Equipartition Research Articles

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.

Kinetic energy equipartition: A tool to characterize quantum thermalization

Kinetic energy equipartition: A tool to characterize quantum thermalization

Analysis of the Dynamic Cushioning Property of Expanded Polyethylene Based on the Stress-Energy Method.

This paper aimed to experimentally clarify the dynamic crushing performance of expanded polyethylene (EPE) and analyze the influence of thickness and dropping height on its mechanical behavior based on the stress-energy method. Hence, a series of impact tests are carried out on EPE foams with different thicknesses and dropping heights. The maximum acceleration, static stress, dynamic stress and dynamic energy of EPE specimens are obtained through a dynamic impact test. Then, according to the principle of the stress-energy method, the functional relationship between dynamic stress and dynamic energy is obtained through exponential fitting and polynomial fitting, and the cushion material constants a, b and c are determined. The maximum acceleration-static stress curves of any thickness and dropping height can be further fitted. By the equipartition energy domain method, the range of static stress can be expanded, which is very fast and convenient. When analyzing the influence of thickness and dropping height on the dynamic cushioning performance curves of EPE, it is found that at the same drop height, with the increase of thickness, the opening of the curve gradually becomes larger. The minimum point on the maximum acceleration-static stress curve also decreases with the increase of the thickness. When the dropping height is 400 mm, compared to foam with a thickness of 60 mm, the tested maximum acceleration value of the lowest point of the specimen with a thickness of 40 mm increased by 45.3%, and the static stress is both 5.5 kPa. When the thickness of the specimen is 50 mm, compared to the dropping height of 300 mm, the tested maximum acceleration value of the lowest point of the specimen with a dropping height of 600 mm increased by 93.3%. Therefore, the dynamic cushioning performance curve of EPE foams can be quickly obtained by the stress-energy method when the precision requirement is not high, which provides a theoretical basis for the design of cushion packaging.

Energy Budget in the Solar Corona

This paper addresses the first direct investigation of the energy budget in the solar corona. Exploiting joint observations of the same coronal plasma by Parker Solar Probe and the Metis coronagraph aboard Solar Orbiter and the conserved equations for mass, magnetic flux, and wave action, we estimate the values of all terms comprising the total energy flux of the proton component of the slow solar wind from 6.3 to 13.3 R ⊙. For distances from the Sun to less than 7 R ⊙, we find that the primary source of solar wind energy is magnetic fluctuations including Alfvén waves. As the plasma flows away from the low corona, magnetic energy is gradually converted into kinetic energy, which dominates the total energy flux at heights above 7 R ⊙. It is found too that the electric potential energy flux plays an important role in accelerating the solar wind only at altitudes below 6 R ⊙, while enthalpy and heat fluxes only become important at even lower heights. The results finally show that energy equipartition does not exist in the solar corona.

Dark Universe phenomenology from Yukawa potential?

We argue that the effect of cold dark matter in the cosmological setup can be explained by the coupling between the baryonic matter particles in terms of the long-range force having a graviton mass mg via the Yukawa gravitational potential. Such a quantum-corrected Yukawa-like gravitational potential is characterized by the coupling parameter α, the wavelength parameter λ, which is related to the graviton mass via mg=ħ/(λc), that determines the range of the force and, finally, a Planck length quantity l0 that makes the potential regular at the centre. The modified Friedmann equations are obtained using Verlinde’s entropic force interpretation of gravity based on the holographic scenario and the equipartition law of energy. The parameter α modifies the Newton’s constant as Geff=G(1+α). Interestingly, we find an equation that relates the dark matter density, dark energy density, and baryonic matter density. It is given by ΩD=2ΩBΩΛ(1+z)3. We argue that dark matter is an apparent effect as no dark matter particle exists in this picture. Furthermore, the dark energy is also related to graviton mass and α; in particular, we point out that the cosmological constant can be viewed as a self-interaction effect between gravitons. We further show a precise correspondence with Verlinde’s emergent gravity theory, and due to the long-range force, the theory can be viewed as a non-local gravity theory. To this end, we performed the phase space analyses and estimated λ≃103 [Mpc] and α≃0.04, respectively. Finally, from these values, for the graviton mass, we get mg≃10−68 kg, and cosmological constant Λ≃10−52m−2. Further, we argue how this theory reproduces the MOND phenomenology on galactic scales via the acceleration of Milgrom a0≃10−10m/s2.

