Power Law Description Research Articles

Analysis of natural planar jump in power-law liquids—A generalized “shallow flow” approach

The study presents a generalized “shallow flow” analysis of natural planar hydraulic jump in power-law liquids. It is based on self-similar velocity profile defined as function of flow behavior index, n, and shows significant improvement over the previous analysis which assumed a quadratic velocity profile and failed for n < 0.5. Thus, the study enables a deeper understanding of the influence of n for shear thinning vis-a-vis shear thickening liquids and emphasizes that the power-law description is adequate for highly shear thinning liquids if the flow parameters are valid over the range of interest.

Aeration and rheology of buttercream icings

Buttercream icings – 4 phase materials comprising sugar, butter (an emulsion) and air – are widely used for coating and decorating sweet goods. The aeration of 2:1 w/w sugar-butter mixtures was investigated in 2 benchtop planetary mixers and the rheology of the resultant viscoplastic material was quantified using lubricated compressive testing (squeeze flow and upsetting) over extensional shear rates of 10−3–100 s−1. The aeration dynamics exhibited asymptotic behaviour which fitted a simple phenomenological model well. Optical microscopy and X-ray microtomography indicated that the air was incorporated as small bubbles of similar size to the sugar particles or as voids generated by the mixer wires. The measured yield stress was strongly dependent on the air volume fraction, φ. The flow behaviour for one batch could be reasonably described as a Herschel-Bulkley fluid whereas a second batch gave overall better agreement with a modified power-law description: in both cases the viscous friction term decreased strongly with φ.

Influences of graphene morphology and contact distance between nanosheets on the effective conductivity of polymer nanocomposites

Herein, the contact distance and effective tunneling conductivity in graphene polymer nanocomposites are expressed assuming the properties of graphene stack and the resistances of all components by graphene dimensions, interphase depth, contact resistance and filler morphology (stacked and well-dispersed nanosheets). In the case of incomplete filler dispersion in the matrix, the volume share, aspect ratio and conduction of stacks are suggested. Also, the contact distance is presented based on a power law description by percolation onset and effective filler amount supposing the properties of stacks. The effects of all parameters on the contact distance and effective conductivity are plotted at various ranges of factors. Undoubtedly, the reasonable impacts of all factors on the contact distance and effective conductivity justify the suggested equations. A higher filler amount, more filler dispersion, lower number of nanosheets in stacks, higher aspect ratio of filler (thinner and larger nanosheets), deeper interphase and larger distance between nanosheets in stacks produce a shorter contact distance, bigger network and less total resistance causing more effective conductivity.

Planetesimal Initial Mass Functions Following Diffusion-regulated Gravitational Collapse

The initial mass function (IMF) of planetesimals is of key importance for understanding the initial stages of planet formation, yet theoretical predictions so far have been insufficient in explaining the variety of IMFs found in simulations. Here, we connect diffusion-tidal-shear limited planetesimal formation within the framework of a Toomre-like instability in the particle midplane of a protoplanetary disk to an analytic prediction for the planetesimal IMF. The shape of the IMF is set by the stability parameter Q p, which in turn depends on the particle Stokes number, the Toomre Q value of the gas, the local dust concentration, and the local diffusivity. We compare our prediction to high-resolution numerical simulations of the streaming instability and planetesimal formation via gravitational collapse. We find that our IMF prediction agrees with numerical results and is consistent with both the paradigm that planetesimals are born big and the power-law description commonly found in simulations.

Thermo-Mechanical and Creep Behaviour of Polylactic Acid/Thermoplastic Polyurethane Blends.

