Lognormal Function Research Articles

A scale-invariant log-normal droplet size distribution below the critical concentration for protein phase separation

Many proteins have been recently shown to undergo a process of phase separation that leads to the formation of biomolecular condensates. Intriguingly, it has been observed that some of these proteins form dense droplets of sizeable dimensions already below the critical concentration, which is the concentration at which phase separation occurs. To understand this phenomenon, which is not readily compatible with classical nucleation theory, we investigated the properties of the droplet size distributions as a function of protein concentration. We found that these distributions can be described by a scale-invariant log-normal function with an average that increases progressively as the concentration approaches the critical concentration from below. The results of this scaling analysis suggest the existence of a universal behaviour independent of the sequences and structures of the proteins undergoing phase separation. While we refrain from proposing a theoretical model here, we suggest that any model of protein phase separation should predict the scaling exponents that we reported here from the fitting of experimental measurements of droplet size distributions. Furthermore, based on these observations, we show that it is possible to use the scale invariance to estimate the critical concentration for protein phase separation.

Fragility of Highway Embankments Exposed to Permanent Deformations Due to Underlying Fault Rupture

Seismic risk expresses the expected degree of damage and loss following a catastrophic event. An efficient tool for assessing the seismic risk of embankments is fragility curves. This research investigates the influence of embankment’s geometry, the depth of rupture occurrence, and the underlying sandy soil’s conditions on the embankment’s fragility. To achieve this, the response of three highway embankments resting on sandy soil was examined through quasi-static parametric numerical analyses. For the establishment of fragility curves, a cumulative lognormal probability distribution function was used. The maximum vertical displacement of the embankments’ external surface and the fault displacement were considered as the damage indicator and the intensity measure, respectively. Damage levels were categorized into three qualitative thresholds: minor, moderate, and extensive. All fragility curves were generated for normal and reverse faults, as well as the combination of those fault types (dip-slip fault). Finally, the proposed curves were verified via their comparison with those provided by HAZUS. It was concluded that embankment geometry and depth of fault rupture appearance do not significantly affect fragility, as exceedance probabilities show minimal differences (<4%). However, an embankment founded on dense sandy soil reveals slightly higher fragility compared to the one founded on loose sand. Differences regarding the probability of exceedance of a certain damage level are restricted by a maximum of 7%.

Novel Mo: CoFe2O4 nanoparticles combustion synthesis for opto-magneto-electrochemical applications: A systematic analysis

In current work, various concentrations (0.0 wt%, 0.10 wt%, 0.25 wt%, 0.50 %,0.75 wt%, and 1 wt%) of Molybdenum (Mo) - doped cobalt ferrite (CFO) nanoparticles (Mo:CFO NPs) were synthesized using the flash combustion approach. The structural analysis of the prepared Mo:CFO was examined by the XRD patterns, and the obtained crystallite size 48.64, 46.72, 22.81, 21.05, 18.03, and 19.31 nm for 0.0 wt%, 0.10 wt%, 0.25 wt%, 0.50 wt%, 0.75 wt%, and 1 wt% Mo:CFO NPs, respectively. The presence of stoichiometry and homogeneity of the prepared Mo:CFO NPs was confirmed by the EDX analysis. The five phonon modes of the prepared Mo:CFO NPs were recorded by FT-Raman spectra, and the phonon modes were observed around 220, 312, 479, 624, and 685 cm−1 that corresponded o T2g(2), Eg, T2g(1), A1g(2), and A1g(1) symmetries, respectively. The grain sizes of the pure CFO and Mo:CFO NPs were evaluated using the images of scanning electron microscopy (SEM) and obtained in the range of 39–61 nm, respectively. The presence of valence states Co (2p), Fe (2p), O (1 s), and Mo (3d) in the prepared 1 wt% Mo:CFO NPs were examined by XPS spectra. The particle sizes ∼26.4 nm and ∼ 16.7 nm were obtained for pure CFO and 1 wt% Mo:CFO NPs using lognormal function fitting. The emission peaks at 445 ± 3, 521 ± 3, and 620 ± 2 nm in the PL spectra were observed by PL spectroscopy. The decrease in saturation magnetization Ms. (70.80–66.54 emu/g) and reduced remanent magnetization Mr. (24.22–18.64 emu/g) of prepared Mo: CFO NPs was observed in the MH analysis by SQUID analysis. The electrochemical study of Mo: CFO NPs (0.0 %, 0.25 %, 0.50 %, and 1.0 %) was done in a three-electrode assembly cell. The capacitance of values 650.0 Fg−1, 800.0 Fg-1, and 810.0 Fg−1 for pure CFO, 0.25 % Mo: CFO, and 0.50 % Mo: CFO were recorded in electrochemical analysis. The highest capacitance of 840.0 Fg−1 was observed for the electrode with 1.0 % Mo: CFO NPs. It was analyzed that the increase in CFO electrodes enhances their performance, and therefore, it can be utilized for multifunctional devices.

