Equipartition Research Articles

Non-Hermitian non-equipartition theory for trapped particles

The equipartition theorem is an elegant cornerstone theory of thermal and statistical physics. However, it fails to address some contemporary problems, such as those associated with optical and acoustic trapping, due to the non-Hermitian nature of the external wave-induced force. We use stochastic calculus to solve the Langevin equation and thereby analytically generalize the equipartition theorem to a theory that we denote the non-Hermitian non-equipartition theory. We use the non-Hermitian non-equipartition theory to calculate the relevant statistics, which reveal that the averaged kinetic and potential energies are no longer equal to kBT/2 and are not equipartitioned. As examples, we apply non-Hermitian non-equipartition theory to derive the connection between the non-Hermitian trapping force and particle statistics, whereby measurement of the latter can determine the former. Furthermore, we apply a non-Hermitian force to convert a saddle potential into a stable potential, leading to a different type of stable state.

Modeling of Radio Supernovae: Including the Effects of Inhomogeneities and Radiative Cooling

The presence of inhomogeneities in a spatially unresolved source is often hard to establish. This limits the accuracy with which the source properties can be determined. It is shown how observed features not expected for a homogeneous model can be used to infer the properties of the inhomogeneities in radio supernovae. Furthermore, the observed consequences of radiative cooling can be seriously affected by inhomogeneities. It is shown that the deduced source properties are very sensitive to the observed value of the cooling frequency; even a lower limit is often useful to constrain its characteristics. It is argued that the main synchrotron emission region in SN 2003L has a small volume filling factor, possibly as low as a few percent. On the contrary, deviations from homogeneity are substantially smaller in SN 2002ap. The observed properties of Type Ib/c radio supernovae in general indicate the volume filling factor to remain rather constant with time for individual sources, but those peaking later at radio frequencies have lower filling factors. The conditions in the main synchrotron component in both SN 2003L and SN 2002ap are consistent with equipartition of energy between relativistic electrons and magnetic fields.

Event-based analysis of convergence to energy equipartition

The paper presents a simple yet rigorous analysis of the transient of the spatial distribution of the kinetic energy of a classical ideal monoatomic dilute gas that passes from non-equilibrium to equilibrium. The proposed approach is event-based, in the sense that the evolution of the system is analyzed as a function of the random number of collisions in a given time interval. Taking a very simple yet realistic model of collision, the paper shows that collisions between atoms lead exponentially to energy equipartition.

Potential vorticity conservation for plasma turbulence in an inhomogeneous magnetic field: Theory and implications

The concept and theory of potential vorticity in drift wave turbulence are extended to the case of an inhomogeneous magnetic field. A one-field magnetic potential vorticity conserving equation is derived via the use of conservative gyrokinetics. The similarity between the corresponding systems for drift wave turbulence and shallow water theory is discussed in detail. Zonal flow physics in an inhomogeneous magnetic field is discussed. In particular, a Charney–Drazin type nonacceleration theorem is derived from the novel system, which conserves magnetic potential vorticity. Extensions of the turbulent equipartition theory to the transport of magnetic potential vorticity are proposed.

Modeling the Multiwavelength Spectral Energy Distributions of the Fermi-4LAC Bright Flat-spectrum Radio Quasars

In this paper, we present a long-term multiwavelength investigation focusing on 12 distinct samples of Fermi-4LAC bright flat-spectrum radio quasars (FSRQs). Detailed variability and spectral analyses of γ-ray, X-ray, and ultraviolet/optical data obtained by the Fermi Large Area Telescope, the Swift X-ray Telescope, and the Swift Ultraviolet and Optical Telescope were performed over a period of about 14 yr, spanning from 2008 October to 2022 October. These analyses provide insights into characterizing the variations within different activity states. To efficiently reproduce the multiwavelength simultaneous/quasi-simultaneous spectral energy distributions (SEDs) of the samples, we propose a novel approach for constraining the model parameters. By analyzing the parameters of the energy spectral curvature (β), the peak frequency (ν pk), the peak luminosity (L pk), the Compton dominance parameter (A C), and the variability timescale (t var) in different activity states, we can estimate the values of the jet radiation region parameters for the samples. Subsequently, we utilize the synchrotron-self-Compton and external Compton processes, employing a logarithmic-parabolic spectral shape to approximate the observed spectra of the sample sources, while considering the induced regime for the physical parameters. The model results show that: (1) by effectively reproducing SEDs in various active states of bright FSRQs, the parameters within the emission region were reasonably constrained; (2) compared to other active states, the emission region of the jet exhibits a reduced radius during the high state, while the magnetic field strength increases during the low state; and (3) for bright FSRQs in a high-activity state, there is an enhancement of the Doppler factor, often exhibiting a tendency toward energy equipartition.

