Exponential Mass Research Articles

Unsteady MHD flow past an oscillating vertical plate with periodic temperature variation and exponential mass diffusion embedded in a porous medium with thermal and mass stratification effects

AbstractThe present study investigates the impacts of both mass and thermal stratification on unsteady magnetohydrodynamic flow past a plate swinging vertically on its own axis while entangled in a porous medium with periodic temperature variation and exponential mass diffusion. The technique of Laplace transform for the unitary Prandtl and Schmidt numbers is employed to obtain the closed‐form solution for the nondimensional system of partial differential equations that govern the system for velocity, temperature, and concentration fields. For various physical factors, such as stratification parameters, phase angle, thermal Grashof number, Darcy number, mass Grashof number, and time on velocity, temperature, concentration, skin‐friction, plate heat flux, and mass flux, numerical computations have been performed, and illustrated in graphs. It has been observed that the steady state is attained more swiftly when stratification is applied to the flow. The desire to enhance the knowledge of fluid flow in many technological as well as environmental circumstances, where such conditions are widely used could be the driving force behind this research. Significant results from the thermal and mass stratification are contrasted with the environment where stratification is absent. Understanding flow mechanisms in both naturally occurring and artificially created environments can be enriched by this innovative method.

Influences of thermal and mass stratification on unsteady magnetohydrodynamics parabolic flow along an infinite vertical plate with periodic temperature variation and exponential mass diffusion in porous medium

AbstractThis study explores the dynamics of unsteady magnetohydrodynamics (MHD) parabolic flow along an infinite vertical plate, emphasizing the effects of thermal and mass stratification in a porous medium subjected to periodic temperature variation and exponential mass diffusion. Utilizing the Laplace transform technique to obtain precise solutions, this study effectively integrates the impacts of both thermal and mass stratification without dependence on approximations. The main goal is to assess how thermal and mass stratification impact MHD flow dynamics, temperature, and concentration profiles under varying conditions. The study provides a thorough comparison of these findings with traditional nonstratified scenarios, presenting a comprehensive analysis of fluid behavior under diverse conditions. The conclusions reveal that thermal and mass stratifications considerably diminish velocity and stabilize temperature distributions, which suggests a damping influence on fluid movement and improved management of diffusion processes. Enhanced Grashof numbers improve heat and mass transfer efficiency, while magnetic and Darcy parameters significantly influence flow resistance and heat transfer characteristics. These conditions also result in higher Nusselt and Sherwood numbers, indicating increased efficiency in heat and mass transfer. In contrast, scenarios without stratification display higher velocities and more unstable temperature and concentration profiles. The findings highlight the critical role of stratification in improving fluid dynamics and increasing the efficiency of heat and mass transfer processes, offering valuable insights for engineering and environmental applications in similar conditions. The main novelty of the research is being the first to use the Laplace transform for exact solutions on combined thermal and mass stratification in MHD flows, enhancing prediction accuracy and process control.

Influences of thermal stratification and chemical reaction on MHD free convective flow along an accelerated vertical plate with variable temperature and exponential mass diffusion in a porous medium

AbstractThis study examines the impacts of thermal stratification and chemical reaction on magnetohydrodynamic (MHD) free convective flow along an accelerated vertical plate with variable temperature and exponential mass diffusion, set within a porous medium. Analytical solutions, utilized, are obtained through the Laplace transform technique to accurately represent the flow's physical mechanism. The research employs advanced mathematical models to analyze the intricate interplay between MHD and convective processes under varying thermal and exponential mass diffusion conditions, offering insights into fluid dynamics that closely simulate real‐world conditions. The study draws a significant conclusion by contrasting the effects of thermal stratification with a nonstratified environment. It has been noted that when stratification is applied to the flow, the steady state is achieved more quickly. The study reveals that thermal stratification reduces fluid velocity and temperature but increases skin friction and the Nusselt number, diverging from nonstratified conditions. It also shows that parameters, like, , and significantly influence velocity, temperature, and concentration in fluid dynamics. This research could be driven by a need to enhance the understanding of fluid flow in various engineering and environmental contexts, where such conditions are prevalent, including geothermal energy extraction, thermal management, chemical processing industries, and environmental control technologies. This novel approach enhances understanding of flow processes in both natural and engineered porous environments.