Assessment of arteriosclerosis based on lognormal fitting

Objective. Pulse pressure waves contain information about human physiology. There is a need for a simple, accurate way to know cardiovascular health in the clinic, so as to realize the implementation of convenient and effective early health monitoring for patients with arteriosclerosis.Approach. This study proposes an arteriosclerosis assessment method based on fitting a lognormal function, along with improving a conventional electronic sphygmomanometer. During the deflation phase of blood pressure measurement, the cuff pressure was kept constant (40 mmHg) and an additional 10 s of pulse signal was acquired. To derive the pulse pressure waveforms for a single cycle, the acquired pulse data of 101 cases were preprocessed in this study, including filtering for noise removal, onset point identification, removal of baseline drift, and normalization. In this study, an improved pulse resolution algorithm is proposed for the multimodal problem of the pulse wave, combining waveform matching and threshold setting, and finally obtaining the resolution parameters of the lognormal function with an average error less than 1.5%.Main results. According to the correlation analysis, the resolved parametersA1,W2,C2,W3, andC3were significantly correlated with brachial-ankle Pulse Wave Velocity, and the absolute correlation range in 0.17-0.53, which can be used as a reference index for arteriosclerosis. An arteriosclerosis assessment model was constructed based on the support vector mechanism, and the prediction accuracy was 91.1%.Significance. This study provides a new solution idea for the arteriosclerosis assessment method as well as the pulse resolution algorithm, which has a greater reference value.

Optimal solution of multiobjective stable environmental economic power dispatch problem considering probabilistic wind and solar PV generation

The Environmental Economic Power Dispatch (EEPD) problem, a widely studied bi-objective nonlinear optimization challenge in power systems, traditionally focuses on the economic dispatch of thermal generators without considering network security constraints. However, environmental sustainability necessitates reducing emissions and increasing the penetration of renewable energy sources (RES) into the electrical grid. The integration of high levels of RES, such as wind and solar PV, introduces stability issues due to their uncertain and intermittent nature. This paper addresses these concerns by formulating and solving the Stable Environmental Economic Power Dispatch (SEEPD) problem, which includes fixed zonal reserve capacity from conventional thermal generators and uncertain reserves from RES. Uncertainties in RES and load demand are modeled using random variable generation techniques, applying Gaussian, Weibull, and log-normal probability density functions (PDFs) for load demand, wind velocity, and solar irradiance, respectively. The stochastic SEEPD problem extends to multiple periods by replicating the single-period problem for each interval in the planning horizon, linking periods through intertemporal ramping costs, physical ramp rate, and fixed zonal reserve constraints on dispatch variables. Multi-Objective Evolutionary Algorithms (MOEAs) have gained importance for solving complex nonlinear problems involving multi-objective functions. This paper applies the latest MOEAs to tackle the proposed SEEPD problem, incorporating stochastic wind and solar PV power sources. Network security constraints, such as transmission line capacities and bus voltage limits, are considered along with constraints on generator capabilities and intertemporal spinning reserves, ramp-up and ramp-down constraints for thermal generators. A bidirectional coevolutionary-based multi-objective evolutionary algorithm is employed, integrating an advanced constraint-handling technique to ensure compliance with system constraints. The simulation results show that the proposed formulation achieves a better trade-off between various conflicting objective functions compared to other state-of-the-art MOEAs.

