Boltzmann distribution

Velocity statistics of a uniformly heated granular fluid PEDRO M. REIS, MARK D. SHATTUCK , Benjamin Levich Institute, City College of New York, CUNY -We report results from an experimental investigation of a uniformly heated granular fluid. We vertically vibrate an ensemble of spheres (diameter D) confined by two horizontal glass plates. The top and bottom plates are separated by 1.6D which ensures a quasi-two dimensional configuration. We show that the use a rough, instead of flat, bottom plate has the considerable advantage of a more effective transfer of momentum from the vertical mode of the cell's vibration onto the motion of individual spheres in the horizontal plane. This allows a greater range of cell's filling fraction, ϕ, to be explored. We study the single particle velocity distributions, f (c), as a function of ϕ and vibration parameters; frequency and amplitude. In agreement with previous studies, we find a consistent overpopulation in the distribution's high energy tails, of the form log f ∼ -c 3/2 . Moreover, we calculate the deviations from a Mawellian, ∆(c) = f (c)/f M B (c) -1, where f M B ∼ exp(-c 2 ). We find ∆(c) to be well described by a 4th-order polynomial which, however, is not the Sonine polynomial commonly used in the solution of the Enskog-Boltzmann equation for inelastic hard spheres driven by a stochastic thermostat.

The existence of novel electrostatic solitary pulses in an electron-positron plasmas with two distinct groups of positrons is demonstrated. It is found that the soliton potential c&n &ssume both the rarefactive and compressive polarities. The results might be useful in understanding the properties of localized disturbances in astrophysical plasmas.

Starting from the classical Boltzmann distribution, we obtain the ion density distribution in the limit of either high temperature/low density (Coulomb interaction energy much less than ion kinetic energy) or low temperature/high density (kinetic energy much less than Coulomb interaction energy), and the trapping force for an ion cloud in Penning ion cyclotron resonance, Paul (quadrupole), and combined (Paul trap in a uniform axial static magnetic field) traps. At equilibrium (total angular momentum conserved), the ion cloud rotates at a constant frequency in Penning and combined traps. In a Penning trap, the maximum ion density is proportional to B 2 /m (B is magnetic field and m is the mass of ions), whereas the maximum ion density in a Paul trap is proportional to (V rf 2 /m⍀ 2 r 0 4 ), with Mathieu equation axial q value Ͻ 0.4 to satisfy the pseudopotential approximation. Ion maximum densities in both Penning and Paul ion traps depend on the trapping field (magnetic or electric) and ion mass, but not on ion charge. In a Penning trap at maximum ion density (zero pressure), the radial (but not the axial) trapping potential is mass dependent, whereas both radial and axial potentials in a Paul trap at maximum ion density are mass dependent. (

We investigate the statistical parity of a class of chaos-generated noises on the escape of strongly damped particles out of a potential well. We show that statistical asymmetry in the chaotic fluctuations can lead to a skewed Maxwell-Boltzmann distribution in the well. Depending on the direction of skew, the Kramers escape rate is enhanced or suppressed accordingly. Based on the Perron-Frobenious equation, we determine an analytical expression for the escape rate's prefactor that accounts for this effect. Furthermore, our perturbative analysis proves that in the zeroth-order limit, the rate of particle escape converges to the Kramers rate.

We consider a system of hard spheres in thermal equilibrium. Using Lanford's result about the convergence of the solutions of the BBGKY hierarchy to the solutions of the Boltzmann hierarchy, we show that in the low-density limit (Boltzmann-Grad limit): (i) the total time correlation function is governed by the linearized Boltzmann equation (proved to be valid for short times), (ii) the self time correlation function, equivalently the distribution of a tagged particle in an equilibrium fluid, is governed by the Rayleigh-Boltzmann equation (proved to be valid for all times). In the latter case the fluid (not including the tagged particle) is to zeroth order in thermal equilibrium and to first order its distribution is governed by a combination of the Rayleigh-Boltzmann equation and the linearized Boltzmann equation (proved to be valid for short times).

