Velocity Probability Distribution Research Articles

Random billiards with wall temperature and associated Markov chains

By a random billiard we mean a billiard system in which the standard rule of specular reflection is replaced with a Markov transition probabilities operator P that gives, at each collision of the billiard particle with the boundary of the billiard domain, the probability distribution of the post-collision velocity for a given pre-collision velocity. A random billiard with microstructure, or RBM for short, is a random billiard for which P is derived from a choice of geometric/mechanical structure on the boundary of the billiard domain, as explained in the text. Such systems provide simple and explicit mechanical models of particle–surface interaction that can incorporate thermal effects and permit a detailed study of thermostatic action from the perspective of the standard theory of Markov chains on general state spaces. The main focus of this paper is on the operator P itself and how it relates to the mechanical and geometric features of the microstructure, such as mass ratios, curvatures, and potentials. The main results are as follows: (1) we give a characterization of the stationary probabilities (equilibrium states) of P and show how standard equilibrium distributions studied in classical statistical mechanics such as the Maxwell–Boltzmann distribution and the Knudsen cosine law arise naturally as generalized invariant billiard measures; (2) we obtain some of the more basic functional theoretic properties of P, in particular that P is under very general conditions a self-adjoint operator of norm 1 on a Hilbert space to be defined below, and show in a simple but somewhat typical example that P is a compact (Hilbert–Schmidt) operator. This leads to the issue of relating the spectrum of eigenvalues of P to the geometric/mechanical features of the billiard microstructure; (3) we explore the latter issue, both analytically and numerically in a few representative examples. Additionally, (4) a general algorithm for simulating the Markov chains is given based on a geometric description of the invariant volumes of classical statistical mechanics. Our description of these volumes may have independent interest.

Breaking through: The effects of a velocity distribution on barriers to dust growth

It is unknown how far dust growth can proceed by coagulation. Obstacles to collisional growth are the fragmentation and bouncing barriers. However, in all previous simulations of the dust-size evolution in protoplanetary disks, only the mean collision velocity has been considered, neglecting that a small but possibly important fraction of the collisions will occur at both much lower and higher velocities. We study the effect of the probability distribution of impact velocities on the collisional dust growth barriers. Assuming a Maxwellian velocity distribution for colliding particles to determine the fraction of sticking, bouncing, and fragmentation, we implement this in a dust-size evolution code. We also calculate the probability of growing through the barriers and the growth timescale in these regimes. We find that the collisional growth barriers are not as sharp as previously thought. With the existence of low-velocity collisions, a small fraction of the particles manage to grow to masses orders of magnitude above the main population. A particle velocity distribution softens the fragmentation barrier and removes the bouncing barrier. It broadens the size distribution in a natural way, allowing the largest particles to become the first seeds that initiate sweep-up growth towards planetesimal sizes.

The 6dFGS peculiar velocity field

AbstractThe 6dF Galaxy Survey (6dFGS) is an all-southern-sky galaxy survey, including 125,000 redshifts and a Fundamental Plane (FP) subsample of 10,000 peculiar velocities. This makes 6dFGS the largest peculiar-velocity sample to date. We have fitted the FP with a tri-variate Gaussian model using a maximum-likelihood approach, and derive the Bayesian probability distribution of the peculiar velocity for each of the 10,000 galaxies. We fit models of the velocity field, including comparisons to the field predicted from the redshift-survey density field, to derive the values of the redshift-space distortion parameter β, the bulk flow and the residual bulk flow in excess of that predicted from the density field. We compare these results to those derived by other authors and discuss the cosmological implications.

Effect of harmonic noise on a Brownian particle in a ratchet periodic potential

A Brownian particle in a ratchet periodic potential driven by harmonic noise, which is produced through a RLC oscillation circuit with Gaussian white noise, is investigated. The mean velocity and stationary probability distribution function (SPDF) of the system are obtained by means of numerical simulations. We also used the power spectrum of the harmonic noise, the peak position and semi-height width of which can be changed by modulating the driving oscillation circuit’s parameters, to analyse contributions of characteristics of the power spectrum to the mean velocity. The results indicate that: (i) appropriate peak position and semi-height width of the harmonic noise’s power spectrum can maximise the particle’s mean velocity; (ii) the SPDF undergoes a state transition from monostability → bistability → tristability → monostability as the Gaussian white noise intensity is increased, and the other parameters of the driving oscillation circuit can also modify the system’s state.

