting process and plays an important role for applications but also for fundamental research in the area of multiphase fluid dynamics. Since the pioneering work by Lucas and Washburn, many research articles have been published which aim at a simplified description of the capillary rise dynamics using complexity-reduced models formulated as ordinary differential equations. Despite the fact that these models are based on profound simplifications, they may still be able to describe the essential physical mechanisms and their interplay. In this study, we focus on the phenomenon of oscillations of the liquid column. The latter has been observed experimentally for liquids with sufficiently small viscosity leading to comparably small viscous dissipation. Back in 1999, Quéré et al. formulated a condition for the appearance of rise height oscillations for an ODE model introduced by Bosanquet in 1923. This model has later been extended to include further dissipative mechanisms. In this work, we extend the mathematical analysis to a larger class of models including additional channels of dissipation. We show that Quéré’s critical condition is generalized to Ω+β<2, where Ω was introduced earlier and β is an additional non-dimensional parameter describing, e.g., contact line friction. A quantitative prediction of the oscillation dynamics is achieved from a linearization of the governing equations. We apply the theory to experimental data by Quéré et al. and, in particular, reveal the oscillatory behavior of dynamics for the nearly critically damped case of ethanol.

An offset and gain error calibration method in high-precision SAR ADCs

Successive approximation (SAR) ADC suffers from kinds of nonidealities such as offset and gain error which affect its absolute quantization range and reduce its performance in the high-precision applications. This paper presents a static offset and an embedded gain error calibration in analog domain, attempting to improve the conversion range error in SAR ADC architecture. The presented calibration technique reuses the CADC array, facilitating an analog-domain calibration within small dissipation.

Sigmoid function generator using stochastic adiabatic superconductor logic

Stochastic-computing-based artificial neural networks (SC-ANNs) can be used to perform hardware- and energy-efficient neuromorphic computing. We have been developing SC-ANNs using an energy-efficient superconductor logic family, namely, adiabatic quantum-flux-parametron (AQFP) logic. AQFP logic is suitable as a building block for SC-ANNs since it can perform stochastic operations with extremely small energy dissipation. In this Letter, we propose and demonstrate a sigmoid function generator (SFG) for AQFP SC-ANNs, which we refer to as the AQFP SFG. An SFG is an important circuit in neural networks that generates outputs from the sum of weighted inputs in accordance with the sigmoid function. The AQFP SFG performs the sigmoid function using a finite state machine based on an AQFP buffer coupled to a flux storage loop. We experimentally demonstrate that the AQFP SFG generates output signals from stochastic bitstreams in accordance with the sigmoid function and that the characteristics of the sigmoid function can be controlled by a bias current. Furthermore, we show that the AQFP SFG operates with small power dissipation due to its simple structure.

Two novel families of multiscale staggered patch schemes efficiently simulate large-scale, weakly damped, linear waves

Many multiscale wave systems exhibit macroscale emergent behaviour, for example, the fluid dynamics of floods and tsunamis. Resolving a large range of spatial scales typically requires prohibitively high computational costs. The small dissipation in wave systems poses a significant challenge to further developing multiscale modelling methods in multiple dimensions. This article develops and evaluates two families of equation-free multiscale methods on novel 2D staggered patch schemes, and demonstrates the power and utility of these multiscale schemes for weakly damped linear waves. A detailed study of sensitivity to numerical roundoff errors establishes the robustness of developed staggered patch schemes. Comprehensive eigenvalue analysis over a wide range of parameters establishes the stability, accuracy, and consistency of the multiscale schemes. Analysis of the computational complexity shows that the measured compute times of the multiscale schemes may be 100,000 times smaller than the compute time for the corresponding (same resolution) full-domain computation. This work provides the essential foundation for efficient large-scale simulation of challenging nonlinear multiscale waves.