S u m m a r y . Economic growth is a result of production and may be calculated by a differential form 5q(K, L), which depends on capital (K) and labor (L). Differential forms in two variables are generally not exact. Accordingly, a general production function q does not exist ex ante, the integral J Sq may not be calculated, unless the path of integration is given. All production functions q x (K,L) depend on a given production process (x). This corresponds to the first law of thermodynamics. The not exact differential form Sq of economic production will become exact by introducing an integrating factor (A), 5q = Xdf. This corresponds to the second law of thermodynamics. The function / exists ex ante and is called entropy (5) in physics and production function ([/) in economics. The factor A will be a mean price level, the mean GDP per capita or standard of living. Production is a Carnot process, which always creates two levels (A). In motors and refrigerators these levels (A) are called hot and cold, in economic systems they are called rich and poor, which explains the two different functions of wealth for normal and for super rich people. The efficiency of the Carnot process grows with the difference in levels (A). For this reason the gap between poor and rich tends to grow permanently.

We try to construct an evolutionary theory of economic and social interaction of heterogeneous agents. Modern physics is helpful for such an attempt, as the recent flourishing of econophysics exemplifies. In this article, we are interested in a specific or more fundamental use of physics rather than in the recent researches of econo-physics. In the first part of this article, we mainly focus on the traditional von Neumann-Sraffa model of production as complex adaptive system and examine the measure of complexity on this model in view of thermodynamical ideas. We then suggest the idea of hierarchical inclusion on this model to define complexity of production. In the latter part, we try to construct an elementary theory of social interaction of heterogeneous agents in view of statistical mechanics.

In an electron-proton plasma, electrostatic solitary waves and hole structures can easily be generated by streaming instability due to the asymmetric inertia between ions and electrons. It has been argued theoretically whether electrostatic solitons and/or hole structures can form in a pair plasma. This paper presents results on the formation of pair electrostatic hole structure in an electron-positron plasma based on one-dimensional electrostatic particle-in-cell simulations. In particular, we show the feature of interlacing electron and positron holes in phase space generated by current-free electron and positron beams streaming in a stationary electron-positron background plasma. The coexistent electron and positron holes are associated with periodic interlacing of positive and negative potentials, respectively. Detailed comparisons between simulation results and linear theory of streaming instability in pair plasmas are made and the thermodynamic state is inferred.

A general theory for nonlinear propagation of one dimensional modified ion-acoustic waves in an unmagnetized electron-positron-ion (e-p-i) degenerate plasma is investigated. This plasma system is assumed to contain relativistic electron and positron fluids, non-degenerate viscous positive ions, and negatively charged static heavy ions. The modified Burgers and Gardner equations have been derived by employing the reductive perturbation method and analyzed in order to identify the basic features (polarity, width, speed, etc.) of shock and double layer (DL) structures. It is observed that the basic features of these shock and DL structures obtained from this analysis are significantly different from those obtained from the analysis of standard Gardner or Burgers equations. The implications of these results in space and interstellar compact objects (viz. nonrotating white dwarfs, neutron stars, etc.) are also briefly mentioned.

Profile consistency is a long-standing mystery of transport in thermonuclear fusion plasmas. This phenomenon is critically tested by studying the system response to off-axis fueling or heating. The present paper investigates the potential of a recently proposed probabilistic transport model to simulate this phenomenon, and it is found that most of the observed phenomenology can be reproduced, at least qualitatively. In particular, the observed differences between tokamaks and stellarators under off-axis heating may find an explanation in this framework.

The combustion stability of a spark ignition engine significantly influences its performances. The cyclic variation is generally evaluated by the fluctuation of in-cylinder peak pressure which changes in both magnitude and position measured from TDC. In this work the cyclic variation of combustion process were analysed as function of crank angles. The different SI engine process phases were investigated. The pressure related data were correlated with cycle resolved visualization measurements. The cycle resolved digital imaging was applied to follow the kernel inception and growth and to study the flame front propagation until the exhaust phase. A custom numerical post-detection procedure was applied to correlate the optical data from the integral luminous signal measured in the combustion chamber with the pressure related parameters .The flame kernel and the abnormal combustion due to the fuel deposits burning resulted particular relevant for the cycle-to-cycle variations. Optical measurements outlined better than pressure related analysis the role of the early and final stages of the combustion process. The experiments were performed in a 400 cm3 single cylinder, port fuel injection, four-stroke spark ignition engine. The engine was optical accessible with the same geometrical parameters as a 1600 cm 3 passenger car engine. The head and the injection system of a commercial engine mounted on a passenger car were used. Standard EURO IV gasoline was used.