The role of probabilities in physics

Although modern physics was born in the XVIIth century as a fully deterministic theory in the form of Newtonian mechanics, the use of probabilistic arguments turned out later on to be unavoidable. Three main situations can be distinguished. (1) When the number of degrees of freedom is very large, on the order of Avogadro's number, a detailed dynamical description is not possible, and in fact not useful: we do not care about the velocity of a particular molecule in a gas, all we need is the probability distribution of the velocities. This statistical description introduced by Maxwell and Boltzmann allows us to recover equilibrium thermodynamics, gives a microscopic interpretation of entropy and underlies our understanding of irreversibility. (2) Even when the number of degrees of freedom is small (but larger than three) sensitivity to initial conditions of chaotic dynamics makes determinism irrelevant in practice, because we cannot control the initial conditions with infinite accuracy. Although die tossing is in principle predictable, the approach to chaotic dynamics in some limit implies that our ignorance of initial conditions is translated into a probabilistic description: each face comes up with probability 1/6. (3) As is well-known, quantum mechanics is incompatible with determinism. However, quantum probabilities differ in an essential way from the probabilities introduced previously: it has been shown from the work of John Bell that quantum probabilities are intrinsic and cannot be given an ignorance interpretation based on a hypothetical deeper level of description.

A Monte Carlo simulation of ion transport at finite temperatures

We have developed a Monte Carlo simulation for ion transport in hot background gases, which is an alternative way of solving the corresponding Boltzmann equation that determines the distribution function of ions. We consider the limit of low ion densities when the distribution function of the background gas remains unchanged due to collision with ions. Special attention has been paid to properly treating the thermal motion of the host gas particles and their influence on ions, which is very important at low electric fields, when the mean ion energy is comparable to the thermal energy of the host gas. We found the conditional probability distribution of gas velocities that correspond to an ion of specific velocity which collides with a gas particle. Also, we have derived exact analytical formulae for piecewise calculation of the collision frequency integrals. We address the cases when the background gas is monocomponent and when it is a mixture of different gases. The techniques described here are required for Monte Carlo simulations of ion transport and for hybrid models of non-equilibrium plasmas. The range of energies where it is necessary to apply the technique has been defined. The results we obtained are in excellent agreement with the existing ones obtained by complementary methods. Having verified our algorithm, we were able to produce calculations for Ar+ ions in Ar and propose them as a new benchmark for thermal effects. The developed method is widely applicable for solving the Boltzmann equation that appears in many different contexts in physics.

Time evolution of interacting vortices under overdamped motion

A system of interacting vortices under overdamped motion, which has been commonly used in the literature to model flux-front penetration in disordered type-II superconductors, was recently related to a nonlinear Fokker-Planck equation, characteristic of nonextensive statistical mechanics, through an analysis of its stationary state. Herein, this connection is extended by means of a thorough analysis of the time evolution of this system. Numerical data from molecular-dynamics simulations are presented for both position and velocity probability distributions P(x,t) and P(v(x),t), respectively; both distributions are well fitted by similar q-Gaussian distributions, with the same index q=0, for all times considered. Particularly, the evolution of the system occurs in such a way that P(x,t) presents a time behavior for its width, normalization, and second moment, in full agreement with the analytic solution of the nonlinear Fokker-Planck equation. The present results provide further evidence that this system is deeply associated with nonextensive statistical mechanics.

Onshore wind farms maintenance optimization using a stochastic model

A specific feature of wind power generation is the stochastic behavior of wind velocity which determines the energy produced, and also influences the turbine degradation process due to the stochastic load suffered by the wind turbine. Thus, wind turbines present a degradation process more complex than the equipment that work under stationary conditions.The selection of a maintenance strategy, that comprises the date to perform a maintenance activity, the maintenance frequency and the duration of such activity have a great influence on the operational cost, that determines the plant viability.Wind velocity probability distribution can be obtained from daily wind data measurements, and by Monte Carlo sampling, it is possible to estimate the power generated. This generation has an associated cost due to the loss of production during maintenance.This work is focused on finding the best maintenance strategy that minimizes the total cost and maximizes the annual energy produced of a wind turbine considering the maintenance frequency as a decision variable.

