Equipartition Research Articles

A metamaterial with negative thermal expansivity and programmable Poisson's ratio based on rotating triangles and quivering rhombi

A mechanical metamaterial with anomalous thermal expansion and Poisson's ratio is introduced herein by inspiration from the tangram puzzle. The on-axes coefficients of thermal expansion have been extracted by matching the energy expressions from the simple harmonic motion and the principle of energy equipartition, while the on-axes Young's moduli were obtained by matching the total spring potential energy at the hinges with the strain energy of the homogenised continuum. The in-plane Poisson's ratio for infinitesimal deformation were derived using engineering strain definition, and were validated for finite deformation using true strain definition. Results reveal that the metamaterial exhibits completely in-plane negative thermal expansivity with high anisotropy, as well as an in-plane Poisson's ratio that toggles between positive and negative signs when the applied stress direction changes between tensile and compressive. The other in-plane Poisson's ratio is undefined for stretching, but can be programmed to exhibit either positive or negative signs during compression by attachment of charges on the tangram pieces. The combined characteristics of this tangram-inspired metamaterial permits it to function in ways that can neither be attained by conventional materials, nor by existing metamaterials that possess only negative properties.

Modeling the acoustic radiation of plates using circular pistons

We propose an equivalent piston model (EPM) for estimating the modal radiation efficiency of complex-shaped plates. In this model, the structure is not discretized into small elementary radiators. Instead, it involves determining the minimum number, size, volume velocity, and position of vibrating pistons that radiate in accordance with the mode shape they emulate, i.e. one piston per lobe within the mode shape, regardless the plate’s shape. For shapes relatively close to rectangles, a specific version of the method is proposed by representing the shape only by two characteristic lengths (EPM-RD), and paving the virtual surface with pistons vibrating at alternate phases along a rectangular grid. The validity of this assumption is evaluated in terms of shape dissymmetry, which can cause residual radiators to appear, and where the dissymmetry can affect the spatial distribution of lobes. With a known spatial distribution mimicking the mode shapes, it is possible to estimate also average values of radiation parameters, such as the average radiation efficiency and time-average mean power. The average radiation efficiency and power follow the Equipartition Theorem and allow the calculation to be performed without cross-terms and using only the modal efficiencies and resonance frequencies. Due to their simplicity and ease of implementation, piston-based equivalent models have great potential for solving radiation problems from flat structures. Any mode shape can be modeled with vibrating pistons and analytical formulations can be easily derived, even for complex geometries. Moreover, with EPM-RD, average values can also be calculated, which provides an estimation of average radiated properties within seconds without requiring any integration or large matrices to be solved.

A new closed analytical solution for the elastodynamic half-space Green’s function

The elastodynamic half-space Green’s function has been the subject of research for more than a century since the Lamb’s classical solution. Here, we revisit the problem and present a new closed analytical solution, in frequency domain, based upon the Principle of Equipartition (EQP) of Energy. This principle asserts that the imaginary parts of the Green’s tensor components equal the average cross-correlations of the fields generated by the uniform incidence of P and S body waves and by Rayleigh surface waves with amplitudes weighted by partition factors. The real part of the Green’s function is the Hilbert transform of the imaginary part. We validate our results by comparing synthetic seismograms of well-known solutions for surface and buried forces and discuss the implications of this new solution. Constructing synthetic diffuse fields is a first step for identifying them in nature.Graphical