Effects of massive central objects on the degree of energy equipartition of globular clusters

ABSTRACT We present an analysis of the degree of energy equipartition in a sample of 101 Monte Carlo numerical simulations of globular clusters (GCs) hosting either a system of stellar-mass black holes (BHS), an intermediate-mass black hole (IMBH) or neither of them. For the first time, we systematically explore the signatures that the presence of BHS or IMBHs produces on the degree of energy equipartition and if these signatures could be found in current observations. We show that a BHS can halt the evolution towards energy equipartition in the cluster centre. We also show that this effect grows stronger with the number of stellar-mass black holes in the GC. The signatures introduced by IMBHs depend on how dominant their masses are to the GCs and for how long the IMBH has co-evolved with its host GCs. IMBHs with a mass fraction below 2 per cent of the cluster mass produce a similar dynamical effect to BHS, halting the energy equipartition evolution. IMBHs with a mass fraction larger than 2 per cent can produce an inversion of the observed mass-dependence of the velocity dispersion, where the velocity dispersion grows with mass. We compare our results with observations of Galactic GCs and show that the observed range of the degree of energy equipartition in real clusters is consistent with that found in our analysis. In particular, we show that some Galactic GCs fall within the anomalous behaviour expected for systems hosting a BHS or an IMBH and are promising candidates for further dynamical analysis.

Nonlinear Perturbation of Random Matrix Theory.

We consider a system of linear oscillators, or quantum states, described by random matrix theory and analyze how its time evolution is affected by a nonlinear perturbation. Our numerical results show that above a certain chaos border a weak or moderate nonlinearity leads to a dynamical thermalization of a finite number of degrees of freedom with energy equipartition over linear eigenmodes as expected from the laws of classical statistical mechanics. The system temperature is shown to change in a broad range from positive to negative values, and the dependence of system characteristics on the initial injected energy is determined. Below the chaos border the dynamics is described by the Kolmogorov-Arnold-Moser integrability. Owing to universal features of random matrix theory we argue that the obtained results describe the generic properties of its nonlinear perturbation.

Dynamics of the galactic component of Abell S1063 and MACS J1206.2−0847

Context.The galactic component in clusters is commonly thought to be generally nonrotating and in a dynamical state different from that of a collisionally relaxed system. In practice, a test for such conditions is often not available.Aims.We consider the member galaxies of the two clusters Abell S1063 and MACS J1206.2−0847 and study the possible presence of mean rotation and some properties of their distribution in phase space. We look for empirical evidence of factors normally found in collisionally relaxed systems and other characteristics of violently relaxed collisionless systems.Methods.Starting from the CLASH-VLT data, we obtained positions, stellar masses, and individual line-of-sight velocities for a large number of galaxies (NAS1063 ≈ 1200 andNM1206 ≈ 650) extending out to ≈1.6 (Abell) and ≈2.5 (MACS) times the radiusr200. We studied the spatial distribution of the galaxy velocities and the properties of the available galaxy sets when divided into stellar mass bins. To test the presence of velocity dispersion anisotropy, we compared the results based on the Jeans equations with those obtained by assuming a specific form of the galaxy distribution function incorporating the picture of violent relaxation, where the total gravitational potential is imposed as set by the available gravitational lensing observations.Results.We find evidence of systematic rotation in both clusters, with significant rotation in each core (within 0.5′ from the center) and no signatures of rotation at large radii. While no signs of energy equipartition were found, there is a clear indication of (stellar) mass segregation. Velocity dispersion anisotropy is present and qualitatively similar to what has been found in violently relaxed collisionless systems. This last conclusion is strengthened by the overall success in matching the observations with the predictions of the physically justified distribution function.

Multi-degree-of-freedom kinetic model and its applications in simulation of three-dimensional nonequilibrium flows

A multi-degree-of-freedom (MDOF) kinetic model is developed to simulate three-dimensional nonequilibrium flows. By modifying collision term of Boltzmann model equation, a multi-scale molecular collision mechanism with translational, rotational and vibrational degrees of freedom is indicated by three coupled relaxation processes. Molecular vibration is modeled by harmonic oscillator. As a result, the degree of freedom of vibration may be directly related to temperature. Internal energy of various degrees of freedom are computed based on the equipartition theorem, and equilibrium distribution functions for each relaxation step are calculated. The present MDOF kinetic model is beneficial for assessing multi-scale evolutions of internal energy and molecular inelastic collisions. Several three-dimensional flow cases, including lid-driven cavity flow and hypersonic flows around a blunt cone at varying angles of attack, are simulated using the current MDOF technique. All numerical results were consistent with the reference data. It was shown that the proposed MDOF kinetic model well explains the vibration–rotation-translation energy exchange of polyatomic molecules in high-temperature gas, and that the current computational technique is a useful engineering tool for nonequilibrium aerodynamic research.

Two-Dimensional Crystals far from Equilibrium.