Dispersion and rotation measures from fast radio burst (FRB) host galaxies based on the TNG50 simulation

Context. Fast radio bursts (FRBs) are poised to become important cosmological tools in the near future, as the number of observed FRBs is increasing rapidly with multiple surveys underway. A large sample of FRBs will soon have available dispersion measures (DMs) and rotation measures (RMs), which can be used to study the cosmic baryon density and the intergalactic magnetic field. However, the observed DM and RM of FRBs consists of multiple contributions that must be quantified to estimate the DM and RM of the intergalactic medium (IGM). Aims. In this paper, we estimate one such contribution to DM and RM, namely, of FRB host galaxies. We show how this contribution changes with redshift, galaxy type, and the stellar mass of the galaxies. We also investigate its dependence on galaxy inclination and on an FRB’s offset from the center of the galaxy. Methods. Using the TNG50 simulation of the IllustrisTNG project, we selected 16 500 galaxies at redshifts of 0≤ ɀ ≤2, with stellar masses in the range of 9 ≤ log(M*/M⊙) ≤ 12. In each galaxy, we calculated the DM and RM contributions of 1000 sightlines; from these, we constructed the DM and RM probability density functions (PDFs). Results. We find that the rest frame DM distributions of all galaxies at a given redshift can be fitted by a log normal function and its median and width increase as a function of redshift. The rest-frame RM distribution is symmetric, with a median RMhost,rf=0 rad m–2 and it can be fitted by a combination of a Lorentzian and two Gaussian functions. The redshift evolution of the distribution width can be fitted by a curved power law. The parameters of these functions change for different subsets of galaxies with different stellar mass, inclination, and FRB offset. These changes are due to an increasing ne with redshift, SFR, and stellar mass. We do find a more ordered B field at lower ɀ compared to higher ɀ, as suggested by the presence of more galaxies with B field reversals and B fields dominated by random B field at higher ɀ. Conclusions. We estimated the FRB host DM and RM contributions, which can be used in the future to isolate the IGM contribution from the observed DM and RM of FRBs. We predict that to constrain a σRM,IGM of 2 rad m–2 to the 95% confidence level, we would need to observe 95 000 FRBs at ɀ = 0.5, but only 9 500 FRBs at ɀ = 2.

The JWST/NIRISS Deep Spectroscopic Survey for Young Brown Dwarfs and Free-floating Planets

The discovery and characterization of free-floating planetary-mass objects (FFPMOs) is fundamental to our understanding of star and planet formation. Here we report results from an extremely deep spectroscopic survey of the young star cluster NGC1333 using Near-InfraRed Imager and Slitless Spectrograph (NIRISS) wide field slitless spectroscopy on the James Webb Space Telescope. The survey is photometrically complete to K ∼ 21, and includes useful spectra for objects as faint as K ∼ 20.5. The observations cover 19 known brown dwarfs, for most of which we confirm spectral types using NIRISS spectra. We discover six new candidates with L-dwarf spectral types that are plausible planetary-mass members of NGC1333, with estimated masses between 5 and 15 M Jup. One, at ∼5 M Jup, shows clear infrared excess emission and is a good candidate to be the lowest-mass object known to have a disk. We do not find any objects later than mid-L spectral type (M ≲ 4 M Jup). The paucity of Jupiter-mass objects, despite the survey’s unprecedented sensitivity, suggests that our observations reach the lowest-mass objects that formed like stars in NGC1333. Our findings put the fraction of FFPMOs in NGC1333 at ∼10% of the number of cluster members, significantly more than expected from the typical log-normal stellar mass function. We also search for wide binaries in our images and report a young brown dwarf with a planetary-mass companion.

Lognormal mode dissociation method based on intrinsic characteristics of aerosol size distribution.

Aerosols significantly affect the transmission of optical signals in the atmosphere, necessitating accurate atmospheric models for the performance evaluation of electro-optic devices. Aerosol size distribution is a critical parameter in these models, and the lognormal function is commonly used to mathematically represent it. This study aims to handle the lack of a solid criterion for determining the number of lognormal modes and introduces an improved scheme that leverages the characteristics of the second derivative (SD) of the Gaussian curve to identify the mode amount and to initialize mode parameters for fitting. The optimization problem is solved using a genetic algorithm, incorporating a goodness-of-fit index to determine the presence of spurious modes. For aerosol size distributions characterized by a single Gaussian peak, mode parameters such as mode radius and width can be straightforwardly identified through the positions of peaks and roots on the SD curve. However, the original mode dissociation method may overlook potential modes in distributions composed of superimposed Gaussian peaks. Numerical tests indicate that such oversights can result in substantial errors in calculating the aerosol extinction coefficient, with relative errors exceeding 100%. The proposed scheme significantly enhances the accuracy of mode dissociation in aerosol size distribution, reducing errors in aerosol extinction coefficient calculations by approximately 40% when applied to data from the Aerosol Robotic Network (AERONET). An additional benefit of this method is its ability to constrain the number of lognormal modes in an aerosol size distribution. Results from applying this scheme to data from selected AERONET sites reveal that over half of the size distributions consist of more than two lognormal modes, highlighting the effectiveness of the proposed approach in capturing complex aerosol behaviors.