The Metastable State of Fermi–Pasta–Ulam–Tsingou Models

Classical statistical mechanics has long relied on assumptions such as the equipartition theorem to understand the behavior of the complicated systems of many particles. The successes of this approach are well known, but there are also many well-known issues with classical theories. For some of these, the introduction of quantum mechanics is necessary, e.g., the ultraviolet catastrophe. However, more recently, the validity of assumptions such as the equipartition of energy in classical systems was called into question. For instance, a detailed analysis of a simplified model for blackbody radiation was apparently able to deduce the Stefan-Boltzmann law using purely classical statistical mechanics. This novel approach involved a careful analysis of a "metastable" state which greatly delays the approach to equilibrium. In this paper, we perform a broad analysis of such a metastable state in the classical Fermi-Pasta-Ulam-Tsingou (FPUT) models. We treat both the α-FPUT and β-FPUT models, exploring both quantitative and qualitative behavior. After introducing the models, we validate our methodology by reproducing the well-known FPUT recurrences in both models and confirming earlier results on how the strength of the recurrences depends on a single system parameter. We establish that the metastable state in the FPUT models can be defined by using a single degree-of-freedom measure-the spectral entropy (η)-and show that this measure has the power to quantify the distance from equipartition. For the α-FPUT model, a comparison to the integrable Toda lattice allows us to define rather clearly the lifetime of the metastable state for the standard initial conditions. We next devise a method to measure the lifetime of the metastable state tm in the α-FPUT model that reduces the sensitivity to the exact initial conditions. Our procedure involves averaging over random initial phases in the plane of initial conditions, the P1-Q1 plane. Applying this procedure gives us a power-law scaling for tm, with the important result that the power laws for different system sizes collapse down to the same exponent as Eα2→0. We examine the energy spectrum E(k) over time in the α-FPUT model and again compare the results to those of the Toda model. This analysis tentatively supports a method for an irreversible energy dissipation process suggested by Onorato et al.: four-wave and six-wave resonances as described by the "wave turbulence" theory. We next apply a similar approach to the β-FPUT model. Here, we explore in particular the different behavior for the two different signs of β. Finally, we describe a procedure for calculating tm in the β-FPUT model, a very different task than for the α-FPUT model, because the β-FPUT model is not a truncation of an integrable nonlinear model.

Encoding dynamical information of protein from normal mode analysis as multigraphs for function prediction by graph neural networks and highlighting functionally-critical residues.

Dynamics of proteins are crucial for its ability to perform its functions, but are often overlooked in the plethora of protein function prediction methods. However, the need for faster screening over sequence-structure-function spaces is ever growing due to the increase in number of known peptide sequences and structures without function annotations. Machine learning methods are commonly implemented to meet the latter need to moderate success, with graph neural networks (GNN) gaining traction in recent years due to its rather intuitive mapping to protein structures. On the other hand, encoding the dynamics of proteins is yet a critical outstanding issue. In our attempt to close this gap, we encode the dynamical information of modal shapes and frequencies, along with the protein contact map, into multigraphs to aid graph neural networks in their prediction. This reduction of mode shapes into graphs is done by calculating the correlation between pairwise residue movements, then weight-averaging the correlation across multiple mode shapes, taking into consideration the equipartition theorem. The anisotropic network model (ANM) and the torsional network model (TNM) are implemented for modal analysis, and the two network models are compared within our framework. The method is tested on various datasets with different sequence similarity cutoffs, and the results compared with existing methods. Furthermore, we are able to extract from our model the dynamically-critical residues, such as hydrogen-binding and ligand-binding sites, for each of the protein's functions. As such, the proposed model provides insight to critical mutation points, facilitating further investigation and understanding of the protein.

Magnetic Fields, Star Formation Rates, and Gas Densities at Sub-kiloparsec Scales in a Pilot Sample of Nearby Galaxies

We have estimated the magnetic field strengths of a sample of seven galaxies using their nonthermal synchrotron radio emission at meter wavelengths, and assuming energy equipartition between magnetic fields and cosmic-ray particles. We tested for deviation of magnetic fields from energy equipartition with cosmic-ray particles, and found that deviations of ∼25% are typical for the sample galaxies. Spatially resolved star formation rates (SFRs) were estimated for the seven galaxies along with five galaxies studied previously. For the combined sample of 12 galaxies, the equipartition magnetic fields (B eq) are correlated with the SFR surface densities (ΣSFR) at sub-kiloparsec scales with B eq ∝ , consistent with model predictions. We estimated gas densities (ρ gas) for a subsample of seven galaxies using archival observations of the CO rotational transitions and the atomic hydrogen (H i) 21 cm line and studied the spatially resolved correlation between the magnetic fields and ρ gas. Magnetic fields and gas densities are found to be correlated at sub-kiloparsec scale as B eq ∝ . This is broadly consistent with models, which typically predict B ∝ .