When driven by nonequilibrium fluctuations, particle systems may display phase transitions and physical behavior with no equilibrium counterpart. We study a two-dimensional particle model initially proposed to describe driven non-Brownian suspensions undergoing nonequilibrium absorbing phase transitions. We show that when the transition occurs at large density, the dynamics produces long-range crystalline order. In the ordered phase, long-range translational order is observed because equipartition of energy is lacking, phonons are suppressed, and density fluctuations are hyperuniform. Our study offers an explicit microscopic model where nonequilibrium violations of the Mermin-Wagner theorem stabilize crystalline order in two dimensions.

Comparison of Eight Classical Lennard-Jones-Based H2 Molecular Models in the Gas Phase at Temperatures and Pressures Relevant to Hydrogen On-Board Storage Tanks.

This work compares eight classical H2 molecular models in the gas phase taken from the existing literature. All models are based on Lennard-Jones (LJ) 12-6 terms for the van der Waals interactions and hence easier to transfer to multiphase molecular simulations than more sophisticated potentials. The H2 potentials tested include one-site, two-site, three-site, and five-site models, with the sites being either the H atoms, the center-of-mass of the H2 molecule, or massless sites. For the multisite models, high-frequency H-H stretching modes can lead to poor equipartition of the kinetic energy, and the timestep for molecular dynamics (MD) simulations should be reduced to maintain a stable numerical integration of the equations of motion. As such, only those models with rigid bonds are considered. In the present case, 600 MD simulations of H2 gas were carried out over a large range of temperatures (-50 to +90 °C) and at densities corresponding to a pressure range of 50 to 2000 bar, which include the operating conditions of on-board storage tanks in hydrogen-fueled vehicles. Most of the models under study were found to reproduce reasonably well the experimental pVT phase diagram as well as the solubility. Discrepancies only became significant at the highest densities tested, and these could be used to rank the different models. All model diffusion coefficients were essentially indistinguishable from experimental results, and as such, kinetically dominated dynamic properties could not be used as a criterion for the choice of model. Among the eight models tested, two of them, i.e., the two-site model of Yang and Zhong and the one-site model derived from Buch performed very well over the range of conditions tested. They represent a good compromise between realism, simplicity, and computational efficiency.

Multiple pendulum and nonuniform distribution of average kinetic energy.

Multiple pendulums are investigated numerically and analytically to clarify the nonuniformity of average kinetic energies of particles. The nonuniformity is attributed to the system having holonomic constraints and it is consistent with the generalized principle of the equipartition of energy. With the use of explicit expression for Hamiltonian of a multiple pendulum, approximate expressions for temporal and statistical average of kinetic energies are obtained, where the average energies are expressed in terms of masses of particles. In a typical case, the average kinetic energy is large for particles near the end of the pendulum and small for those near the root. Moreover, the exact analytic expressions for the average kinetic energy of the particles are obtained for a double pendulum.

Persistent nonequilibrium effects in generalized Langevin dynamics of nonrelativistic and relativistic particles.

Persistent nonequilibrium effects such as the memory of the initial state, the ballistic diffusion, and the break of the equipartition theorem and the ergodicity in Brownian motions are investigated by analytically solving the generalized Langevin equationof nonrelativistic Brownian particles with colored noise. These effects can also be observed in the Brownian motion of relativistic particles by numerically solving the generalized Langevin equationfor specially chosen memory kernels. Our analyses give rise to think about the possible anomalous motion of heavy quarks in relativistic heavy-ion collisions.

Rayleigh–Jeans thermalization vs beam cleaning in multimode optical fibers

Classical nonlinear waves exhibit, as a general rule, an irreversible process of thermalization toward the Rayleigh–Jeans equilibrium distribution. On the other hand, several recent experiments revealed a remarkable effect of spatial organization of an optical beam that propagates through a graded-index multimode optical fiber (MMF), a phenomenon termed beam self-cleaning. Our aim here is to evidence the qualitative impact of disorder (weak random mode coupling) on the process of Rayleigh–Jeans thermalization by considering two different experimental configurations. In a first experiment, we launch speckle beams in a relatively long MMF. Our results report a clear and definite experimental demonstration of Rayleigh–Jeans thermalization through light propagation in MMFs, over a broad range of kinetic energy (i.e., degree of spatial coherence) of the injected speckle beam. In particular, the property of energy equipartition among the modes is clearly observed in the condensed regime. The experimental results also evidence the double turbulence cascade process: while the power flows toward the fundamental mode (inverse cascade), the energy flows toward the higher-order modes (direct cascade). In a 2nd experiment, a coherent laser beam is launched into a relatively short MMF length. It reveals an effect of beam cleaning driven by an incipient process of Rayleigh–Jeans thermalization. As discussed through numerical simulations, the fast process of Rayleigh–Jeans thermalization observed in the 1st experiment can be attributed due to a random phase dynamics among the modes, which is favored by the injection of a speckle beam and the increased impact of disorder in the long fiber system.