The 3D Kinematics of the Orion Nebula Cluster. II. Mass-dependent Kinematics of the Inner Cluster

We present the kinematic analysis of 246 stars within 4′ from the center of Orion Nebula Cluster (ONC), the closest massive star cluster with active star formation across the full mass range, which provides valuable insights in the formation and evolution of star cluster on an individual-star basis. High-precision radial velocities and surface temperatures are retrieved from spectra acquired by the NIRSPEC instrument used with adaptive optics (NIRSPAO) on the Keck II 10 m telescope. A 3D kinematic map is then constructed by combining with the proper motions previously measured by the Hubble Space Telescope Advanced Camera for Surveys/WFPC2/WFC3IR and Keck II NIRC2. The measured root-mean-squared velocity dispersion is 2.26 ± 0.08 km s−1, significantly higher than the virial equilibrium’s requirement of 1.73 km s−1, suggesting that the ONC core is supervirial, consistent with previous findings. Energy equipartition is not detected in the cluster. Most notably, the velocity of each star relative to its neighbors is found to be negatively correlated with stellar mass. Low-mass stars moving faster than their surrounding stars in a supervirial cluster suggests that the initial masses of forming stars may be related to their initial kinematic states. Additionally, a clockwise rotation preference is detected. A weak sign of inverse mass segregation is also identified among stars excluding the Trapezium stars, although it could be a sample bias. Finally, this study reports the discovery of four new candidate spectroscopic binary systems.

Brownian Particle in a Poisson‐Shot‐Noise Active Bath: Exact Statistics, Effective Temperature, and Inference

AbstractThe dynamics of an overdamped Brownian particle in a thermal bath that contains a dilute solution of active particles is studied. The particle moves in a harmonic potential and experiences Poisson shot‐noise kicks with specified amplitude distribution due to moving active particles in the bath. From the Fokker–Planck equation for the particle dynamics, the stationary solution for the displacement distribution is derived along with the moments characterizing mean, variance, skewness, and kurtosis, as well as finite‐time first and second moments. An effective temperature is also computed through the fluctuation–dissipation theorem and show that equipartition theorem holds for all zero‐mean kick distributions, including those leading to non‐Gaussian stationary statistics. For the case of Gaussian‐distributed active kicks, a re‐entrant behavior from non‐Gaussian to Gaussian stationary states and a heavy‐tailed leptokurtic distribution across a wide range of parameters are found as seen in recent experimental studies. Further analysis reveals statistical signatures of the irreversible dynamics of the particle displacement in terms of the time asymmetry of cross‐correlation functions. Fruits of the work is the development of an compact inference scheme that may allow experimentalists to extract the rate and moments of underlying shot‐noise solely from the statistics the particle position.

Eigenvalue sensitivity from eigenstate geometry near and beyond arbitrary-order exceptional points

Systems with an effectively non-Hermitian Hamiltonian display an enhanced sensitivity to parametric and dynamic perturbations, which arises from the nonorthogonality of their eigenstates. This enhanced sensitivity can be quantified by the phase rigidity, which mathematically corresponds to the eigenvalue condition number and physically also determines the Petermann factor of quantum noise theory. Here, we derive an exact nonperturbative expression for this sensitivity measure that applies to arbitrary eigenvalue configurations. The expression separates spectral correlations from additional geometric data and retains a simple asymptotic behavior close to exceptional points (EPs) of any order, while capturing the role of additional states in the system. This reveals that such states can have a sizable effect even if they are spectrally well separated and identifies the specific matrix whose elements determine this nonperturbative effect. The employed algebraic approach, which follows the eigenvectors-from-eigenvalues school of thought, also provides direct insights into the geometry of the states near an EP. For instance, it can be used to show that the phase rigidity follows a striking equipartition principle in the quasidegenerate subspace of a system. Published by the American Physical Society 2024