There is a great need to develop biodegradable thermoplastics for a variety of applications in a wide range of temperatures. In this work, we prepare polymer blends from polylactic acid (PLA) and thermoplastic polyurethane (TPU) via a melting blend method at 200 °C and study the creep deformation of the PLA/TPU blends in a temperature range of 10 to 40 °C with the focus on transient and steady-state creep. The stress exponent for the power law description of the steady state creep of PLA/TPU blends decreases linearly with the increase of the mass fraction of TPU from 1.73 for the PLA to 1.17 for the TPU. The activation energies of the rate processes for the steady-state creep and transient creep decrease linearly with the increase of the mass fraction of TPU from 97.7 ± 3.9 kJ/mol and 59.4 ± 2.9 kJ/mol for the PLA to 26.3 ± 1.3 kJ/mol and 25.4 ± 1.7 kJ/mol for the TPU, respectively. These linearly decreasing trends can be attributed to the weak interaction between the PLA and the TPU. The creep deformation of the PLA/TPU blends consists of the contributions of individual PLA and TPU.

Extreme COVID-19 waves reveal hyperexponential growth and finite-time singularity

Coronavirus disease 2019 (COVID-19) has rapidly spread throughout our planet, bringing human lives to a standstill. Understanding the early transmission dynamics of a wave helps plan intervention strategies such as lockdowns that mitigate further spread, minimizing the adverse impact on humanity and the economy. Exponential growth of infections was thought to be the defining feature of an epidemic in its initial growth phase. Here we show that, contrary to common belief, early stages of extreme COVID-19 waves have an unbounded growth and finite-time singularity accompanying a hyperexponential power-law. The faster than exponential growth phase is hazardous and would entail stricter regulations to minimize further spread. Such a power-law description allows us to characterize COVID-19 waves better using single power-law exponents, rather than using piecewise exponentials. Furthermore, we identify the presence of log-periodic patterns decorating the power-law growth. These log-periodic oscillations may enable better prediction of the finite-time singularity. We anticipate that our findings of hyperexponential growth and log-periodicity will enable accurate modeling of outbreaks of COVID-19 or similar future outbreaks of other emergent epidemics.

Oscillatory Microrheology, Creep Compliance and Stress Relaxation of Biological Cells Reveal Strong Correlations as Probed by Atomic Force Microscopy

The mechanical properties of cells are important for many biological processes, including wound healing, cancers, and embryogenesis. Currently, our understanding of cell mechanical properties remains incomplete. Different techniques have been used to probe different aspects of the mechanical properties of cells, among them microplate rheology, optical tweezers, micropipette aspiration, and magnetic twisting cytometry. These techniques have given rise to different theoretical descriptions, reaching from simple Kelvin-Voigt or Maxwell models to fractional such as power law models, and their combinations. Atomic force microscopy (AFM) is a flexible technique that enables global and local probing of adherent cells. Here, using an AFM, we indented single retinal pigmented epithelium cells adhering to the bottom of a culture dish. The indentation was performed at two locations: above the nucleus, and towards the periphery of the cell. We applied creep compliance, stress relaxation, and oscillatory rheological tests to wild type and drug modified cells. Considering known fractional and semi-fractional descriptions, we found the extracted parameters to correlate. Moreover, the Young’s modulus as obtained from the initial indentation strongly correlated with all of the parameters from the applied power-law descriptions. Our study shows that the results from different rheological tests are directly comparable. This can be used in the future, for example, to reduce the number of measurements in planned experiments. Apparently, under these experimental conditions, the cells possess a limited number of degrees of freedom as their rheological properties change.

Unraveling the implications of finite specimen size on the interpretation of dynamic experiments for polycrystalline aluminum through direct numerical simulations