The plumes accompanying 1064 nm nanosecond pulsed laser ablation of barium in vacuum at three moderate incident laser fluences in the range of 5.3–10.8 J/cm2 have been studied using both wavelength and time resolved optical emission spectroscopy and time-of-flight laser-induced fluorescence. Neutral atoms and both singly and doubly charged monatomic cations in excited states up to near the corresponding ionization limits are identified in the optical emission spectra. The population distributions of low-lying (≤1.41 eV) “dark” states of Ba atoms measured by laser-induced fluorescence revel that the metastable D3J and D12 abundances in the plume are higher than predictions based on assuming a Boltzmann distribution. The D3J and D12 populations are seen, respectively, to decrease slightly and nearly no vary with raising fluence, which contrasts with the increasing trend that is observed in the ground-state Ba(S10) population. At all fluences, the time-of-flight distributions of the wh...

The plumes accompanying 1064 nm nanosecond pulsed laser ablation of barium in vacuum at three moderate incident laser fluences in the range of 5.3–10.8 J/cm2 have been studied using both wavelength and time resolved optical emission spectroscopy and time-of-flight laser-induced fluorescence. Neutral atoms and both singly and doubly charged monatomic cations in excited states up to near the corresponding ionization limits are identified in the optical emission spectra. The population distributions of low-lying (≤1.41 eV) “dark” states of Ba atoms measured by laser-induced fluorescence revel that the metastable D3J and D12 abundances in the plume are higher than predictions based on assuming a Boltzmann distribution. The D3J and D12 populations are seen, respectively, to decrease slightly and nearly no vary with raising fluence, which contrasts with the increasing trend that is observed in the ground-state Ba(S10) population. At all fluences, the time-of-flight distributions of the wh...

The long-standing deficit of measured versus predicted solar neutrino fluxes is reexamined in light of possible astrophysical solutions. In the last decade, solar neutrino flux and helioseismic measurements have greatly strengthened the case for non-astrophysical solutions. But some model-independent tests remain open.

A variant of the Metropolis algorithm is proposed that allows parallel processing. Rather than generating a single candidate point, as in the Metropolis algorithm, for each chain iteration a number of candidates are generated. Energy calculations for each of these candidates can be carried out in parallel. This algorithm would be advantageous in fitting model parameters to data in a Bayesian context, where the forward model calculations (the analog of the energy calculations), are time consuming.

A generalized form of Wien's displacement law and the blackbody radiation laws of (a) Rayleigh and Jeans, (b) Rayleigh, (c) Wien and Paschen, (d) Thiesen and (e) Planck are derived using principles of dimensional analysis. This kind of scaling is expressed in a strictly mathematical manner employing dimensional π-invariants analysis sometimes called Buckingham's π-theorem. It is shown that in the case of the classical radiation law of Rayleigh and Jeans only one π number occurs that has to be considered as a non-dimensional universal constant. This π number may be determined theoretically or/and empirically. It is also shown that dimensional π-invariants analysis yields a generalized form of Wien's displacement law. In this instance two π numbers generally occur. Consequently, a universal function is established that is indispensable to avoid the so-called Rayleigh-Jeans catastrophe in the ultraviolet. Unfortunately, such a universal function cannot be inferred from dimensional arguments. It has to be derived theoretically or/and empirically, too. It is shown that such a similarity function can be deduced on the basis of heuristic principles, when criteria like the maximum condition regarding the generalized form of Wien's displacement law, the requirement of the power law of Stefan and Boltzmann, and Ehrenfest's arguments regarding the red and the violet requirements are adopted.

A projectile ( 84 Kr 36 ) having kinetic energy around 1 A GeV was used to expose NIKFI BR-2 emulsion target. A total of 700 inelastic events are used in the present studies on projectile fragments. The emission angle of the projectile fragments are strongly affected by charge of the other projectile fragments emitted at same time with different emission angle is observed. The angular distribution studies show symmetrical nature for lighter charge projectile fragments. The symmetrical nature decreased with the charge of projectile fragments. At ~4o of emission angle for double charge projectile fragments, the momentum transfer during interaction is similar for various target species of emulsion were observed. We also observed a small but significant amplitude peaks on both side of the big peak for almost all light charge projectile fragments having different delta angle values. It reflects that there are few percent of projectile fragments that are coming from the decay of heavy projectile fragments or any other process.