Analytical modeling of acoustic exponential materials and physical mechanism of broadband anti-reflection

Spatially exponential distributions of material properties are ubiquitous in many natural and engineered systems, from the vertical distribution of the atmosphere to acoustic horns and anti-reflective coatings. These media seamlessly interface different impedances, enhancing wave transmission and reducing internal reflections. This work advances traditional transfer matrix theory by integrating analytical solutions for acoustic exponential materials, which possess exponential density and/or bulk modulus, offering a more accurate predictive tool and revealing the physical mechanism of broadband anti-reflection for sound propagation in such non-uniform materials. Leveraging this method, we designed an acoustic dipole array that effectively mimics exponential mass distribution. Through experiments with precisely engineered micro-perforated plates, we demonstrate an ultra-low reflection rate of about 0.86% across a wide frequency range from 420 Hz to 10,000 Hz. Our modified transfer matrix approach underpins the design of exponential materials, and our layering strategy for stacking acoustic dipoles suggests a pathway to more functional gradient acoustic metamaterials.

Understanding the universal dust attenuation scaling relation of star-forming galaxies

ABSTRACT Star-forming galaxies (SFGs) adhere to a surprisingly tight scaling relation of dust attenuation parametrized by the infrared excess (IRX≡ LIR/LUV), being jointly determined by the star formation rate (SFR), galaxy size (Re), metallicity (Z/Z⊙), and axial ratio (b/a). We examine how these galaxy parameters determine the effective dust attenuation and give rise to the universal IRX relation, utilizing a simple two-component star-dust geometry model in which dust in the dense and diffuse interstellar medium (ISM) follows exponential mass density profiles, connected with but not necessarily identical to the stellar mass profiles. Meanwhile, empirical relations are adopted to link galaxy properties, including the gas–star formation relation, the dust-to-stellar size relation, as well as the dust-to-gas ratio versus metallicity relation. By fitting a large sample of local SFGs with the model, we obtain the best-fitting model parameters as a function of metallicity, showing that the two-component geometry model is able to successfully reproduce the dependence of IRX on SFR, Re, b/a at given Z/Z⊙, as well as the dependence of power-law indices on metallicity. Moreover, we also retrieve constraints on the model geometry parameters, including the optical depth of birth clouds (BCs), BC-to-total dust mass fraction, BC covering factor of UV-emitting stars, and star-to-total dust disc radius ratio, which all evolve with galaxy metallicity. Finally, a consistent picture of how the star-dust geometry in SFGs evolves with galaxy metallicity is discussed.

Thermal and Mass Stratification Effects on Unsteady Flow Past an Accelerated Infinite Vertical Plate with Variable Temperature and Exponential Mass Diffusion in Porous Medium

This study looks at how thermal and mass stratification affect the unsteady flow past an infinitely fast-moving vertical plate when the temperature is changing and there is exponential mass diffusion in a porous medium. By applying the Laplace transformation method, we determine the solutions to the equations that govern the system for the case of unitary Prandtl and Schmidt numbers. Graphical representations of the concentration, temperature, and velocity profiles, as well as the Nusselt Number, Sherwood number, and the Skin friction are provided to facilitate discussion of the cause of the different variables. To see the effects of thermal and mass stratification on the fluid flow, we compare the classical solution (Fluid with out stratification) with the primary solution (Fluid with the stratification) by using graph. The combined effects of the two stratification lead to a quicker approach to steady states. The outcomes can be helpful for heat exchange design and other engineering applications.

Analysis of vibrational resonance in an oscillator with exponential mass variation

This study investigated vibrational resonance (VR) in a Duffing-type oscillator with position-dependent mass (PDM) distribution defined by spatially varying exponential function. The role of two PDM parameters, the fixed rest mass m0 and nonlinear strength k on observed resonances was investigated from the analytical and numerical computation of response amplitude Q, which is a measure of the amplification of a low-frequency (LF) signal through the introduction and modulation of a high-frequency (HF) signal in a weakly driven nonlinear system. The method of direct separation of motion was used to analytically compute the response amplitude, while the numerically computed response amplitude was obtained from the Fourier spectrum of the output signal. Single resonance peaks with good agreement between the numerically and the analytically computed responses were observed for the traditional HF-induced VR and the PDM-induced resonances. The results demonstrated that spatial mass perturbation can play the roles of HF signals typically used in traditional VR setups. The results of this investigation corroborate earlier reports that stated PDM parameters can complement the HF signal to control the observed resonance peaks. However, the exponentially varying PDM parameters did not initiate double or multiple resonances as reported for other mass distributions such as the regular mass function and the doubly-singular mass function. This study communicates that the nature of the PDM distribution actually determines the possibility of generating new peaks from observed resonances.