Normal and Pressure-shear plate impact (NPI and PSPI) experiments are popular experimental techniques for studying the mean-field macroscopic behavior of polycrystalline metals under high-rate dynamic loading. However, since both configurations rely upon geometry for subjecting the specimen to high strain rates, these experiments often involve a limited specimen size. Moreover, because of the inherent heterogeneities present within polycrystalline metals, it is difficult to ascertain if the size of the specimen and/or regions where measurements are made are sufficiently large for making representative inferences about the mean-field macroscopic properties from single-point velocity measurements. In the present study, we quantify the expected measurement variability on observable point measurements in NPI and PSPI experiments by carrying out direct numerical simulations (DNS) of statistically representative polycrystalline microstructures subjected to dynamic compression and compression-shear loading. In particular, we consider the role of specific material heterogeneities (e.g. the grain-to-grain difference in size, crystallographic orientation) on dispersion in the normal and transverse particle velocity records and on local fluctuations in key state variables (e.g. velocity, accumulated plastic strain) by incorporating these effects directly into a representative synthetic microstructure geometry and crystalline description of pure polycrystalline aluminum. The form of the present study is a large parametric investigation, consisting of ten ensembles of one hundred simulations. Each of the thousand simulations reflects a randomly realized synthetic microstructure in one of five cases of decreasing average grain size for the two loading configurations. Our analysis of the DNS results demonstrates that for both of these experimental configurations, the grain size directly correlates with the coefficient of variation (CV) in simulated point measurements, showing a convergent decrease in CV to zero (i.e. particle velocity record approaches the mean-field value) with decreasing grain size. Remarkably, the magnitude of variations in the particle velocity record is shown to be largest where the deviatoric stresses are most significant. In the case of NPI, this occurs at the elastic and plastic wavefront, whereas, in the case of PSPI, the magnitude of fluctuations are approximately constant throughout the experimental window time. The reasoning for the scatter in particle velocity due to the heterogeneous microstructure is demonstrated to be dependent on the mechanisms for accommodating deformation and on the interaction of reflection waves generated at sites of heterogeneities occurring at the scale of grains. Lastly, we develop a power-law description for the magnitude of scattering versus characteristic length, which provides a statistical framework for assessing the required number of grains per characteristic specimen dimension for minimizing scatter within these two experimental configurations (NPI, PSPI).

Python-Based Program for Error Structure Analysis and Regression of Impedance Data

The measurement model described in this presentation is based on a program developed in the 1990s by Prof. Mark Orazem and his group using a Matlab front-end and FORTRAN executables.[1] The objective of the present work was to completely rebuild the original measurement model into a stand-alone program. This program was to have more features, be more user-friendly, and be free of proprietary code in order to enable its distribution. The new code was written from the ground up in Python, although some portions of the backend libraries, such as numpy, use C.The working part of the program has five tabs. The File Tools and Conversion part of the program is used to convert files to a program-usable format. The Measurement Model part of the program is used to fit a Voigt measurement model to the impedance data. The Voigt measurement model is the foundation of the error structure analysis performed by the program. The measurement model can be used to filter replicated measurements to extract the standard deviation of the stochastic part of the error structure.[2] These are presented in the Error File Preparation part of the program. In the Error Analysis part of the program, a simple model can be fit to the standard deviation of the stochastic part of the error structure. The model of the stochastic error structure can be used to weight regressions of the impedance spectra. The measurement model may be used as well to determine the portion of the measurement that satisfies the Kramers-Kronig relations.[3] The approach employing a properly-weighted nonlinear regression has been shown to be more sensitive to failures of causality than the linear Voigt model regression developed by Boukamp and employed by many instrument vendors.[4] The final tab of the working part of the program, Custom Formula Fitting, allows regression of arbitrary functions to impedance data. Rather than employ electrical circuit components such as resistors and capacitors,[5],[6] this part of the program is equation-based. The formula is input as Python code and may make use of imported Python packages. The package numpy is already imported. As in standard Python, the imaginary number can be accessed as 1j. The equation-based approach enables regression of models that include look-up tables for convective diffusion,[7] integral functions for the power-law description of oxide films,[8] and de Levie formulas for porous electrodes.[9] The objective of this effort is to provide a program that can allow the users to access the power of the measurement model concept and to fit custom process models. The program will be distributed as a Windows executable.