Improved partitioning between matrix and macropore flow: Novel bimodal lognormal functions for water retention and hydraulic conductivity in pumice and non-pumice soils

Dual-porosity models have been shown to improve models of soil–water movement and enhance the water balance of structured soils. In this study we introduce novel, continuous, closed-form, bimodal, lognormal functions for soil–water retention, θ(ψ), and unsaturated hydraulic conductivity, K(ψ), enhancing traditional models and significantly improving predictions, particularly for pumice soils. These functions incorporate the thresholds for (a) water pressure, (b) soil water content, and (c) unsaturated conductivity, which accurately differentiates macropores from matrix pores. Validation using 313 observation points from laboratory data shows an increase in the Nash–Sutcliffe efficiency coefficient of θ(ψ) from 0.94 to 0.97, and for K(ψ) from 0.83 to 0.95. The model also requires six constant, semi-empirical tortuosity parameters and provides a physically constrained approach for the hydraulic parameters that reduces non-uniqueness risks. The derived functions yield improved predictions and enable the computation of macropore soil water content and flow contributions, with potential applications for the advancement of preferential flow modelling.

Long-time behavior and quasi-density function for a stochastic epidemic model with relapse and reinfection

A stochastic susceptible–infected–recovered–infected (SIRI) epidemic model with relapse and reinfection is established in this paper. First, we prove that the solution to the epidemic model is unique and globally positive. Next, we determine some sufficient conditions for the extinction of the disease when [Formula: see text] and for the persistence in mean in the case of [Formula: see text]. Furthermore, we prove the existence of at least one ergodic stationary distribution of the stochastic model if [Formula: see text]. Additionally, by solving the corresponding three-dimensional Fokker–Planck equation, it is theoretically shown that the epidemic model has a log-normal probability density function when [Formula: see text], then we obtain the exact expression of density function of the stationary distribution. Finally, we give some numerical simulations to support our theoretical results.

Stationary distribution and probability density function of a stochastic COVID-19 infections model with general incidence

With the sudden outbreak of COVID-19 all over the world, vaccination was one of the most effective treatments. In this paper, a stochastic COVID-19 model with vaccination is developed. We first prove that the stochastic model has a unique global solution by constructing the corresponding Lyapunov function. Then, sufficient condition for ergodic stationary distribution υ(⋅) is provided under the condition R0s>1. Moreover, by applying the six-dimensional Fokker–Planck equation, we obtain the explicit expression of the probability density function near quasi-local equilibrium Q∗, which has the form of lognormal density function. Finally, numerical simulations are presented to verify the above conclusions.

Dynamics and probability density function of a stochastic COVID-19 epidemic model with nonlinear incidence

In this paper, a stochastic SEIQ model with nonlinear incidence rate is proposed. The existence and uniqueness of global positive solution of the stochastic model are proved. Then, by constructing some Lyapunov functions, we derive a sufficient condition for the ergodic stationary distribution when [Formula: see text] is greater than one. By solving a four-dimensional Fokker–Planck equation, we get the exact expression of log-normal probability density function of stationary distribution for the stochastic model. Sufficient condition for the extinction of the exposed and infected population is also provided. Finally, numerical simulations are presented to verify the above theoretical results.

Log-Normal Waiting Time Widths Characterize Dynamics

Many astronomical phenomena, including Fast Radio Bursts and Soft Gamma Repeaters, consist of brief distinct aperiodic events. The intervals between these events vary randomly, but there are periods of greater activity, with shorter mean intervals, and of lesser activity, with longer mean intervals. A single dimensionless parameter, the width of a log-normal function fitted to the distribution of waiting times between events, quantifies the variability of the activity. This parameter describes its dynamics in analogy to the critical exponents and universality classes of renormalization group theory. If the distribution of event strengths is a power law, the width of the log-normal fit is independent of the detection threshold and is a robust measure of the dynamics of the phenomenon.