Kinetic Temperature of Structures for Resilience, Instability, and Failure Analysis of Building Systems

From theory, calibration and application of the equipartition theorem of statistical physics to structural failure and instability analysis, we introduce the kinetic temperature of structures as an order parameter to ascertain equilibrium and out-of-equilibrium states in structural mechanics. Set within the framework of molecular dynamics-based structural mechanics, this is achieved by connecting the set of momentum balance equations to an outside bath reservoir maintained at a reference temperature history through the Nosé-Hoover thermostat. The problem thus comes down to solving the momentum balance equation with a dissipative mass damping term, which evolves in function of the difference in temperature between the structure’s kinetic temperature/energy and the bath temperature. Following the Zeroth Law of Thermodynamics, it is recognized that a structure is in (thermal) equilibrium as long as the structure’s kinetic temperature attains the bath temperature; whereas it is out-of-equilibrium when the open system (structure plus bath) exhibits a sustained temperature difference. In this case, the structure has exhausted its fluctuation-dissipation capacity, which is indicative—for structures—of a progressive failure and instability. The implementation of the kinetic temperature as an order parameter in structural failure and instability analysis is illustrated for a prototype five-storey building subject to excessive wind and fire loads. It is suggested that the proposed order parameter becomes an integral part of the structural engineering toolbox for resilience studies of buildings and structures.

Stationary states of activity-driven harmonic chains.

We study the stationary state of a chain of harmonic oscillators driven by two active reservoirs at the two ends. These reservoirs exert correlated stochastic forces on the boundary oscillators which eventually leads to a nonequilibrium stationary state of the system. We consider three most well-known dynamics for the active force, namely, the active Ornstein-Uhlenbeck process, run-and-tumble process, and active Brownian process, all of which have exponentially decaying two-point temporal correlations but very different higher-order fluctuations. We show that, irrespective of the specific dynamics of the drive, the stationary velocity fluctuations are Gaussian in nature with a kinetic temperature which remains uniform in the bulk. Moreover, we find the emergence of an "equipartition of energy" in the bulk of the system-the bulk kinetic temperature equals the bulk potential temperature in the thermodynamic limit. We also calculate the stationary distribution of the instantaneous energy current in the bulk which always shows a logarithmic divergence near the origin and asymmetric exponential tails. The signatures of specific active driving become visible in the behavior of the oscillators near the boundary. This is most prominent for the RTP- and ABP-driven chains where the boundary velocity distributions become non-Gaussian and the current distribution has a finite cutoff.

EDIG-CHANGES I: extended Hα emission from the extraplanar diffuse ionized gas (eDIG) around CHANG-ES galaxies

ABSTRACT The extraplanar diffuse ionized gas (eDIG) represents the cool/warm ionized gas reservoir around galaxies. We present spatial analysis of the Hα images of 22 nearby edge-on spiral galaxies taken with the Apache Point Observatory 3.5-m telescope (eDIG-CHANGES). We conduct an exponential fit to the vertical Hα intensity profiles of the galaxies, of which 16 can be decomposed into thin + thick disk components. The median value of the Hα scale height of the thick disk is $\langle h_{\rm H\alpha }\rangle =1.13\pm 0.14\rm ~kpc$. We further examine the dependence of hHα on the stellar mass, SFR, and SFR surface density (SFRSD) of the galaxies. We find a tight sublinear correlation between hHα and SFR, expressed in hHα ∝ SFRα, where α ≈ 0.29. Moreover, the offset of individual galaxies from the best-fit SFR-hHα relation, expressed in hHα/SFRα, shows significant anti-correlation with SFRSD. We further compare the vertical extension of the eDIG to multi-wavelength measurements of other CGM phases. We find the eDIG slightly more extended than the neutral gas. This indicates the existence of some extended ionizing sources, in addition to the leaking photons from the disk star formation regions. Most galaxies have an X-ray scale height smaller than Hα, suggesting the majority of the X-ray photons are actually from the thick disk instead of the extended CGM. hHα is comparable to the L-band radio continuum scale height. This indicates that the thermal and non-thermal electrons have similar spatial distributions, a natural result if both are transported outwards by a galactic wind. This further indicates the thermal gas, cosmic rays, and magnetic field may be close to energy equipartition.

Strain correlation functions in isotropic elastic bodies: large wavelength limit for two-dimensional systems.