Discovery of a Nearby Young Brown Dwarf Disk

Abstract We report the discovery of the youngest brown dwarf with a disk at 102 pc from the Sun, WISEA J120037.79−784508.3 (W1200−7845), via the Disk Detective citizen science project. We establish that W1200−7845 is located in the Myr old ε Cha association. Its spectral energy distribution (SED) exhibits clear evidence of an infrared (IR) excess, indicative of the presence of a warm circumstellar disk. Modeling this warm disk, we find the data are best fit using a power-law description with a slope α = −0.94, which suggests that it is a young, Class II type disk. Using a single blackbody disk fit, we find and . The near-IR spectrum of W1200−7845 matches a spectral type of M6.0 0.5, which corresponds to a low surface gravity object, and lacks distinctive signatures of strong Paβ or Brγ accretion. Both our SED fitting and spectral analysis indicate that the source is cool (T eff = 2784–2850 K), with a mass of 42–58 M Jup, well within the brown dwarf regime. The proximity of this young brown dwarf disk makes the system an ideal benchmark for investigating the formation and early evolution of brown dwarfs.

Decaying two-dimensional turbulence undergoes statistical heating

An emergent property of decaying two-dimensional turbulence is shown to be a weak but persistent statistical heating, or tendency towards clustering of like-signed vortices. The rate of this heating, which is driven by changes to the vortex population due to vortex mergers and straining events, provides a strong constraint on both vortex decay laws and kinetic theories of vortex interactions. A quasi-equilibrium statistical theory determines the energy spectrum from just the vortex interaction energy and bulk information about the number density of the vortices. The emergent heating rate is shown to determine the upscale transfer of energy, and is a powerful constraint on power-law descriptions of the evolving vortex population.

The GLEAM 4-Jy (G4Jy) Sample: I. Definition and the catalogue

Abstract The Murchison Widefield Array (MWA) has observed the entire southern sky (Declination, $\delta< 30^{\circ}$ ) at low radio frequencies, over the range 72–231MHz. These observations constitute the GaLactic and Extragalactic All-sky MWA (GLEAM) Survey, and we use the extragalactic catalogue (EGC) (Galactic latitude, $|b| >10^{\circ}$ ) to define the GLEAM 4-Jy (G4Jy) Sample. This is a complete sample of the ‘brightest’ radio sources ( $S_{\textrm{151\,MHz}}>4\,\text{Jy}$ ), the majority of which are active galactic nuclei with powerful radio jets. Crucially, low-frequency observations allow the selection of such sources in an orientation-independent way (i.e. minimising the bias caused by Doppler boosting, inherent in high-frequency surveys). We then use higher-resolution radio images, and information at other wavelengths, to morphologically classify the brightest components in GLEAM. We also conduct cross-checks against the literature and perform internal matching, in order to improve sample completeness (which is estimated to be $>95.5$ %). This results in a catalogue of 1863 sources, making the G4Jy Sample over 10 times larger than that of the revised Third Cambridge Catalogue of Radio Sources (3CRR; $S_{\textrm{178\,MHz}}>10.9\,\text{Jy}$ ). Of these G4Jy sources, 78 are resolved by the MWA (Phase-I) synthesised beam ( $\sim2$ arcmin at 200MHz), and we label 67% of the sample as ‘single’, 26% as ‘double’, 4% as ‘triple’, and 3% as having ‘complex’ morphology at $\sim1\,\text{GHz}$ (45 arcsec resolution). We characterise the spectral behaviour of these objects in the radio and find that the median spectral index is $\alpha=-0.740 \pm 0.012$ between 151 and 843MHz, and $\alpha=-0.786 \pm 0.006$ between 151MHz and 1400MHz (assuming a power-law description, $S_{\nu} \propto \nu^{\alpha}$ ), compared to $\alpha=-0.829 \pm 0.006$ within the GLEAM band. Alongside this, our value-added catalogue provides mid-infrared source associations (subject to 6” resolution at 3.4 $\mu$ m) for the radio emission, as identified through visual inspection and thorough checks against the literature. As such, the G4Jy Sample can be used as a reliable training set for cross-identification via machine-learning algorithms. We also estimate the angular size of the sources, based on their associated components at $\sim1\,\text{GHz}$ , and perform a flux density comparison for 67 G4Jy sources that overlap with 3CRR. Analysis of multi-wavelength data, and spectral curvature between 72MHz and 20GHz, will be presented in subsequent papers, and details for accessing all G4Jy overlays are provided at https://github.com/svw26/G4Jy.