This document describes a simple nonequilibrium energy exchange and chemical reaction model to be used in a computational fluid dynamics calculation for a hydrogen plasma excited by microwaves. The model takes into account the exchange between the electrons and excited states of molecular and atomic hydrogen. Specifically, electron-translation, electron-vibration, translation-vibration, ionization, and dissociation are included. The model assumes three temperatures, translational/rotational, vibrational, and electron, each describing a Boltzmann distribution for its respective energy mode. The energy from the microwave source is coupled to the energy equation via a source term that depends on an effective electric field which must be calculated outside the present model. This electric field must be found by coupling the results of the fluid dynamics and kinetics solution with a solution to Maxwell's equations that includes the effects of the plasma permittivity. The solution to ...

We report the results of a numerical investigation, performed in the frame of dynamical systems' theory, for a realistic model of a ionic crystal for which, due to the presence of long-range Coulomb interactions, the Gibbs distribution is not well defined. Taking initial data with a Maxwell-Boltzmann distribution for the mode-energies E k , we study the dynamical fluctuations, computing the moduli of the the energychanges |E k (t) -E k (0)|. The main result is that they follow Tsallis distributions, which relax to distributions close to Maxwell-Boltzmann ones; indications are also given that the system remains correlated. The relaxation time τ depends on specific energy ε, and for the curve τ vs, ε one has two results. First, there exists an energy threshold ε 0 , above which the curve has the form τ • ε h ,

We investigate the statistical parity of a class of chaos-generated noises on the escape of strongly damped particles out of a potential well. We show that statistical asymmetry in the chaotic fluctuations can lead to a skewed Maxwell-Boltzmann distribution in the well. Depending on the direction of skew, the Kramers escape rate is enhanced or suppressed accordingly. Based on the Perron-Frobenious equation, we determine an analytical expression for the escape rate's prefactor that accounts for this effect. Furthermore, our perturbative analysis proves that in the zeroth-order limit, the rate of particle escape converges to the Kramers rate.

We consider a quantum particle coupled (with strength λ) to a spatial array of independent non-interacting reservoirs in thermal states (heat baths). Under the assumption that the reservoir correlations decay exponentially in time, we prove that the motion of the particle is diffusive at large times for small, but finite λ. Our proof relies on an expansion around the kinetic scaling limit (λ ց 0, while time and space scale as λ -2 ) in which the particle satisfies a Boltzmann equation. We also show an equipartition theorem: the distribution of the kinetic energy of the particle tends to a Maxwell-Boltzmann distribution, up to a correction of O(λ 2 ).

A rigorous theoretical investigation has been made of multi-dimensional instability of obliquely propagating electrostatic solitary structures in a hot, magnetized, non-thermal dusty plasma which consists of a negatively charged hot dust ¯uid, Boltzmann distributed electrons and nonthermally distributed ions. The Zakharov±Kuznetsov equation for the electrostatic solitary structures that exist in such a dusty plasma system is derived by the reductive perturbation method. The multidimensional instability of these solitary waves is also studied by the small-k (long wavelength plane wave) perturbation expansion method. The nature of these solitary structures, the instability criterion and their growth rate depending on dusttemperature, external magnetic ®eld and obliqueness are discussed. The implications of these results to some space and astrophysical dusty plasma situations are brie¯y mentioned.

Boltzmann machine is a powerful machine learning model with many real-world applications, for example by constructing deep belief networks. Statistical inference on a Boltzmann machine can be carried out by sampling from its posterior distribution. However, uniform sampling from such a model is not trivial due to an extremely multi-modal distribution. Quantum computers have the promise of solving some non-trivial problems in an efficient manner. We explored the application of a D-Wave quantum annealer to generate samples from a restricted Boltzmann machine. The samples are further improved by Markov chains in a hybrid quantum-classical setup. We demonstrated that quantum annealer samples can improve the performance of Gibbs sampling compared to random initialization. The hybrid setup is considerably more efficient than a pure classical sampling. We also investigated the impact of annealing parameters (temperature) to improve the quality of samples. By increasing the amount of classical processing (Gibbs updates) the benefit of quantum annealing vanishes, which may be justified by the limited performance of today's quantum computers compared to classical.