Strain correlation functions in two-dimensional isotropic elastic bodies are shown both theoretically (using the general structure of isotropic tensor fields) and numerically (using a glass-forming model system) to depend on the coordinates of the field variable (position vector r in real space or wavevector q in reciprocal space) and thus on the direction of the field vector and the orientation of the coordinate system. Since the fluctuations of the longitudinal and transverse components of the strain field in reciprocal space are known in the long-wavelength limit from the equipartition theorem, all components of the correlation function tensor field are imposed and no additional physical assumptions are needed. An observed dependence on the field vector direction thus cannot be used as an indication for anisotropy or for a plastic rearrangement. This dependence is different for the associated strain response field containing also information on the localized stress perturbation.

Gaussian fluctuations in the equipartition principle for Wigner matrices

Abstract The total energy of an eigenstate in a composite quantum system tends to be distributed equally among its constituents. We identify the quantum fluctuation around this equipartition principle in the simplest disordered quantum system consisting of linear combinations of Wigner matrices. As our main ingredient, we prove the Eigenstate Thermalisation Hypothesis and Gaussian fluctuation for general quadratic forms of the bulk eigenvectors of Wigner matrices with an arbitrary deformation.

Conformational dynamics and mechanical properties of biomimetic RNA, DNA, and RNA-DNA hybrid nanotubes: an atomistic molecular dynamics study.

With the nanotechnology boom, artificially designed nucleic acid nanotubes have aroused interest due to their practical applications in nanorobotics, vaccine design, membrane channels, drug delivery, and force sensing. In this paper, computational study was performed to investigate the structural dynamics and mechanical properties of RNA nanotubes (RNTs), DNA nanotubes (DNTs), and RNA-DNA hybrid nanotubes (RDHNTs). So far, the structural and mechanical properties of RDHNTs have not been examined in experiments or theoretical calculations, and there is limited knowledge regarding these properties for RNTs. Here, the simulations were carried out using the equilibrium molecular dynamics (MD) and steered molecular dynamics (SMD) approaches. Using in-house scripting, we modeled hexagonal nanotubes composed of six double-stranded molecules connected by four-way Holliday junctions. Classical MD analyses were performed on the collected trajectory data to investigate structural properties. Analyses of the microscopic structural parameters of RDHNT indicated a structural transition from the A-form to a conformation between the A- and B-forms, which may be attributable to the increased rigidity of RNA scaffolds compared to DNA staples. Comprehensive research on the elastic mechanical properties was also conducted based on spontaneous thermal fluctuations of nanotubes and employing the equipartition theorem. The Young's modulus of RDHNT (E = 165 MPa) and RNT (E = 144 MPa) was found to be almost the same and nearly half of that found for DNT (E = 325 MPa). Furthermore, the results showed that RNT was more resistant to bending, torsional, and volumetric deformations than DNT and RDHNT. We also used non-equilibrium SMD simulations to acquire comprehensive knowledge of the mechanical response of nanotubes to tensile stress.

Diffuse radio source candidate in CIZA J1358.9−4750

AbstractWe report on results of our upgraded Giant Metrewave Radio Telescope (uGMRT) observations for an early-stage merging galaxy cluster, CIZA J1358.9−4750 (CIZA1359), in Band-3 (300–500 MHz). We achieved the image dynamic range of ∼38000 using the direction dependent calibration and found a candidate of diffuse radio emission at 4σrms significance. The flux density of the candidate at 400 MHz, 24.04 ± 2.48 mJy, is significantly positive compared to noise, where its radio power, 2.40 × 1024 W Hz−1, is consistent with those of typical diffuse radio sources of galaxy clusters. The candidate is associated with a part of the X-ray shock front at which the Mach number reaches its maximum value of $\mathcal {M}\sim 1.7$. The spectral index (Fν ∝ να) of the candidate, α = −1.22 ± 0.33, is in agreement with an expected value derived from the standard diffusive shock acceleration (DSA) model. However, such a low Mach number with a short acceleration time would require seed cosmic rays supplied from active galactic nucleus (AGN) activities of member galaxies, as suggested in some other clusters. Indeed, we found seven AGN candidates inside the diffuse source candidate. Assuming the energy equipartition between magnetic fields and cosmic rays, the magnetic field strength of the candidate was estimated to be 2.1 μG. We also find head–tail galaxies and radio phoenixes or fossils near CIZA1359.