Modeling of Electric Current Assisted Sintering: An extended fluid-like approach for the description of powders rheological behavior

A general theoretical framework for investigating the rheological behavior of powder undergoing electric current assisted consolidation is proposed in this work. The most relevant phenomena occurring during ECAS process are taken into account. Consolidating powders are assumed to behave as a viscous fluid. Specifically, powder viscosity is expressed by the Bird–Carreau–Yasuda’s model, which includes the power-law creep description typically adopted in the ECAS modeling literature. The proposed stress-strain constitutive law allows one avoiding a priori assumptions regarding the powder rheological behavior. The resulting system of differential equations cannot be solved analytically, so that the finite-element method (FEM) numerical technique is adopted.A numerical investigation of the effect of some representative model parameters on powder consolidation is performed and the possible different rheological regimes correspondingly taking place are identified. Influence of the applied mechanical load and zero-shear rate viscosity is studied and a suitable approach to map out the powder rheological behavior is presented. It is found that powder rheological response is not only determined by the intrinsic material properties but also depends upon the process operating conditions. As an example, high values of mechanical load along with low values of zero-shear rate viscosity induce a nonlinear viscous flow while Newtonian (linear) flow takes place for low values of mechanical load associated to high values of zero-shear rate viscosity.The proposed model should be considered as a first attempt to extend the rheological theory of sintering recently proposed in the field of electric current assisted consolidation to a broader range of shear rates, which, in turn, extends the application of this theory to a wider variety of materials and operating conditions.

Formula omitted]-Weibull distributions describing commercial service routes

We present an investigation of the mode of road transport in Brazil combining tools of complex networks and real-data. Our analysis involves a data-set based on the service routes inscribed on the Brazilian Transport Agency database. Although connectivity distributions of road networks worldwide are usually claimed as described by a power-law fashion, we report a better fit for the Brazilian case offered by the q-Weibull distribution. In our approach nodes assume the role of localities, whereas links represent service routes among them. Interestingly, a rapid drop takes place on the tail of the data distribution for a particular range of the number of outgoing connections. The mechanism responsible for driving this drop is revealed by investigating the spectral centrality of the network and different patterns of disassortative mixture, for both incoming and outgoing distributions. Besides a discussion about a power law description, we report a contrast with two different distributions. They are interpolated by the q-Weibull one: the Weibull and the q-exponential distributions. Moreover, we study the reciprocity of this network, which reflects the influence of mutual links over dynamical processes. This kind of analysis is indispensable for studies on human mobility, shipping, and a multi-modal perspective.

Diminishing Cognitive Capacities in an Ever Hotter World: Evidence From an Applicable Power-Law Description.

Modeling and evaluating a series of power law descriptions for boundary conditions of undiminished cognitive capacities under thermal stress. Thermal stress degrades cognition, but precisely which components are affected, and to what degree, has yet to be fully determined. With increasing global temperatures, this need is becoming urgent. Power-law distributions have proven their utility in describing differing natural mechanisms, including certain orders of human performance, but never as a rationalization of stress-altered states of attention. From a survey of extant empirical data, absolute thresholds for thermal tolerance for varying forms of cognition were identified. These thresholds were then modeled using a rational power-law description. The implications of the veracity of that description were then identified and analyzed. Cognitive performance thresholds under thermal stress are advanced as power-law relationships, t = f(T) = c[(T - Tref)/Tref]-α. Coherent scaling parameters for diverse cognitive functionalities are specified that are consistent with increases in deep (core) body temperature. Therefore, scale invariance provides a "universal constant," viz, 20% detriment in mental performance per 10% increase in T deviation, from a comfortable reference temperature Tref. We know the thermal range within which humans can survive is quite narrow. The presented power-law descriptions imply that if making correct decisions is critical for our future existence, then our functional thermal limits could be much more restricted than previously thought. We provide our present findings, such that others can both assess and mitigate the effects of adverse thermal loads on cognition, in whatever human scenario they occur.