3D stellar motion in the axisymmetric Galactic potential and the e–z resonances

Context. The full phase-space information on the kinematics of a huge number of stars provided by Gaia Data Release 3 increases the demand for a better understanding of the 3D stellar dynamics. Aims. In this paper, we investigate the possible regimes of motion of stars in the axisymmetric approximation of the Galactic potential, applying a 3D observation-based model developed elsewhere. The model consists of three components: the axisymmetric disc, the central spheroidal bulge, and the spherical halo of dark matter. The axisymmetric disc model is divided into thin and thick stellar discs and H I and H2 gaseous disc subcomponents, by combining three Miyamoto-Nagai disc profiles of any model order (1, 2, or 3) for each disc subcomponent, to reproduce a radially exponential mass distribution. The physical and structural parameters of the Galaxy components are adjusted by observational kinematic constraints. Methods. The phase space of the two-degrees-of-freedom model was studied by means of the Poincaré and dynamical mapping, the dynamical spectrum method, and the direct numerical integrations of the Hamiltonian equations of motion. Results. For the chosen physical parameters, the nearly circular (close to the rotation curve) and low-altitude stellar behaviour is composed of two weakly coupled simple oscillations, radial and vertical motions. The amplitudes of the vertical oscillations of these orbits gradually increase with the growing Galactocentric distances, in concordance with the exponential mass decay assumed. However, for increasing planar eccentricities, e, and the altitudes over the equatorial disc, z, new regimes of stellar motion emerge as a result of the beating between the radial and vertical oscillation frequencies, which we refer to as e–z resonances. The corresponding resonant motion produces the characteristic sudden increase or decrease in the amplitude of the vertical oscillation, bifurcations in the dynamical spectra, and the chains of islands of stable motion in the phase space. Conclusions. The results obtained can be useful in understanding and interpreting the features observed in the stellar 3D distribution around the Sun.

Analytical Solution of the Schrödinger Equation with an Exponential Type Mass Depending on the Spatial Variable

In the present work, we proceed to study the Schrödinger equation with dependent mass position. From the resulting partial differential equations, we obtain exact analytical solutions governed by Bessel functions. The exact solution of Schrödinger's equation for a particle with dependent position of the mass (PDM) is a "half-harmonic potential" defined in a Hilbert space. The harmonic oscillator is carried by the wave function ψ(x) through the Bessel function. The magnitude of ψ(x) increases dramatically as the values of the spatial coordinate become larger and larger. This growth is all the more marked as the mass parameter increases. It appears that the wave function has the same asymptotic behavior as the solution of Airy's equation of the second kind. Their oscillations have the same amplitudes near infinity. The mass parameter has a great importance and influence on the mass and also on the wave potential. It is a control parameter in the Schrödinger equations (PDM).

The Sensitivity of Convective Cloud Ensemble Statistics to Horizontal Grid Spacing in Idealized RCE Simulations

Abstract In this work, cloud ensemble statistics are extracted from idealized radiative–convective equilibrium simulations performed at horizontal grid spacings Δ ranging from 2 km to 125 m. At the coarsest resolution, convection remains randomly distributed in space such that the equilibrium statistical mechanics theory proposed by Craig and Cohen in 2006 (CC06; assumes Poisson distributed clouds and exponential mass flux distributions) remains valid. Using classical organization metrics, clustering is already observed at Δ = 1 km, but substantial deviations between the simulated cloud ensemble statistics and CC06 are only observed for grid spacings Δ < 500 m. At these resolutions, the cloud mass flux distributions exhibit heavy tails and cloud counts become overdispersed (higher variance than a Poisson distribution). These changes in ensemble statistics are accompanied by a shift in subcloud organization patterns as well as with the fact that individual cloudy updrafts start to be resolved. Consequently, a horizontal grid spacing no larger than 250 m is recommended, not only to properly resolve the dynamics of individual convective clouds, but also to capture the mesoscale organization of the cloud ensemble. Finally, it is shown that the CC06 theory and our high-resolution results including mesoscale organization may be reconciled if one considers 1) areas smaller than approximately 2 km in size, corresponding roughly to the narrow bands along which clouds develop almost randomly; and 2) individual cloud cores instead of cloud objects, core mass fluxes being shown to generally follow exponential distributions.