We study the intensity distribution of the A 2Δ-Χ 2Π system of CH molecule at 430 nm in a low pressure plasma jet. This system shows an overlap of vibrational bands with Δν = 0. By comparing simulated and experimental emission spectra, we obtain rotational and vibrational temperatures using Boltzmann plots or some thermometer functions. The thermometer . functions are the integrated intensities of line-like transitions composed of several rotational transitions. The result of the Boltzmann plots and the thermometer functions method that we propose are in good agreement.

Estimation of Distribution Algorithms (EDA) have been proposed as an extension of genetic algorithms. In this paper the major design issues of EDA's are discussed using an interdisciplinary framework, the minimum relative entropy (MinRel) approximation. We assume that the function to be optimized is additively decomposed (ADF). The interaction graph GADF of the ADF is used to create exact or approximate factorizations of the Boltzmann distribution. The relation between the Factorized Distribution Algorithm FDA and the MinRel approximation is shown. We present a new algorithm, derived from the Bethe-Kikuchi approach developed in statistical physics. It minimizes the relative entropy KLD(q|p β ) to the Boltzmann distribution p β by solving a difficult constrained optimization problem. We present in detail the concave-convex minimization algorithm CCCP to solve the optimization problem. The two algorithms are compared using popular benchmark problems (2-d grid problems, 2-d Ising spin glasses, Kaufman's nk function.) We use instances up to 900 variables.

We present a theory of population based optimization methods using approximations of search distributions. We prove convergence of the search distribution to the global optima for the factorized distribution algorithm (FDA) if the search distribution is a Boltzmann distribution and the size of the population is large enough. Convergence is defined in a strong sense--the global optima are attractors of a dynamical system describing mathematically the algorithm. We investigate an adaptive annealing schedule and show its similarity to truncation selection. The inverse temperature b is changed inversely proportionally to the standard deviation of the population. We extend FDA by using a Bayesian hyper-parameter. The hyper-parameter is related to mutation in evolutionary algorithms. We derive an upper bound on the hyper-parameter to ensure that FDA still generates the optima with high probability. We discuss the relation of the FDA approach to methods used in statistical physics to approximate a Boltzmann distribution and to belief propagation in probabilistic reasoning. In the last part are sparsely connected. Our empirical results are as good or even better than any other method used for this problem.

Estimation of Distribution Algorithms (EDA) have been proposed as an extension of genetic algorithms. In this paper we explain the relationship of EDA to algorithms developed in statistics, artificial intelligence, and statistical physics. The major design issues are discussed within a general interdisciplinary framework. It is shown that maximum entropy approximations play a crucial role. All proposed algorithms try to minimize the Kullback-Leibler divergence KLD between the unknown distribution p(x) and a class q(x) of approximations. However, the Kullback-Leibler divergence is not symmetric. Approximations which suppose that the function to be optimized is additively decomposed (ADF) minimize KLD(q||p), the methods which learn the approximate model from data minimize KLD(p||q). This minimization is identical to maximizing the log-likelihood. In the paper three classes of algorithms are discussed. FDAuses the ADF to compute an approximate factorization of the unknown distribution....

The Factorized Distribution Algorithm (FDA) is an evolutionary algorithm which combines mutation and recombination by using a distribution. The distribution is estimated from a set of selected points. In general, a discrete distribution defined for n binary variables has 2n parameters. Therefore it is too expensive to compute. For additively decomposed discrete functions (ADFs) there exist algorithms which factor the distribution into conditional and marginal distributions. This factorization is used by FDA. The scaling of FDA is investigated theoretically and numerically. The scaling depends on the ADF structure and the specific assignment of function values. Difficult functions on a chain or a tree structure are solved in about O(n√n) operations. More standard genetic algorithms are not able to optimize these functions. FDA is not restricted to exact factorizations. It also works for approximate factorizations as is shown for a circle and a grid structure. By using results from Ba...

In this paper the optimal population size N is empirically computed for the ONEMAX function and truncation selection. N depends on the size of the problem n, the probability p 0 of the advantageous allele in the initial population and the selection intensity I. The dependency of N on I is very complex. By numerically tting the data the following formula could be obtained: N = 1 + f (I) p n ln n (1= p p 0 ; 1). Furthermore it is shown that the minimal number of function evaluations FE needed to nd the optimum is fairly constant in the range 1:0 I 1:4.