From General Economy to Poetic Economy: Expenditure and Equipartition of Energy in Paul Valéry’s Course in Poetics

Resumo Recuperando registros estenográficos inéditos e organizando as notas de aula dispersas conservadas pela Biblioteca Nacional da França, William Marx publica em dois volumes o Cours de Poétique, de Paul Valéry, no início de 2023. Finalmente pudemos saber como se desenvolveram nas aulas do poeta as reflexões sobre a economia do ato poético e as relações entre produção e consumo do poema, anunciadas nas pouquíssimas partes do curso até então conhecidas: a proposta de sua candidatura ao cargo de professor (publicada com o título “De l’enseignement de la Poétique au Collège de France”) e a “Primeira aula do curso de poética”, um dos ensaios de Variété mais lidos de Valéry. Abordaremos essas reflexões recentemente publicadas, articulando-as com algumas passagens da “Poiética”, elaborada pelo escritor em seus Cahiers, e em diálogo e contraponto com comunicações e textos recentes de William Marx, Antoine Compagnon e Jonathan Culler, que também buscaram entender como esses dois volumes do Cours de Poétique desdobram a economia da experiência poética, na produção e no consumo.

TeV Neutrinos and Hard X-Rays from Relativistic Reconnection in the Corona of NGC 1068

The recent discovery of astrophysical neutrinos from the Seyfert galaxy NGC 1068 suggests the presence of nonthermal protons within a compact “coronal” region close to the central black hole. The acceleration mechanism of these nonthermal protons remains elusive. We show that a large-scale magnetic reconnection layer, of the order of a few gravitational radii, may provide such a mechanism. In such a scenario, rough energy equipartition between magnetic fields, X-ray photons, and nonthermal protons is established in the reconnection region. Motivated by recent 3D particle-in-cell simulations of relativistic reconnection, we assume that the spectrum of accelerated protons is a broken power law, with the break energy being constrained by energy conservation (i.e., the energy density of accelerated protons is at most comparable to the magnetic energy density). The proton spectrum is dnp/dEp∝Ep−1 below the break and dnp/dEp∝Ep−s above the break, with IceCube neutrino observations suggesting s ≃ 3. Protons above the break lose most of their energy within the reconnection layer via photohadronic collisions with the coronal X-rays, producing a neutrino signal in good agreement with the recent observations. Gamma rays injected in photohadronic collisions are cascaded to lower energies, sustaining the population of electron–positron pairs that makes the corona moderately Compton thick.

Magnetogenesis in a Collisionless Plasma: From Weibel Instability to Turbulent Dynamo

We report on a first-principles numerical and theoretical study of plasma dynamo in a fully kinetic framework. By applying an external mechanical force to an initially unmagnetized plasma, we develop a self-consistent treatment of the generation of “seed” magnetic fields, the formation of turbulence, and the inductive amplification of fields by the fluctuation dynamo. Driven large-scale motions in an unmagnetized, weakly collisional plasma are subject to strong phase mixing, which leads to the development of thermal pressure anisotropy. This anisotropy triggers the Weibel instability, which produces filamentary “seed” magnetic fields on plasma-kinetic scales. The plasma is thereby magnetized, enabling efficient stretching and folding of the fields by the plasma motions and the development of Larmor-scale kinetic instabilities such as the firehose and mirror. The scattering of particles off the associated microscale magnetic fluctuations provides an effective viscosity, regulating the field morphology and turbulence. During this process, the seed field is further amplified by the fluctuation dynamo until energy equipartition with the turbulent flow is reached. By demonstrating that equipartition magnetic fields can be generated from an initially unmagnetized plasma through large-scale turbulent flows, this work has important implications for the origin and amplification of magnetic fields in the intracluster and intergalactic mediums.