Log-Gaussian Distribution and Evolution of Minimum Variation Timescales for SGR J1550-5418

The major observation of the X-ray burst (XRB) in association with the fast radio burst (FRB) FRB 200428 originating from SGR J1935+2154 on 2020 April 28 substantiates magnetars as one of the origins of FRBs. However, it is still unclear why only a few XRBs in association with FRBs have been observed solely in this magnetar. In this study, we investigate the minimum variation timescales (MVTs) of XRBs from SGR J1550-5418, an active magnetar similar to SGR J1935+2154, and conduct a comprehensive comparison of their MVT properties. Our analysis reveals that the distribution of the MVT for SGR J1550-5418 follows a lognormal Gaussian function with the mean 6.4 ± 0.4 ms. Interestingly, we detect an evolution in the MVTs, i.e., the fitted mean of the MVT distribution in 2008 October is about 3 times larger than that in 2009, indicating potential variations in radiation regions possibly attributed to changes in magnetic field topology. Besides, we do not find a significant correlation between the MVT and either duration or energy. Finally, there is no significant difference between the MVT distributions across different energy ranges for SGR J1550-5418 and SGR J1935+2154, which supports a common physical origin.

Randomness of optical turbulence generated by Rayleigh–Bénard convection using intensity statistics

The experimental study of optical turbulence proves difficult due to challenges in generating controllable conditions in a laboratory environment. Confined water tanks that produce Rayleigh–Bénard (RB) convection are one method to generate optical turbulence using a controllable temperature gradient. It is of utmost concern to quantify the properties of the optical turbulence generated for characterization of other optical applications such as imaging, sensing, or communications. In this experimental study a Gaussian beam is propagated through a RB water tank where two intensity measurements are made at the receiver’s pupil and focal plane. The pupil and focal plane results include quantification of the intensity fluctuation distribution, scintillation distribution, and refractive index structure constant at various values of the temperature gradient. The angle of arrival fluctuations is also calculated at the focal plane to obtain a second estimate of C n 2. The pupil plane estimate for C n 2 using scintillation index and focal plane angle of arrival fluctuations is compared to preliminary predictions of C n 2 as a function of RB temperature gradient showing C n 2∼ΔT4/3. The outcomes of the study confirm that the RB process produces intensity fluctuations that follow gamma–gamma and log-normal probability density functions. Estimates of the refractive index structure constant C n 2 produce the same trends with different magnitudes when measured from the pupil and focal plane.

A novel derivative search political optimization algorithm for multi-area economic dispatch incorporating renewable energy

Multi-area economic dispatch (MAED) incorporating renewable energy has become an important issue in the power system optimization. Existing intelligent optimization algorithms often suffer from poor solution accuracy or slow convergence when dealing with MAED problems. In this paper, a novel derivative search-based political optimization (DSPO) algorithm is proposed to handle the MAED problem incorporating renewable energy including wind and solar energy. In the renewable energy modeling, the Weibull and log-normal probability density functions are used to calculate available wind and solar power respectively. In order to improve the search performance, DSPO adopts two strategies: leader guide strategy and derivative search mechanism. The former adds the leader’s global optimal information which can direct candidate solutions to more promising regions and speed up convergence. The latter derives neighborhood solutions around some high-quality solutions to improve the exploitation ability. The DSPO algorithm is applied to solve four MAED problems which take into account valve point effect, prohibited operating zone, power loss and so on. The simulation results show that the DSPO algorithm achieves the overall best results in terms of convergence speed, solution accuracy and stability when compared with several well-established algorithms.

Measurement of acceleration and angular velocity of particles during a 3D silo discharge

Accurate measurement of the three-dimensional (3D) movement of discrete particles is crucial for comprehending complex granular rheology in silos. In this paper, the acceleration and angular velocity of particles in 3D silos are measured by using a spherical detector based on inertial technology and magnetic positioning technology. The acceleration of particles is the largest in the center of silos, which suggest that the resistance generated by friction and extrusion is the smallest. Surprisingly, the angular velocity distribution follows lognormal function except for particles near the outlet. The correlation between acceleration and angular velocity is opposite in different flow regions. It reveals for the first time that the extent to which the resultant force on the particles affects their rotational motion is related to the flow pattern. These results have practical significance for regulating the granular flow pattern and optimizing the structural design of silos.