The Hamiltonian Monte Carlo algorithm, or alternately called hybrid Monte Carlo is Markov chain Monte Carlo technique, which combines a Gibbs sampling update with the Metropolis acceptance-rejection rule. The Hamiltonian Monte Carlo algorithm simulates the distribution using Hamiltonian dynamics which, involves gradient information to investigate the distribution space, and thus has better convergence properties than Metropolis-Hastings and Gibbs sampling algorithms. The Hamiltonian Monte Carlo algorithm suffers from random walk in generating the dynamics momentum, and an additional error when the dynamics is simulated using constant step-size. This thesis investigates three approaches to improve the performance of the Hamiltonian Monte Carlo algorithm. The first approach enhances the Hamiltonian Monte Carlo by suppressing random walk in the Gibbs sampling using ordered over-relaxation. The second approach investigates the simulation of the Hamiltonian dynamics using an adaptive step-size to reduce the error of the simulation. The third proposal is to combine the two versions into one algorithm. Bergen, Norway -June 2008, Mohammed Alfaki I am very grateful to my supervisor Dr. Sam Subbey for the privilege of having excellent guidance and constant support during this work. My knowledege has benefited greatly from his expertise, enthusiasm and encouragement. I also wish to express my warmest gratitude to my co-supervisor Prof. Dag Haugland for the numerous invaluable suggestions during this work and for giving valuable suggestions to the manuscript. I also thank the administration staff at the department of Informatics. I am enormously grateful to the Norwegian state educational loan fund for the support during my master study.

The Korteweg de Vries-Burgers (KdV-B) (1+1) equation ∂ψ ∂t + a ψ ∂ψ ∂x + b ∂ 3 ψ ∂x 3 = c ∂ 2 ψ ∂x 2 , incorporating constant (real) coefficients representing nonlinearity (a), dispersion (b) and dissipation (c), is a long known paradigm in e.g. plasma physics, where it can be derived from plasma fluid-dynamical models, so that all coefficients depend parametrically on the plasma composition. For a positive dispersion coefficient b (value), which is the general case in beam-free electron-ion plasma, this PDE possesses analytical solutions representing "shock"-shaped traveling waves with a characteristic kink (or anti-kink) soliton-like profile, for negative (or positive, respectively) values of the nonlinearity coefficient (a). In a plasma context, these excitations represent a monotonic transition between two (different) asymptotic values of the electrostatic potential ϕ, associated with a monopolar (i.e. bell-shaped) disturbance of the electric field (E). Contrary to widespread belief (based on a beam-free plasma description), an investigation of nonlinear electrostatic waves in beam-permeated plasmas reveals that the sign(s) of all (any) of the coefficients (a, b or c) may be reversed, independently from each other, depending on the beam velocity (value). In the light of this result, the analytical solutions have been reexamined in an effort to elucidate their applicability in plasma-physical scenarios (e.g., reconnection jets and other planetary plasma environments) in terms of the combined sign(s) of the various coefficients involved in the KdV-B equation. Different types of excitations are demonstrated to exist and the influence of the various coefficients on the solution's propagation characteristics is examined.

Nuclear reactions in stars occur between nuclei in the high-energy tail of the energy distribution and are sensitive to possible deviations from the standard equilibrium thermal-energy distribution. We are able to derive strong constraints on such deviations by using the detailed helioseismic information of the solar structure. If a small deviation is parameterized with a factor exp{-δ(E/kT ) 2 }, we find that δ should lie between -0.005 and +0.002. However, even values of δ as small as 0.003 would still give important effects on the neutrino fluxes.

Nuclear reactions in stars occur between nuclei in the high-energy tail of the energy distribution and are sensitive to possible deviations from the standard equilibrium thermal-energy distribution. We are able to derive strong constraints on such deviations by using the detailed helioseismic information of the solar structure. If a small deviation is parameterized with a factor exp{-δ(E/kT ) 2 }, we find that δ should lie between -0.005 and +0.002. However, even values of δ as small as 0.003 would still give important effects on the neutrino fluxes.

Nuclear reactions in stars occur between nuclei in the high-energy tail of the energy distribution and are sensitive to possible deviations from the standard equilibrium thermal-energy distribution. We are able to derive strong constraints on such deviations by using the detailed helioseismic information of the solar structure. If a small deviation is parameterized with a factor exp{-δ(E/kT ) 2 }, we find that δ should lie between -0.005 and +0.002. However, even values of δ as small as 0.003 would still give important effects on the neutrino fluxes.