Correspondence between the Energy Equipartition Theorem in Classical Mechanics and Its Phase-Space Formulation in Quantum Mechanics.

In classical physics, there is a well-known theorem in which it is established that the energy per degree of freedom is the same. However, in quantum mechanics, due to the non-commutativity of some pairs of observables and the possibility of having non-Markovian dynamics, the energy is not equally distributed. We propose a correspondence between what is known as the classical energy equipartition theorem and its counterpart in the phase-space formulation in quantum mechanics based on the Wigner representation. Further, we show that in the high-temperature regime, the classical result is recovered.

Some discussions on equipartition theorem

During the development of physics, we can see that the Equipartition Theorem (EPT) has been redefined many times. In this paper, we start with the description of the historical development of the theorem along with the various definitions given by different scientists. Then we found the expressions for classical, quantum, and discrete energy systems and redefined EPT.

Universal thermal corrections to symmetry-resolved entanglement entropy and full counting statistics

In the presence of a global symmetry, the entanglement entropy of a quantum system can be decomposed among the individual symmetry sectors, dubbed symmetry-resolved entanglement entropy. For a conformal field theory with Abelian symmetry, it obeys the equipartition theorem in the scaling limit, i.e., at the leading order, the entanglement is distributed equally between different symmetry sectors. In this work, we examine the thermal corrections to a single interval symmetry-resolved Rényi and entanglement entropies for two-dimensional conformal field theories on a circle. Using a low-temperature expansion of the thermal density matrix, we find that in addition to the mass gap and the degeneracy of the first excited state, these universal corrections depend also on the four-point correlation function of the primary fields. We also obtain thermal corrections to the full counting statistics of the ground state and define the probability fluctuations function which scales as {e}^{-2pi {Delta}_{psi}beta /L} , where ∆ψ is the scaling dimension of the lowest weight states. As an example, we explicitly evaluate the thermal corrections to the symmetry-resolved entanglement entropy and FCS for the spinless fermions and find a term breaking entanglement equipartition at order (log l)−2.

Generalised energy equipartition in electrical circuits

In this brief note, we demonstrate a generalised energy equipartition theorem for a generic electrical circuit with Johnson-Nyquist (thermal) noise. From quantum mechanical considerations, the thermal modes have an energy distribution dictated by Planck's law. For a resistive circuit with some inductance, it is shown that the real part of the admittance is proportional to a probability distribution function which modulates the contributions to the system's mean energy from various frequencies of the Fourier spectrum. Further, we analyse the case with a capacitor connected in series with an inductor and a resistor. The results resemble superstatistics, i.e. a superposition of two statistics and can be reformulated in the energy representation. The correct classical limit is obtained as $\hbar \rightarrow 0$.

Dynamics of intermediate mass black holes in globular clusters

Context. We recently introduced a new method for simulating collisional gravitational N-body systems with approximately linear time scaling with N. Our method is based on the multi-particle collision (MPC) scheme, previously applied in fluid dynamics and plasma physics. We were able to simulate globular clusters with a realistic number of stellar particles (at least up to several times 106) on a standard workstation. Aims. We simulated clusters hosting an intermediate mass black hole (IMBH), probing a broad range of BH-cluster and BH–average-star mass ratios, unrestricted by the computational constraints that affect direct N-body codes. Methods. We set up a grid of hybrid particle-in-cell-MPC N-body simulations using our implementation of the MPC method, MPCDSS. We used either single mass models or models with a Salpeter mass function (a single power law with an exponent of −2.35), with the IMBH initially sitting at the centre. The force exerted by and on the IMBH was evaluated with a direct sum scheme with or without softening. For all simulations we measured the evolution of the Lagrangian radii and core density and velocity dispersion over time. In addition, we also measured the evolution of the velocity anisotropy profiles. Results. We find that models with an IMBH undergo core collapse at earlier times, the larger the IMBH mass the shallower they are, with an approximately constant central density at core collapse. The presence of an IMBH tends to lower the central velocity dispersion. These results hold independently of the mass function of the model. For the models with Salpeter MF, we observed that equipartition of kinetic energies is never achieved, even long after core collapse. Orbital anisotropy at large radii appears to be driven by energetic escapers on radial orbits, triggered by strong collisions with the IMBH in the core. We measured the wander radius, that is the distance of the IMBH from the centre of mass of the parent system over time, finding that its distribution has positive kurtosis. Conclusions. Among the results we obtained, which mostly confirm or extend previously known trends that had been established over the range of parameters accessible to direct N-body simulations, we underline that the leptokurtic nature of the IMBH wander radius distribution might lead to IMBHs presenting as off-centre more frequently than expected, with implications on observational IMBH detection.