Population balance modeling of formation and breakage of nanoparticle agglomerates in a spouted bed

A population balance framework for fluidization of Al2O3 nano-agglomerates in a ProCell-type spouted bed considering agglomeration and breakage phenomena is developed. An equipartition of kinetic energy (EKE kernel) is used to model the agglomeration phenomena. For breakage, two different fragment distribution functions are considered, the Dirac delta function and a function with confined uniform binary breakage. The fractal dimension property is incorporated in both the agglomeration and breakage models. The model parameters are extracted from batch fluidization experiments conducted by varying the air pressure of the top nozzle. The developed model shows a good agreement with experimental observation in regard of the temporal change in the area and volume-weighted average size of nano-agglomerates and of their final size distribution. The developed population balance model is simulated to study the influence of model parameters on the evolution of average size and coefficient of variation of the nano-agglomerates.

Two modes of motions for a single disk on the vibration stage

The motion of a single granular particle is important for understanding their collective motions on vibration stage, but it remains poorly studied for simple shaped particles, such as a disk. Here, we systematically measure the motions of a single disk with different vibration amplitudes A at a fixed vibration frequency f or a fixed acceleration a. The distributions, time-correlations, and power spectra of displacements per step, mean squared displacements and couplings for translational and rotational motions are measured. These analyses reveal that the motions randomly switch between active and inactive modes. Both a and f are important to particle’s motions and the fraction of active mode. The translational and rotational kinetic energy deviates from Boltzmann distribution and violates the equipartition theorem in each mode. We find three types of motion: rolling, lying flat, and fluttering, which give rise to active and inactive modes. The translational and rotational mean squared displacements, autocorrelations, and power spectra at different a collapse in active modes, because the disk rolls along its rim with a fixed inclination angle though our system is under vibration and confinement. The nonzero cross-correlations between particle’s translational and rotational motions indicate that only translational motions are insufficient for understanding dense particle systems.

Thermal physics in the data age—Students judge the applicability of the equipartition theorem

Providing students of introductory thermal physics with a plot of the heat capacities of many low density gases as a function of temperature allows them to look for systematic trends. Specifically, large amounts of heat capacity data not only allow students to discover the equipartition theorem but also point to its limited applicability. Computer code to download and plot the temperature-dependent heat capacity data is provided.

Equipartition of modal energy in an ideal string vibrating in the presence of a boundary obstacle

Equipartition of modal energy in an ideal string vibrating in the presence of a boundary obstacle

Simulations of two-temperature jets in galaxy clusters

Context. Non-radiating protons in the radio lobes play an essential role in shaping the jet morphology, as demonstrated in recent radio and X-ray observations. However, since protons and electrons are not always in energy equilibrium due to weak Coulomb coupling, it is difficult to estimate the energy contribution of protons for the inflation of radio lobes. Aims. The focus of this study is to examine the effect of the variable model for electron heating by turbulence and shock waves on the thermal energy distribution of electrons and protons. Methods. We performed two-temperature three-dimensional magnetohydrodynamic (3D MHD) simulations of sub-relativistic jets in the galaxy cluster, while varying the jet magnetization parameters. Because the energy partition rate between electrons and protons in shock and turbulence is determined by plasma kinetic scale physics, our global simulations include electron instantaneous heating sub-grid models for shock waves and turbulence. Results. We find that most of the bulk kinetic energy of the jet is converted into the thermal energy of protons through both shocks and turbulence. Thus, protons are energetically dominant. Meanwhile, thermal electrons stored in the lobe evolve toward energy equipartition with magnetic energy through turbulent dissipation. We further estimated the radio power and the mechanical jet power of radio lobes following the same method used for radio and X-ray observations, then we compared these powers with that of the observed radio jets. The two-temperature model quantitatively explains the radiatively inefficient radio cavities, but it cannot reproduce the radiatively efficient cavity, even for strongly magnetized jets. This implies that a significant population of pair-plasma is needed to explain radiatively efficient radio cavities.