Kappa-distributed velocities in plasmas are common in a wide variety of settings, from low-density to high-density plasmas. To date, they have been found mainly in space plasmas, but are recently being considered also in the modelling of laboratory plasmas. Despite being routinely employed, the origin of the kappa distribution remains, to this day, unclear. For instance, deviations from the Maxwell-Boltzmann distribution are sometimes regarded as a signature of the non-additivity of the thermodynamic entropy, although there are alternative frameworks such as superstatistics where such an assumption is not needed. In this work we recover the kappa distribution for particle velocities from the formalism of non-equilibrium steady-states, assuming only a single requirement on the dependence between the kinetic energy of a test particle and that of its immediate environment. Our results go beyond the standard derivation based on superstatistics, as we do not require any assumption about the existence of temperature or its statistical distribution, instead obtaining them from the requirement on kinetic energies. All of this suggests that this family of distributions may be more common than usually assumed, widening its domain of application in particular to the description of plasmas from fusion experiments. Furthermore, we show that a description of kappa-distributed plasma is simpler in terms of features of the superstatistical inverse temperature distribution rather than the traditional parameters κ and the thermal velocity v th .

Models in econophysics, i.e., the emerging field of statistical physics that applies the main concepts of traditional physics to economics, typically consist of large systems of economic agents who are characterized by the amount of money they have. In the simplest model, at each time step, one agent gives one dollar to another agent, with both agents being chosen independently and uniformly at random from the system. Numerical simulations of this model suggest that, at least when the number of agents and the average amount of money per agent are large, the distribution of money converges to an exponential distribution reminiscent of the Boltzmann-Gibbs distribution of energy in physics. The main objective of this paper is to give a rigorous proof of this result and show that the convergence to the exponential distribution is universal in the sense that it holds more generally when the economic agents are located on the vertices of a connected graph and interact locally with their neighbors rather than globally with all the other agents. We also study a closely related model where, at each time step, agents buy with a probability proportional to the amount of money they have, and prove that in this case the limiting distribution of money is Poissonian.

The restricted Boltzmann machine (RBM) is a generative neural network that can learn in an unsupervised way. This machine has been proven to help understand complex systems, using its ability to generate samples of the system with the same observed distribution. In this work, an Ising system is simulated, creating configurations via Monte Carlo sampling and then using them to train RBMs at different temperatures. Then, 1. the ability of the machine to reconstruct system configurations and 2. its ability to be used as a detector of configurations at specific temperatures are evaluated. The results indicate that the RBM reconstructs configurations following a distribution similar to the original one, but only when the system is in a disordered phase. In an ordered phase, the RBM faces levels of irreproducibility of the configurations in the presence of bimodality, even when the physical observables agree with the theoretical ones. On the other hand, independent of the phase of the system, the information embodied in the neural network weights is sufficient to discriminate whether the configurations come from a given temperature well. The learned representations of the RBM can discriminate system configurations at different temperatures, promising interesting applications in real systems that could help recognize crossover phenomena.

Using the recently derived Dissipation Theorem and a corollary of the Transient Fluctuation Theorem (TFT), namely the Second Law Inequality, we derive the unique time independent, equilibrium phase space distribution function for an ergodic Hamiltonian system in contact with a remote heat bath. We prove under very general conditions that any deviation from this equilibrium distribution breaks the time independence of the distribution. Provided temporal correlations decay, and the system is ergodic, we show that any nonequilibrium distribution that is an even function of the momenta, eventually relaxes (not necessarily monotonically) to the equilibrium distribution. Finally we prove that the negative logarithm of the microscopic partition function is equal to the thermodynamic Helmholtz free energy divided by the thermodynamic temperature and Boltzmann's constant. Our results complement and extend the findings of modern ergodic theory and show the importance of dissipation in the process of relaxation towards equilibrium.

The extension of Einstein's model of a two-level atom to relativistic velocities is considered. It is shown that the approach based on the usual Fokker-Planck equation is not correct. Consequently there is no inconsistency between the Planck spectrum and the Boltzmann distribution contrary to the statement of T.H. Boyer. It is also indicated how to modify the Fokker-Planck equation in order to include the relativistic corrections.