Bekenstein Bound and Non-Commutative Canonical Variables

A universal upper limit on the entropy contained in a localized quantum system of a given size and total energy is expressed by the so-called Bekenstein bound. In a previous paper [Buoninfante, L. et al. 2022], on the basis of general thermodynamic arguments, and in regimes where the equipartition theorem still holds, the Bekenstein bound has been proved practically equivalent to the Heisenberg uncertainty relation. The smooth transition between the Bekenstein bound and the holographic bound suggests a new pair of canonical non-commutative variables, which could be thought to hold in strong gravity regimes.

Deciphering the radio–star formation correlation on kpc scales

Given the multiple energy-loss mechanisms of cosmic-ray (CR) electrons in galaxies, the tightness of the infrared (IR)–radio continuum correlation is surprising. As the radio continuum emission at GHz frequencies is optically thin, this offers the opportunity to obtain unbiased star formation rates (SFRs) from radio-continuum flux-density measurements. The calorimeter theory can naturally explain the tightness of the far-infrared (FIR)–radio correlation but makes predictions that do not agree with observations. Noncalorimeter models often have to involve a conspiracy to maintain the tightness of the FIR–radio correlation. We extended a published analytical model of galactic disks by including a simplified prescription for the synchrotron emissivity. The galactic gas disks of local spiral galaxies, low-z starburst galaxies, high-z main sequence star-forming galaxies, and high-z starburst galaxies are treated as turbulent clumpy accretion disks. The magnetic field strength is determined by the equipartition between the turbulent kinetic and the magnetic energy densities. Our fiducial model, which includes neither galactic winds nor CR electron secondaries, reproduces the observed radio continuum spectral energy distributions of most (∼70%) of the galaxies. Except for the local spiral galaxies, fast galactic winds can potentially make the conflicting models agree with observations. The observed IR–radio correlations are reproduced by the model within 2σ of the joint uncertainty of model and data for all datasets. The model agrees with the observed SFR–radio correlations within ∼4σ. Energy equipartition between the CR particles and the magnetic field only approximately holds in our models of main sequence star-forming galaxies. If a CR electron calorimeter is assumed, the slope of the IR–radio correlation flattens significantly. Inverse Compton losses are not dominant in the starburst galaxies because in these galaxies not only the gas density but also the turbulent velocity dispersion is higher than in normal star-forming galaxies. Equipartition between the turbulent kinetic and magnetic field energy densities then leads to very high magnetic field strengths and very short synchrotron timescales. The exponents of our model SFR–radio correlations at 150 MHz and 1.4 GHz are very close to one.

3D code for MAgneto-Thermal evolution in Isolated Neutron Stars, MATINS: the magnetic field formalism

ABSTRACT The long-term evolution of the internal, strong magnetic fields of neutron stars needs a specific numerical modelling. The diversity of the observed phenomenology of neutron stars indicates that their magnetic topology is rather complex and 3D simulations are required, for example, to explain the observed bursting mechanisms and the creation of surface hotspots. We present MATINS, a new 3D numerical code for magnetothermal evolution in neutron stars, based on a finite-volume scheme that employs the cubed-sphere system of coordinates. In this first work, we focus on the crustal magnetic evolution, with the inclusion of realistic calculations for the neutron star structure, composition, and electrical conductivity assuming a simple temperature evolution profile. MATINS follows the evolution of strong fields (1014 − 1015 Gauss) with complex non-axisymmetric topologies and dominant Hall-drift terms, and it is suitable for handling sharp current sheets. After introducing the technical description of our approach and some tests, we present long-term simulations of the non-linear field evolution in realistic neutron star crusts. The results show how the non-axisymmetric Hall cascade redistributes the energy over different spatial scales. Following the exploration of different initial topologies, we conclude that during a few tens of kyr, an equipartition of energy between the poloidal and toroidal components happens at small-scales. However, the magnetic field keeps a strong memory of the initial large scales, which are much harder to be restructured or created. This indicates that large-scale configuration attained during the neutron star formation is crucial to determine the field topology at any evolution stage.