Ratcheting assessment of 304 steel samples by means of two kinematic hardening rules coupled with isotropic hardening descriptions

Ratcheting response of 304 stainless steel samples was predicted by means of the Ohno–Wang (O–W) and the Ahmadzadeh–Varvani (A–V) kinematic hardening rules in the presence of isotropic hardening rules of Lee and Zavrel (Iso-LZ), Chaboche (Iso-C), and Kang (Iso-K). Expansion and translation of yield surfaces were studied when isotropic hardening rules were adhered to kinematic hardening rules. The backstress increments in the O–W model were developed based on the Armstrong–Fredrick (A–F) model and activated through the power law description of the O–W model. The dynamic recovery term in the A–V model possessed an internal variable to control the variation of backstress, yield surface translation, and plastic strain accumulation over loading process. The inclusion of isotropic hardening models involved an exponential function to achieve a saturated value of yield stress while controlling the rate of evolution through an exponent, β, and parameter Q representing saturated value of isotropic internal variable R. The influence of isotropic models coupled with the kinematic hardening rules of O–W and A–V on ratcheting response of steel samples was discussed.

On the Nuances in the Power Law Description and Interpretation of High Homologous Temperature Creep and Superplasticity Data

“Power law’’ representation is used to describe minimum creep rate and “steady state” superplastic deformation. In creep isothermal log stress – log strain rate relationship is linear for so long as the rate controlling mechanism remains unchanged. During optimal superplastic flow the slope of this curve changes even when there is no change in the rate controlling mechanism, i.e. the stress exponent, n, at a constant temperature and grain size is a function of strain rate. For a constant rate controlling mechanism, in both the phenomena, n decreases with increasing temperature. Grain size has no effect on creep, but its effect is significant in superplasticity. Therefore, analyzing creep and superplasticity data by treating n for any given mechanism as a constant independent of stress and temperature is questionable. In this analysis stress is normalized with respect to a reference stress, rather than the shear modulus. The microstructure dependence comes through the Buckingham Pi theorem. When contribution from microstructure terms to isothermal strain rate is constant, Laurent’s theorem helps generate a set of values for n. It is shown that the simplest solution, viz. n is independent of stress, but is a linear function of temperature, describes steady state creep. (The case n is independent of both stress and temperature follows as a special case.) The second simplest solution, viz. n is a linear function of both temperature and stress corresponds to steady state superplasticity. Using the equations, the values of n, activation energies for the rate controlling processes and strain rates at different temperatures and stresses could be estimated for both creep and superplasticity. The analysis is validated using experimental results concerning many systems. iiThis lecture is dedicated to the sacred memory of late Prof. Oleg D. Sherby.

New constraints on time-dependent variations of fundamental constants using Planck data