Equipartition of Rotational and Translational Energy in a Dense Granular Gas

Experiments quantifying the rotational and translational motion of particles in a dense, driven, 2D granular gas floating on an air table reveal that kinetic energy is divided equally between the two translational and one rotational degrees of freedom. This equipartition persists when the particle properties, confining pressure, packing density, or spatial ordering are changed. While the translational velocity distributions are the same for both large and small particles, the angular velocity distributions scale with the particle radius. The probability distributions of all particle velocities have approximately exponential tails. Additionally, we find that the system can be described with a granular Boyle's law with a vanderWaals-like equation of state. These results demonstrate ways in which conventional statistical mechanics can unexpectedly apply to nonequilibrium systems.

Statistical Fluid Mechanics and Statistical Mechanics of Fluid Turbulence

Turbulence in a fluid has two mutually contradictory aspects, that is, the apparent ran-domeness as a whole and the intrinsic determinicity due to the fluid-dynamical equations. In the traditional approarch to this subject, the velocity field of turbulence has been represented by the mean products of the turbulent velocities at several points and a time, and the equations governing the dynamical system composed of the mean velocity products are solved to obtain the mean velocity products as the deterministic functions in time. The works along this approach constitute the main body of turbulence research as outlined by Monin and Yaglom (1975) under the title of 'Statistical Fluid Mechanics'. It should be noted, however, that this approach has the diffculty of unclosedness of the equations for the mean velocity products due to the nonlinearity of the Navier-Stokes equations and an appropriate closure hypothesis is still awaited.In the statistical approach to turbulence, on the other hand, the random velocity field of turbulence is represented by the joint-probability distributions of the multi-point turbulent velocities. This approach has been taken by Lundgren (1967) and Monin (1967) independently, and the system of equations for the probability distributions of the multi-point velocities has been formulated. This system of equations is also unclosed, but in this case the problem is much easier to deal with since the unclosedness is not due to the nonlinearity but to the higher-order derivatives in the viscous term of the Navier-Stokes equations. Tatsumi (2001) proposed the cross-independence closure hypothesis for this purpose and proved its validity for homogeneous isotropic turbulence. More recently, the theory has extended by Tatsumi (2011) to general inhomogeneous turbulence and the closure is shown to provide an exact closure. This theory is outlined and discussed in the later part of this paper.

Long range boundary effect of 2D intermediate number density vibro-fluidized granular media in micro-gravity

We present a micro-gravity experimental study of the statistical properties of intermediate number density vibro-fluidized inelastic spheres in a rectangular container. It is found that although when taking all the particles into account, the probability distributions of velocities both along and perpendicular to the vibration direction are exponential and symmetric, when dividing particles along the vibration direction into different bins, the local velocity distributions are found to deviate measurably from a symmetric distribution for the velocity component in the vibrating direction. The skewness analysis of the local distribution profiles for vx and vy shows that the local distribution of vx remains symmetric, however, the skewness of the distribution profile in vy changes nearly linear from positive to negative with skew = 0 near the center bin. This indicates a long range boundary effect of the asymmetry in vy. We further studied the hydrodynamic profiles granular pressure px and py, and temperature Tx and Ty in positive and negative components such as p+x and p−x, and Tx+ and Tx−, in accordance with the sign of velocity components. The profiles for the two components are found different along the y direction. Along vibration direction granular medium is found inhomogeneous and anisotropic not only in the particle number densities, but also in vy, py and Ty. This suggests new hydrodynamical modeling is needed for such vibro-fluidized granular systems.

Probability distribution of measured debris-flow velocity in Jiangjia Gully, Yunnan Province, China

Debris flow moves in the form of surge waves and consists of dozens or even hundreds of surges that are separated in time and space and have a variety of appearances, as exemplified in Jiangjia Gully, China. Observations there indicate that the deposit is made up by superposition of successive surges and deposit of a single surge is in effect a "frozen" surge. Then the study of debris flow is reduced to the study of surge sequence, which leads to a probabilistic picture of debris flow. This study attempts to find the probability distribution of velocity of surge using a huge data set of Jiangjia Gully. Statistics of the data shows that the velocity satisfies the Weibull distribution, which is believed to be universally valid because the distribution parameters vary little between events, with the shape parameter being well related to the average of velocity. It follows that the same distribution applies also to other quantities of debris flow, such as the flow depth and the discharge. Therefore, the distribution can be used to assess the magnitude and overflow range of a potential debris flow, as well as to the parameter calculation for engineering design.