Fluctuation relations at large scales in three-dimensional hydrodynamic turbulence

It has long been conjectured that, in three-dimensional turbulence, velocity modes at scales larger than the forcing scale follow equilibrium dynamics. Recent numerical and experimental evidence shows that such modes share the same mean energy and therefore support this claim, but equilibrium dynamics does not reduce to equipartition of energy. In this work, a large set of direct numerical simulations is carried out to investigate if fluctuation-dissipation relations and the fluctuation theorem also apply at these scales. These two results link out-of-equilibrium properties of a forced system with its behavior at equilibrium. Both relations are verified quantitatively by the results of our simulations, further supporting that large-scale modes display equilibrium dynamics. They provide new tools to characterize both the mean value and the fluctuations of the injected energy by a large-scale force acting on turbulence driven by small scale random noise.

A near magnetic-to-kinetic energy equipartition flare from the relativistic jet in AO 0235 + 164 during 2013–2019

ABSTRACT We present the multiwavelength flaring activity of the blazar AO 0235 + 164 during its recent active period from 2013 to 2019. From a discrete correlation function analysis, we find a significant ($\gt 95~{{\ \rm per\ cent}}$) correlation between radio and gamma-ray light curves with flares at longer wavelengths following flares at shorter wavelengths. We identify a new jet component in 43 GHz Very Long Baseline Array data that was ejected from the radio core on MJD $57246^{+26}_{-30}$ (2015 August 12), during the peak of the 2015 radio flare. From the analysis of the jet component, we derived a Doppler factor of δvar = 28.5 ± 8.4, a bulk Lorentz factor of $\Gamma =16.8^{+3.6}_{-3.1}$, and an intrinsic viewing angle of $\theta _{\rm v}=1.42^{+1.07}_{-0.52}\textrm {~degrees}$. Investigation of the quasi-simultaneous radio data revealed a partially absorbed spectrum with the turnover frequency varying in the range of 10−70 GHz and the peak flux density varying in the range of 0.7−4 Jy. We find the synchrotron self-absorption magnetic field strength to be $B_{\rm SSA}=15.3^{+12.6}_{-14.0}\textrm {~mG}$ at the peak of the 2015 radio flare, which is comparable to the equipartition magnetic field strength of $B_{\rm EQ}=43.6^{+10.6}_{-10.4}\textrm {~mG}$ calculated for the same epoch. Additional analysis of the radio emission region in the relativistic jet of AO 0235 + 164 suggests that it did not significantly deviate from equipartition during its recent flaring activity.

A metamaterial with sign-programmable thermal expansivity and Poisson’s ratio constructed from a hybrid of rotating and non-rotating rigid units

A 2D mechanical metamaterial is introduced herein using a hybrid of rotating parallelograms and non-rotating rhombi. The in-plane Poisson’s ratio under on-axes loading were derived by considering the rotation of the parallelograms. The on-axes coefficients of thermal expansion were extracted by matching the simple harmonic motion energy with the principle of energy equipartition, while the on-axes Young’s moduli were formulated by equating the total stored rotational spring potential energy at the connecting points with the strain energy of deformation for the equivalent continuum of the metamaterial. Results indicate that the Poisson’s ratio and CTE are primarily governed by the separation angles between the rigid units and, to a smaller extent, by the shapes of the rotating rigid units. In addition to the aforementioned parameters, the effective Young’s moduli of this metamaterial is directly proportional to the rotational elastic restraint at the hinges that connect adjacent rigid units. Due to the instability of the metamaterial under a specific geometrical condition, charges are incorporated into the rhombi in order to programme the sign of the Poisson’s ratio.

The Role of Stratified Turbulence in the Cold Summer Mesopause Region

AbstractThe summer mesopause at middle and high latitudes is the coldest place on Earth, and atmospheric gravity waves are responsible for the emergence of this extreme thermal phenomenon. Although the main physical mechanism behind the latter is understood, a deeper insight into it can be gained from the investigation of the mesoscale energy spectrum. In this work, we decompose the frequency spectra into divergent and rotational parts and find that their energy contributions are equipartitioned at high frequencies. This mesoscale energy equipartition is a feature of stratified turbulence and illustrates the complexity of the mesoscale dynamics in the summer mesopause region. We also analyze the power spectra of observed and simulated mesoscale zonal and meridional winds at middle latitudes in the Southern Hemisphere and show that stratified turbulence plays a role in the mesopause region during summer.