Exact solutions of an exponential type position dependent mass problem

The Schrödinger equation under the application of a position-dependent mass (PDM) with an exponential form is presented. Several physical models are carried out by choosing different external potential fields including the free field or a confined hard-all potential, the linear potential plus an attractive centrifugal-like term and harmonic oscillator. The eigenfunction of the first case is given by a Bessel function. The calculations of the second case are found to have the Airy function and we are able to get a general form of energy levels based on the zeros of Airy function of the first kind. The last case is found that the eigenfunctions are given by the popular associated Laguerre function. To provide a better physical insight into the solutions, some figures are plotted graphically.

The physics governing the upper truncation mass of the globular cluster mass function

ABSTRACT The mass function of globular cluster (GC) populations is a fundamental observable that encodes the physical conditions under which these massive stellar clusters formed and evolved. The high-mass end of star cluster mass functions are commonly described using a Schechter function, with an exponential truncation mass Mc, *. For the GC mass functions in the Virgo galaxy cluster, this truncation mass increases with galaxy mass (M*). In this paper, we fit Schechter mass functions to the GCs in the most massive galaxy group ($M_{\mathrm{200}} = 5.14 \times 10^{13} \, {\rm M}_{\odot }$) in the E-MOSAICS simulations. The fiducial cluster formation model in E-MOSAICS reproduces the observed trend of Mc, * with M* for the Virgo cluster. We therefore examine the origin of the relation by fitting Mc, * as a function of galaxy mass, with and without accounting for mass loss by two-body relaxation, tidal shocks and/or dynamical friction. In the absence of these mass-loss mechanisms, the Mc, *-M* relation is flat above $M_* \gt 10^{10}\, {\rm M}_{\odot }$. It is therefore the disruption of high-mass GCs in galaxies with $M_{*}\sim 10^{10} \, {\rm M}_{\odot }$ that lowers the Mc, * in these galaxies. High-mass GCs are able to survive in more massive galaxies, since there are more mergers to facilitate their redistribution to less-dense environments. The Mc, * − M* relation is therefore a consequence of both the formation conditions of massive star clusters and their environmentally dependent disruption mechanisms.

Breaking down chipping and fragmentation in sediment transport: the control of material strength

Abstract. As rocks are transported, they primarily undergo two breakdown mechanisms: fragmentation and chipping. Fragmentation is catastrophic breakup by fracture in the bulk – either by subcritical crack growth under repeated collisions, or from a single high-energy (supercritical) collision – and produces angular shards. Chipping is a distinct low-energy mechanism of impact attrition that involves shallow cracking; this process rounds river pebbles in a universal manner under bed-load transport. Despite its geophysical significance, the transition from chipping to fragmentation is not well studied. Here, we examine this transition experimentally by measuring the shape and mass evolution of concrete particles of varying strength, subject to repeated collisions in a rotating drum. For sufficiently strong particles, chipping occurred and was characterized by the following: attrition products were orders of magnitude smaller than the parent; attrition rate was insensitive to material strength; and particles experienced monotonic rounding toward a spherical shape. As strength decreased, we observed the onset of a subcritical cracking regime associated with fragmentation: mass of attrition products became larger and more varied; attrition rate was inversely proportional to material strength; and shape evolution fluctuated and became non-monotonic. Our results validate conceptual and numerical models for impact attrition: chipping follows “Sternberg's law” of exponential mass loss through time; for fragmentation, the lifetime of particles increases nonlinearly with material strength, consistent with “Basquin's law” of fatigue failure. We suggest that bedrock erosion models must be clarified to incorporate distinct attrition mechanisms, and that pebble or bedrock-channel shape may be utilized to deduce the operative mechanism in a given environment.