On the probability distribution function of the velocity field and its derivative in multi-scale turbulence

Using a combination of active and passive grids, we produce an inhomogeneous turbulent flow with strongly non-Gaussian velocity probability distribution (density) functions (pdfs). This new, multi-scale flow shares features with the turbulent jet in freestream turbulence, the near field of turbulence-generating grids, and atmospheric flows. In particular, we study unskewed velocity distributions with both super- and sub-Gaussian tails and demonstrate the linear dependence of the normalized odd and even moments of the distributions on the first and second spatial derivatives of the turbulence r.m.s. fields, respectively. We also note bimodal behavior of the pdfs. The experiments demonstrate a clear effect of the flow organization and the large-scale intermittency on the velocity derivative pdfs and the small scales of the turbulence. The work is motivated by the complex wind fields, and associated, intermittent high stresses, encountered by wind turbines. We also draw comparisons to recent studies of multi-scale turbulence produced by fractal grids.

Three-dimensional synthetic turbulence constructed by spatially randomized fractal interpolation

A spatially randomized fractal interpolation algorithm to construct three-dimensional synthetic turbulence from original coarse field is reported. As in the one-dimensional case by Ding et al. (Phys. Rev. E 82, 036311, 2010), during the fractal interpolation, positions mapping between large and small scale cubes are chosen randomly, and the stretching factors are drawn from a log-Poisson random multiplicative process. A linear combination function defined as the base part in fractal interpolation and a theoretical energy spectrum model for fully developed turbulence are introduced into the procedure. Statistical analysis shows that the synthetic field displays some properties very close to the direct numeric simulated field, such as probability distributions of velocity, velocity gradient, velocity increment, and the anomalous scaling behavior of the longitude velocity structure functions, which follows the SL94 model precisely. After a short time using direct numeric simulation with the synthetic field as initial data, the typical local dynamical structures described by the teardrop shape of the Q-R plane for empirical turbulence can be reproduced.

Entropy Theory for Distribution of One-Dimensional Velocity in Open Channels

Assuming time-averaged velocity as a random variable, this study develops an entropy theory for deriving the one-dimensional distribution of velocity in open channels. The theory includes five parts: (1) Tsallis entropy; (2) the principle of maximum entropy (POME); (3) the specification of information on velocity for constraints; (4) the maximization of entropy; and (5) the probability distribution of velocity and its entropy. An application of the entropy theory is illustrated by deriving a one-dimensional velocity distribution in open channels in which the dimension is vertical or the flow depth. The derived distribution is tested with field and laboratory observations and is compared to Chiu’s velocity distribution derived from Shannon entropy. The agreement between velocity values are computed with the entropy-based distribution.

Estimation of the Probability Distribution of Wave Velocity in Wood Poles

Ultrasonic testing is a nondestructive technique commonly used for in situ condition assessment of wood poles. A transmitter and a receiver are used in a transillumination configuration to measure the first arrival of compressional waves (P-waves). The P-wave velocity is computed using the distance between the transducers and the travel time. The condition assessment of wood poles is inferred from the comparison of the measured wave velocity and a reference velocity that depends on the wood species. A wave velocity smaller than the reference value indicates a reduction in the strength of the wood. The elastic and mechanical properties of wood (elastic moduli, mass density, and Poisson’s ratios) are random variables that vary significantly for the same wood species; consequently, the P-wave velocity in wood poles is also a random variable. A better understanding of wave propagation in an orthotropic material, including the uncertainty in the mechanical properties of wood poles, is required to improve the reliability of ultrasonic tests. This paper presents a new methodology to evaluate the probability distribution of the P-wave velocity in wood poles. This methodology is founded on results from numerical simulations, laboratory tests, a simplified model of P-wave propagation in an infinite cylindrical orthotropic medium, and the consideration of the uncertainty in the mechanical properties of wood. The condition assessment of wood poles is improved by comparing the measured wave velocities Vp at different receiver locations and the corresponding probability distributions of the wave velocity Vp for sound wood poles.