Type IIP supernova SN2016X in radio frequencies

Context. The study of radio emission from core-collapse supernovae (SNe) probes the interaction of the ejecta with the circumstellar medium (CSM) and reveals details of the mass-loss history of the progenitor. Aims. We report observations of the type IIP supernova SN 2016X during the plateau phase, at ages between 21 and 75 days, obtained with the Karl G. Jansky Very Large Array (VLA) radio observatory. Methods. We modelled the radio spectra as self-absorbed synchrotron emission, and we characterised the shockwave and the mass-loss rate of the progenitor. We also combined our results with previously reported X-ray observations to verify the energy equipartition assumption. Results. The properties of the shockwave are comparable to other type IIP supernovae. The shockwave expands according to a self-similar law R ∝ tm with m = 0.76 ± 0.08, which is notably different from a constant expansion. The corresponding shock velocities are approximately 10700–8000 km s−1 during the time of our observations. The constant mass-loss rate of the progenitor is Ṁ = (7.8 ± 0.9) × 10−7 α−8/19 (ϵB/0.1)−1 M⊙ yr−1, for an assumed wind velocity of 10 km s−1. We observe spectral steepening in the optically thin regime at the earlier epochs, and we demonstrate that it is caused by electron cooling via the inverse Compton effect. We show that the shockwave is characterised by a moderate deviation from energy equipartition by a factor of ϵe/ϵB ≈ 28, being the second type IIP supernova to show such a feature.

Equipartitions and Mahler volumes of symmetric convex bodies

Following ideas of Iriyeh and Shibata we give a short proof of the three-dimensional Mahler conjecture for symmetric convex bodies. Our contributions include, in particular, simple self-contained proofs of their two key statements. The first of these is an equipartition (ham sandwich type) theorem which refines a celebrated result of Hadwiger and, as usual, can be proved using ideas from equivariant topology. The second is an inequality relating the product volume to areas of certain sections and their duals. Finally we give an alternative proof of the characterization of convex bodies that achieve the equality case and establish a new stability result.

The Expansion of the X-Ray Nebula Around η Car

The massive colliding wind binary system η Car is embedded in an X-ray emitting region having a characteristic temperature of a few million degrees, associated with ejecta produced during the 1840s, and in earlier outbursts. We use CHANDRA X-ray imaging observations obtained over the past two decades to directly measure the expansion of the X-ray nebula for the first time. A combined CHANDRA/ACIS image shows a faint, nearly uniform elliptic structure. This faint elliptical “shell” has a similar orientation and shape as the Homunculus nebula but is about 3 times larger. We measure proper motions of brighter regions associated with the X-ray emitting ring. We compare spectra of the soft X-ray emitting plasma in CHANDRA/ACIS and XMM-Newton PN observations and show that the PN observations indicate a decline in X-ray flux which is comparable to that derived from NICER observations. We associate the diffuse elliptical emission surrounding the bright X-ray “ring” with the blast wave produced during the Great Eruption. We suggest that the interaction of this blast wave with pre-existing clumps of ejecta produces the bright, broken X-ray emitting ring. We extrapolate the trend in X-ray energy back to the time of the Great Eruption using a simple model and show that the X-ray energy was comparable to the kinetic energy of the Homunculus, suggesting equipartition of energy between fast, low-density ejecta and slower, dense ejecta.

Magnetic field properties in star formation: A review of their analysis methods and interpretation

Linearly polarized emission from dust grains and molecular spectroscopy is an effective probe of the magnetic field topology in the interstellar medium and molecular clouds. The longstanding Davis-Chandrasekhar-Fermi (DCF) method and the recently developed Histogram of Relative Orientations (HRO) analysis and the polarization-intensity gradient (KTH) method are widely used to assess the dynamic role of magnetic fields in star formation based on the plane-of-sky component of field orientations inferred from the observations. We review the advances and limitations of these methods and summarize their applications to observations. Numerical tests of the DCF method, including its various variants, indicate that its largest uncertainty may come from the assumption of energy equipartition, which should be further calibrated with simulations and observations. We suggest that the ordered and turbulent magnetic fields of particular observations are local properties of the considered region. An analysis of the polarization observations using DCF estimations suggests that magnetically trans-to-super-critical and averagely trans-to-super-Alfvénic clumps/cores form in sub-critical clouds. High-mass star-forming regions may be more gravity-dominant than their low-mass counterparts due to higher column density. The observational HRO studies clearly reveal that the preferential relative orientation between the magnetic field and density structures changes from parallel to perpendicular with increasing column densities, which, in conjunction with simulations, suggests that star formation is ongoing in trans-to-sub-Alfvénic clouds. There is a possible transition back from perpendicular to random alignment at higher column densities. Results from observational studies using the KTH method broadly agree with those of the HRO and DCF studies.