Survival and mass growth of cold gas in a turbulent, multiphase medium

ABSTRACT Astrophysical gases are commonly multiphase and highly turbulent. In this work, we investigate the survival and growth of cold gas in such a turbulent, multiphase medium using three-dimensional hydrodynamical simulations. Similar to previous work simulating coherent flow (winds), we find that cold gas survives if the cooling time of the mixed gas is shorter than the Kelvin–Helmholtz time of the cold gas clump (with some weak additional Mach number dependence). However, there are important differences. Near the survival threshold, the long-term evolution is highly stochastic, and subject to the existence of sufficiently large clumps. In a turbulent flow, the cold gas continuously fragments, enhancing its surface area. This leads to exponential mass growth, with a growth time given by the geometric mean of the cooling and the mixing time. The fragmentation process leads to a large number of small droplets which follow a scale-free dN/dm ∝ m−2 mass distribution, and dominate the area covering fraction. Thus, whilst survival depends on the presence of large ‘clouds’, these in turn produce a ‘fog’ of smaller droplets tightly coupled to the hot phase which are probed by absorption line spectroscopy. We show with the aid of Monte Carlo simulations that the simulated mass distribution emerges naturally due to the proportional mass growth and the coagulation of droplets. We discuss the implications of our results for convergence criteria of larger scale simulations and observations of the circumgalactic medium.

Thermal buckling of functionally graded piezomagnetic micro- and nanobeams presenting the flexomagnetic effect

Galerkin weighted residual method (GWRM) is applied and implemented to address the axial stability and bifurcation point of a functionally graded piezomagnetic structure containing flexomagneticity in a thermal environment. The continuum specimen involves an exponential mass distributed in a heterogeneous media with a constant square cross section. The physical neutral plane is investigated to postulate functionally graded material (FGM) close to reality. Mathematical formulations concern the Timoshenko shear deformation theory. Small scale and atomic interactions are shaped as maintained by the nonlocal strain gradient elasticity approach. Since there is no bifurcation point for FGMs, whenever both boundary conditions are rotational and the neutral surface does not match the mid-plane, the clamp configuration is examined only. The fourth-order ordinary differential stability equations will be converted into the sets of algebraic ones utilizing the GWRM whose accuracy was proved before. After that, by simply solving the achieved polynomial constitutive relation, the parametric study can be started due to various predominant and overriding factors. It was found that the flexomagneticity is further visible if the ferric nanobeam is constructed by FGM technology. In addition to this, shear deformations are also efficacious to make the FM detectable.

3D spatial domain gravity inversion with growing multiple polygonal cross-sections and exponential mass density contrast

An automatic 3D spatial domain inversion technique is developed to estimate basement depths of sedimentary basins from observed gravity anomalies using a prescribed exponential mass density contrast. A collage of vertical polygonal cross-sections, each one with unit thickness, in which the density contrast differs exponentially with depth describes the model space. The proposed technique estimates the optimum depth ordinates of the vertices of polygonal cross-sections from a given set of gravity anomalies following predefined convergence criteria. Initial depths to basement interface at plurality of observations are calculated presuming that the density contrast within the Bouguer slab at each observation is also varying exponentially with depth. A previously reported algorithm that make use of both analytic and numeric approaches to compute the gravity response of such 3D model space with exponential mass density contrast is adopted for forward modelling. The proposed inversion is efficient even when the gravity anomalies are available at non-uniform spatial grid intervals. Recovery of basement depths with modest error from a set of gravity anomalies attributable to a synthetic model in the presence of pseudorandom noise and also the fact that the estimated depth structure of the Almazan Basin in NE Spain correlates reasonably well with the information derived from seismic data demonstrates the applicability of the proposed inversion method. The snags associated with other existing density models in the analysis of gravity anomalies are demonstrated on both synthetic and real field anomalies.

Newtonian fractional-dimension gravity and disk galaxies

This paper continues previous work on a novel alternative model of gravity, based on the theory of fractional-dimension spaces applied to Newton's law of gravitation. In particular, our Newtonian Fractional-Dimension Gravity is now applied to axially-symmetric structures, such as thin/thick disk galaxies described by exponential, Kuzmin, or other similar mass distributions. As in the case of spherically-symmetric structures, which was studied in previous work on the subject, we examine a possible connection between our model and Modified Newtonian Dynamics, a leading alternative gravity model, which accounts for the observed properties of galaxies and other astrophysical structures without requiring the dark matter hypothesis. By relating the MOND acceleration constant $a_{0} \simeq 1.2 \times 10^{ -10}\mbox{m}\thinspace \mbox{s}^{ -2}$ to a natural scale length $l_{0}$ of our model, namely $a_{0} \approx GM/l_{0}^{2}$ for a galaxy of mass $M$, and by using the empirical Radial Acceleration Relation, we are able to explain the connection between the observed radial acceleration $g_{obs}$ and the baryonic radial acceleration $g_{bar}$ in terms of a variable local dimension $D$. As an example of this methodology, we provide a detailed rotation curve fitting for the case of the field dwarf spiral galaxy NGC 6503.