Multistep kinetics in self-induced sol–gel process of amorphous anhydride formation during thermal dehydration of potassium tetraborate tetrahydrate

AbstractThe physico-geometrical mechanism and kinetics of the multistep thermal dehydration of potassium tetraborate tetrahydrate was investigated as a model reaction to produce amorphous anhydrate via the self-induced sol–gel process. The thermal dehydration is composed of three consecutive dehydration steps: (1) a surface reaction in the solid-state accompanied by crack formation; (2) a rapid mass loss process accompanied by liquefaction to form the aggregate of the gel powders; and (3) the dehydration of gel powder aggregate to form a glassy anhydride. The changes in the contribution of the individual dehydration steps to the overall process according to the sample particle size and the heating rate (β) were identified as specific features of the multistep dehydration, which was characterized quantitatively using mathematical deconvolution analysis with log-normal four-parameter functions. The difficulty in determining the apparent activation energy (Ea) of the individual dehydration steps using isoconversional analysis due to the changes in the contribution depending on β values was addressed using modulated temperature thermogravimetry. Using the contributions and the apparent Ea of the individual dehydration steps as initial values, the kinetic description of the multistep thermal dehydration was refined through a kinetic deconvolution analysis using the cumulative kinetic equation. As a result, the individual dehydration steps were kinetically characterized as: (1) a surface reaction described by the first-order rate law with Ea,1 of approximately 68 kJ mol−1; (2) a reaction accompanied by liquefaction controlled by an autocatalytic rate behavior with Ea,2 of approximately 123 kJ mol−1; (3) a diffusion-controlled dehydration of gel powder with Ea,3 of approximately 82 kJ mol−1.

Assessing groundwater drought vulnerability through baseflow separation and index-based analysis under climate change projections

Groundwater is an essential resource, providing drinking water to millions of people globally and supporting agriculture, industry, and ecosystems. It acts as a drought buffer and provides a stable source of water during dry periods. However, groundwater is often underestimated and overexploited, leading to aquifer depletion and other negative impacts. Groundwater is invisible and difficult to measure, which makes it challenging for sustainable management. This study employed the Soil & Water Assessment Tool (SWAT) hydrological model to simulate streamflow in an eastern Afghan watershed using satellite-based and reanalysis data. Variables from five Coupled model intercomparison project phase 6 (CMIP6) general circulation models (GCMs), including minimum and maximum temperature and precipitation, were bias-corrected (downscaled) using Quantile Mapping (QM). The results from these five GCMs were then compared based on statistical criteria to determine the most reliable model, and the selected model was used to derive future data (i.e., 2021 to 2050) under different climate scenarios after applying QM. Using the future climate data and the developed SWAT model, streamflow values were projected for future periods. Considering that there are various methods to estimate groundwater recharge and the most commonly used technique involves baseflow separation procedures, this study used seven baseflow separation methods to estimate groundwater recharge values based on streamflow time series from the baseline period and different climate scenarios. To evaluate the groundwater drought characteristics, a recently proposed index called the Groundwater Recharge Drought Index (GRDI) and the run theory method were used. Results indicated that the SWAT model performed well in simulating and predicting streamflow. The sixth version of the Model for Interdisciplinary Research on Climate (MIROC6) provided the highest accuracy compared to other climate models. Moreover, it was observed that the baseflow separation methods significantly affected the estimation of groundwater recharge, thus the obtained results from all developed methods were considered. The log-normal function was found to provide the best correlation for estimating GRDI. The findings revealed a gradual decrease in drought duration in the coming years. However, the magnitude and severity of droughts are projected to increase, especially under pessimistic climate scenarios.

Визначення крупності продуктів подрібнення

Mathematical approaches to particle size determination in closed grinding cycles are considered. The features of average particle size calculation for different fractions with account for the grinding kinetics are shown. Particle size calculation algorithms for the entire fraction range are proposed. Particular attention is paid to output determination for fractions of arbitrarily small particles. A particle size determination method based on a lognormal distribution function is shown. In choosing the mathematical approach, the process requirements are taken into account. The basis of in-flow noncontact particle size control is the acoustic monitoring of the process and the established relationships between the particle size and the acoustic characteristics. The signal amplitude during material transportation in the energy carrier flow and jet grinding was found as a function of the particle size and grinding conditions. In order to determine the fractional composition of a mixture, the frequency characteristics of acoustic signals and their variation during the transportation of narrow fractions and mixtures were considered. The analysis of the amplitude-frequency characteristics of acoustic signals during the compressed-air transportation of narrow fractions in the jet mill channels confirmed the presence of signals with frequencies characteristic for each fraction. These frequencies were experimentally related to the particle size of a fraction in a mixture. These studies form a basis for a noncontact method of determining the particle size distribution of a material in an air flow, in particular in jet grinding. The results may be used for engineering and technological calculations in mineral dressing and the development of process equipment for the chemical industry, construction, mining, and metallurgy.