Performance of rocking frames with friction tension-only devices

The implementation of a new friction tension-only “GripNGrab” device attached to a rocking steel frame is described. The device, when subject to significant tension dissipates energy via sliding in the frictional component. When the device is loaded in the compression direction, almost no compressive force is carried, but displacement occurs in the ratchetting component. This absence of any significant compressive force within the dissipative system means that the rocking frame will always recentre after uplift from earthquake shaking. A 9 m tall 4.75m wide 3-storey steel concentrically braced rocking frame is designed for low-damage seismic performance. Restoring forces are provided by (i) gravity, (ii) friction “GripNGrab” (GNG) tension-only dissipation devices at the base, and (iii) beam-slab effects. The initial fundamental period of the structure was 0.16s. The initial structure used a 10mm GNG ratchet pitch, and had a GNG strength to not slide under serviceability level shaking. Elastic, pushover, cyclic pushover, as well as time history analyses, with different shaking intensities are conducted using OpenSEES software. The scope of work is limited to a single building and a single ground motion. Parameters varied included the presence of beam-slab effects, and the GNG device stiffness, strength and tooth pitch. It is shown that the full behaviour of the frame could be understood considering cyclic pushover analysis. The peak uplift displacement was conservatively estimated from the peak roof displacement using rigid body mechanics and the tension-only device provided no resistance to full frame recentring. For the frames considered, cumulative uplift displacements, necessary to determine the inelastic displacement capacity of the tension only device, were up to 28 times the peak uplift displacement, not necessarily occurring at the maximum shaking intensity. Maximum frame base shear force demands were up to 1.43 times that from pushover analysis. When the beam-slab, connecting the rocking frame to the rest of the structure, increased the lateral force resistance, the base shear increased significantly, reduced peak roof displacements, and increased the effective number of peak uplift displacement cycles (NPUDc). For large shaking intensities, yielding of the beam-slab occurred resulting in permanent peak roof and uplift displacements. The GNG device strength, stiffness and tooth pitch variations for the cases studied did not significantly affect the response. Initial stiffness, and secant stiffness, based methods to predict the response of rocking frames were non-conservative for these short-period structures with small energy dissipation, and a simple improvement to match the behaviour was developed for the case studied based on the R-T-m relationship for a range of shaking intensity.

Simulated diffusion spreadability for characterizing the structure and transport properties of two-phase materials

Time-dependent diffusion processes between phases of heterogeneous materials are ubiquitous in a variety of contexts in the physical, chemical, and biological sciences. Examples of such materials include composites, porous materials, geological media, cellular solids, polymer blends, colloids, gels, and biological media. The recently developed diffusion spreadability, S(t), provides a direct link between time-dependent interphase diffusive transport and the microstructure of two-phase materials across length scales (Torquato, 2021); thus making S(t) a powerful dynamic means for classifying all statistically homogeneous microstructures, spanning from anti-hyperuniform to hyperuniform. It was shown that the small-, intermediate-, and long-time behaviors of S(t) are directly determined by the small-, intermediate-, and large-scale structural features of the material. Moreover, the spreadability can be applied as a physical-property based tool for microstructural characterization in the absence of or as supplement to scattering information. In this work, we develop a computationally efficient algorithm for ascertaining S(t) directly from digitized representations of material microstructures via random-walk techniques. Our algorithm yields the time-dependent local walker concentration field c(x,t), a quantity not previously examined in the context of the spreadability, enabling us to compute the entropy production rate ṡ(t) of the associated diffusion process, which is a quantity related to the rate of energy dissipation. We also derive exact analytical expressions for ṡ(t), and find that hyperuniform materials have smaller dissipation than any nonhyperuniform materials. Lastly, we use our algorithm to compute, for the first time, the more general case of the spreadability in which the phase diffusion coefficients are distinct and provide a method for extracting the effective diffusion coefficient of the two-phase material from such data. We apply our algorithm to a variety of two- and three-dimensional simulated model (non)hyperuniform microstructures to assess their large-scale structures and diffusion properties. Given previously identified connections between the spreadability and certain nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI) procedures, our algorithm is also pertinent to such experimental studies. Overall, our algorithm has practical use in the discovery and design of heterogeneous materials with desirable time-dependent diffusion properties.