Observations of the CMB today allow us to answer detailed questions about the properties of our Universe, targeting both standard and non-standard physics. In this paper, we study the effects of varying fundamental constants (i.e., the fine-structure constant, $\alpha_{\rm EM}$, and electron rest mass, $m_{\rm e}$) around last scattering using the recombination codes CosmoRec and Recfast++. We approach the problem in a pedagogical manner, illustrating the importance of various effects on the free electron fraction, Thomson visibility function and CMB power spectra, highlighting various degeneracies. We demonstrate that the simpler Recfast++ treatment (based on a three-level atom approach) can be used to accurately represent the full computation of CosmoRec. We also include explicit time-dependent variations using a phenomenological power-law description. We reproduce previous Planck 2013 results in our analysis. Assuming constant variations relative to the standard values, we find the improved constraints $\alpha_{\rm EM}/\alpha_{\rm EM,0}=0.9993\pm 0.0025$ (CMB only) and $m_{\rm e}/m_{\rm e,0}= 1.0039 \pm 0.0074$ (including BAO) using Planck 2015 data. For a redshift-dependent variation, $\alpha_{\rm EM}(z)=\alpha_{\rm EM}(z_0)\,[(1+z)/1100]^p$ with $\alpha_{\rm EM}(z_0)\equiv\alpha_{\rm EM,0}$ at $z_0=1100$, we obtain $p=0.0008\pm 0.0025$. Allowing simultaneous variations of $\alpha_{\rm EM}(z_0)$ and $p$ yields $\alpha_{\rm EM}(z_0)/\alpha_{\rm EM,0} = 0.9998\pm 0.0036$ and $p = 0.0006\pm 0.0036$. We also discuss combined limits on $\alpha_{\rm EM}$ and $m_{\rm e}$. Our analysis shows that existing data is not only sensitive to the value of the fundamental constants around recombination but also its first time derivative. This suggests that a wider class of varying fundamental constant models can be probed using the CMB.

Temperature-field phase diagram of geometrically frustrated CePdAl

From the electrical resistivity and ac susceptibility measurements down to $T$ = 40 mK, a rich temperature-field phase diagram has been constructed for the geometrically frustrated heavy-fermion compound CePdAl. In the limit of zero temperature, the field-induced suppression of the antiferromagnetism in this compound results in a sequence of phase transitions/corssovers, rather than a single critical point, in a critical region of field 3$-$5 T. In the lowest temperature region, a power-law description of the temperature-dependent resistivity yields $A$$\cdot$$T^n$ with $n$$>$2 for all applied fields. However, a quadratic-in-temperature resistivity term can be extracted when assuming an additional electron scattering channel from magnon-like excitations, with $A$ diverging upon approaching the critical regime. Our observations suggest CePdAl to be a new playground for exotic physics arising from the competition between Kondo screening and geometrical frustration.

Statistical Distribution Laws for Metallic Mineral Deposit Sizes

Given that expanding industrial societies rely on the continuing supply of metallic raw materials, it is of interest to be able to generate estimates of the likely magnitude of undiscovered metal resources. Statistical predictions of this kind require that one knows the frequency distribution of the metal content (or tonnage) of mineral deposits. The distribution of metal content among deposits of different sizes may be treated empirically (i.e., without explicit formulation of a density function), or formally, starting from a density function that is known to represent the distribution of metal in known deposits. The empirical approach has been applied to the prediction of undiscovered resources of some metals, most notably copper. The focus here is on the formal approach, which may be simpler and more general than the empirical one. It has long been suspected that the distribution of metal tonnages is heavy-tailed, in the sense that a relatively few very large deposits are likely to contain much of the metal endowment. There is, however, uncertainty about the precise nature of the distribution of metal tonnages. Lognormal and power law models have been proposed. Under some conditions, which are examined here, the choice of one or the other of these two models may result in widely diverging estimates of the likely magnitude and distribution of undiscovered resources. However, a detailed statistical analysis of the available deposit tonnage data for 20 metals reveals that it is not possible to decide between the two models. Using conservative significance levels, both models fit the data equally well. Two different procedures are used to attempt to decide whether either of the two models is more plausible: likelihood ratio tests and power law fits to lognormally distributed synthetic data sets that mimic the natural data sets. Neither of the two procedures can detect any preference for one or the other of the two models, for any of the metals discussed here. This uncertainty may arise from the relatively small sizes of the samples available, which are limited to known and reasonably well-characterized metallic mineral deposits. There is at present no statistical justification to choose between a lognormal and a power law description of metal distribution in mineral deposits. This uncertainty should be kept in mind when preparing, and interpreting, estimates of the magnitude of undiscovered mineral resources.