Central Himalayan tree-ring isotopes reveal increasing regional heterogeneity and enhancement in ice mass loss since the 1960s

Abstract. Tree-ring δ18O values are a sensitive proxy for regional physical climate, while their δ13C values are a strong predictor of local ecohydrology. Utilizing available ice-core and tree-ring δ18O records from the central Himalaya (CH), we found an increase in east–west climate heterogeneity since the 1960s. Further, δ13C records from transitional western glaciated valleys provide a robust basis for reconstructing about 3 centuries of glacier mass balance (GMB) dynamics. We reconstructed annually resolved GMB since 1743 CE based on regionally dominant tree species of diverse plant functional types. Three major phases became apparent: positive GMB up to the mid-19th century, the middle phase (1870–1960) of slightly negative but stable GMB, and an exponential ice mass loss since the 1960s. Reasons for accelerated mass loss are largely attributed to anthropogenic climate change, including concurrent alterations in atmospheric circulations (weakening of the westerlies and the Arabian Sea branch of the Indian summer monsoon). Multi-decadal isotopic and climate coherency analyses specify an eastward declining influence of the westerlies in the monsoon-dominated CH region. Besides, our study provides a long-term context for recent GMB variability, which is essential for its reliable projection and attribution.

Can Leaf Shape be Represented by the Ratio of Leaf Width to Length? Evidence from Nine Species of Magnolia and Michelia (Magnoliaceae)

Leaf shape is closely related to economics of leaf support and leaf functions, including light interception, water use, and CO2 uptake, so correct quantification of leaf shape is helpful for studies of leaf structure/function relationships. There are some extant indices for quantifying leaf shape, including the leaf width/length ratio (W/L), leaf shape fractal dimension (FD), leaf dissection index, leaf roundness index, standardized bilateral symmetrical index, etc. W/L ratio is the simplest to calculate, and recent studies have shown the importance of the W/L ratio in explaining the scaling exponent of leaf dry mass vs. leaf surface area and that of leaf surface area vs. leaf length. Nevertheless, whether the W/L ratio could reflect sufficient geometrical information of leaf shape has been not tested. The FD might be the most accurate measure for the complexity of leaf shape because it can characterize the extent of the self-similarity and other planar geometrical features of leaf shape. However, it is unknown how strongly different indices of leaf shape complexity correlate with each other, especially whether W/L ratio and FD are highly correlated. In this study, the leaves of nine Magnoliaceae species (>140 leaves for each species) were chosen for the study. We calculated the FD value for each leaf using the box-counting approach, and measured leaf fresh mass, surface area, perimeter, length, and width. We found that FD is significantly correlated to the W/L ratio and leaf length. However, the correlation between FD and the W/L ratio was far stronger than that between FD and leaf length for each of the nine species. There were no strong correlations between FD and other leaf characteristics, including leaf area, ratio of leaf perimeter to area, fresh mass, ratio of leaf fresh mass to area, and leaf roundness index. Given the strong correlation between FD and W/L, we suggest that the simpler index, W/L ratio, can provide sufficient information of leaf shape for similarly-shaped leaves. Future studies are needed to characterize the relationships among FD and W/L in leaves with strongly varying shape, e.g., in highly dissected leaves.

Chirality production with mass effects — Schwinger pair production and the axial Ward identity

The anomalous generation of chirality with mass effects via the axial Ward identity and its dependence on the Schwinger mechanism is reviewed, utilizing parity violating homogeneous electromagnetic background fields. The role vacuum asymptotic states play on the interpretation of expectation values is examined. It is discussed that observables calculated with an in–out scattering matrix element predict a scenario under Euclidean equilibrium. A notable ramification of which is a vanishing of the chiral anomaly. In contrast, it is discussed observables calculated under an in–in, or real-time, formalism predict a scenario out-of equilibrium, and capture effects of mean produced particle–antiparticle pairs due to the Schwinger mechanism. The out-of equilibrium chiral anomaly is supplemented with exponential quadratic mass suppression as anticipated for the Schwinger mechanism. Similar behavior in and out-of equilibrium is reviewed for applications including the chiral magnetic effect and chiral condensate.