Coherent Vorticity Simulation of Three-Dimensional Forced Homogeneous Isotropic Turbulence

Coherent vorticity simulation (CVS) is a multiscale method to compute incompressible turbulent flows based on the wavelet filtered Navier–Stokes equations. At each time step the vorticity field is decomposed into two orthogonal components using an orthogonal wavelet basis: the coherent vorticity, corresponding to the coefficients whose modulus is larger than a threshold, and the remaining incoherent vorticity. The threshold value only depends on the total enstrophy, which evolves in time, and on the maximal resolution, which remains constant. The induced coherent velocity is computed from the coherent vorticity using the Biot–Savart kernel. To compute the flow evolution one only retains the coherent wavelet coefficients and some of their neighbors in space, scale, and direction, which define the safety zone. Two different strategies are studied to minimize at each time step the number of degrees of freedom to be computed, either by increasing the threshold value or by reducing the width of the safety zone. Their efficiency is compared for a three-dimensional forced homogeneous isotropic turbulent flow at initial Taylor microscale Reynolds number . The quality of the results is assessed in comparison to a direct numerical simulation of the same flow. It is found that, as long as a safety zone is present, CVS well preserves the statistical predictability of the turbulent flow (even the vorticity and velocity probability distribution functions) with a reduced number of degrees of freedom.

RECOILING MASSIVE BLACK HOLES IN GAS-RICH GALAXY MERGERS

The asymmetric emission of gravitational waves produced during the coalescence of a massive black hole (MBH) binary imparts a velocity "kick" to the system that can displace the hole from the center of its host. Here we study the trajectories and observability of MBHs recoiling in three (one major, two minor) gas-rich galaxy merger remnants that were previously simulated at high resolution, and in which the pairing of the MBHs had been shown to be successful. We run new simulations of MBHs recoiling in the major merger remnant with Mach numbers in the range 1<M<6, and use simulation data to construct a semi-analytical model for the orbital evolution of MBHs in gas-rich systems. We show that: 1) in major merger remnants the energy deposited by the moving hole into the rotationally supported, turbulent medium makes a negligible contribution to the thermodynamics of the gas. This contribution becomes significant in minor merger remnants, potentially allowing for an electromagnetic signature of MBH recoil; 2) in major merger remnants, the combination of both deeper central potential well and drag from high-density gas confines even MBHs with kick velocities as high as 1200 km/s within 1 kpc from the host's center; 3) kinematically offset nuclei may be observable for timescales of a few Myr in major merger remnants in the case of recoil velocities in the range 700-1,000 km/s; 4) in minor mergers remnants the effect of gas drag is weaker, and MBHs with recoil speeds in the range 300-600 km/s will wander through the host halo for longer timescales. When accounting for the probability distribution of kick velocities, however, we find that the likelihood of observing recoiling MBHs in gas-rich galaxy mergers is very low, typically below 10^-5 - 10^-6.

Transport of Brownian particles confined to a weakly corrugated channel

We investigate the average velocity of Brownian particles driven by a constant external force when constrained to move in two-dimensional, weakly corrugated channels. We consider both the geometric confinement of the particles between solid walls as well as the soft confinement induced by a periodic potential. Using perturbation methods we show that the leading order correction to the marginal probability distribution of particles in the case of soft confinement is equal to that obtained in the case of geometric confinement, provided that the (configuration) integral over the cross-section of the confining potential is equal to the width of the solid channel. We then calculate the probability distribution and average velocity in the case of a sinusoidal variation in the width of the channels. The reduction on the average velocity is larger in the case of soft channels at small Péclet numbers and for relatively narrow channels and the opposite is true at large Péclet numbers and for wide channels. In the limit of large Péclet numbers the convergence to bulk velocity is faster in the case of soft channels. The leading order correction to the average velocity and marginal probability distribution agree well with Brownian Dynamics simulations for the two types of confinement and over a wide range of Péclet numbers.

Entropy Theory for Two-Dimensional Velocity Distribution

Assuming time-averaged velocity as a random variable, this study develops an entropy theory for deriving two-dimensional (2D) distribution of velocity in open channels. The theory comprises five parts: (1) Tsallis entropy; (2) principle of maximum entropy (POME); (3) specification of information on velocity in terms of constraints; (4) maximization of entropy; and (5) derivation of the probability distribution of velocity. The entropy theory is then combined with a hypothesis on the cumulative distribution function of velocity in terms of flow depth to derive a 2D velocity distribution. The derived distribution is tested using field as well as laboratory observations reported in the literature and is compared with known velocity distributions. Agreement between velocity values computed using the entropy-based distribution and observed values is found satisfactory. Also, the derived distribution compares favorably with known distributions.