Gamma-ray decay schemes have been measured with bare and Compton-suppressed Ge detectors at low-energy resonances (Ep < 340 keV) in the (p, 3') reactions on 25Mg, 26Mg and 27A1. Altogether 58 new decay branches have been observed and a new 26Mg(p, "y)27Al resonance has been found at Ep = 154.5 ± 1.0 keV. The new branchings lead to J~; T determinations (or limitations) for two states in 26A1 and four states in 28Si. The absolute strengths of the 25Mg(p, 5,)26AI and 26Mg(p, 3,)27A1 resonances have also been obtained, and the uncertainties of the stellar rates, deduced from the available data for both reactions, are significantly reduced. Some astrophysical consequences are discussed.

This work introduces a link-based covariance measure between the nodes of a weighted directed graph where a cost is associated to each arc. To this end, a probability distribution on the (usually infinite) countable set of paths through the graph is defined by minimizing the total expected cost between all pairs of nodes while fixing the total relative entropy spread in the graph. This results in a Boltzmann distribution on the set of paths such that long (high-cost) paths occur with a low probability while short (low-cost) paths occur with a high probability. The sum-over-paths (SoP) covariance measure between nodes is then defined according to this probability distribution: two nodes are considered as highly correlated if they often co-occur together on the same-preferably shortpaths. The resulting covariance matrix between nodes (say n nodes in total) is a Gram matrix and therefore defines a valid kernel on the graph. It is obtained by inverting a n × n matrix depending on the costs assigned to the arcs. In the same spirit, a betweenness score is also defined, measuring the expected number of times a node occurs on a path. The proposed measures could be used for various graph mining tasks such as computing betweenness centrality, semi-supervised classification of nodes, visualization, etc, as shown in the experimental section.

Tezpur University Tezpur, Napaam-784 028, Assam, India &lt;em&gt;E&lt;/em&gt;-&lt;em&gt;mail&lt;/em&gt; : ramesh@tezu.ernet.in, nkgour1@tezu.ernet.in Department of Chemistry, Indian Institute of Technology, Guwahati-781 039, Assam, India Department of Physics, Girijananda Chowdhury Institute of Management and Technology, Guwahati-781 017, Assam, India &lt;em&gt;Manuscript received 06 December 2017, accepted 20 December 2017&lt;/em&gt; &lt;strong&gt;A detailed density functional studies of the reaction of n-butyl nitrate initiated by OH radical as well as&lt;/strong&gt; &lt;strong&gt;atmospheric life-time of the titled molecule have been carried out by using hybrid functional M06-2X &lt;/strong&gt; &lt;strong&gt;method along with 6-311+G(d,p) basis set. We obtain the possible H-abstraction channels of the reaction of &lt;/strong&gt; &lt;strong&gt;n-butyl nitrate with OH radical on the potential energy surface along with the thermochemistry associated &lt;/strong&gt; &lt;strong&gt;wit...

A theoretical study on the kinetic and thermochemistry of the reactions of CH 3 C(O)OCH 3 with Cl atoms is performed using DFT and ab initio methods. The formation of pre and post-reactive complexes indicates an indirect hydrogen-abstraction reaction. The calculated total rate coefficient value k Cl = 2.61 Â 10 À12 cm 3 molecule À1 s À1 is in good agreement with the experimental value of k Cl = 2.79 Â 10 À12 cm 3 molecule À1 s À1 at 298 K. Our calculations suggest that the H abstraction from-OCH 3 group is kinetically and thermodynamically more favorable than the-CH 3 C(O) group. The atmospheric lifetime of CH 3 C(O)OCH 3 is found to be 44 days.

The Khinchin theorem provides the condition that a stationary process is ergodic, in terms of the behavior of the corresponding correlation function. Many physical systems are governed by non-stationary processes in which correlation functions exhibit aging. We classify the ergodic behavior of such systems and provide a generalization of Khinchin's theorem. Our work quantifies deviations from ergodicity in terms of aging correlation functions. Using the framework of the fractional Fokker-Planck equation we obtain a simple analytical expression for the two-time correlation function of the particle displacement in a general binding potential, revealing universality in the sense that the binding potential only enters into the prefactor through the first two moments of the corresponding Boltzmann distribution. We discuss applications to experimental data from systems exhibiting anomalous dynamics.

We introduce a fractional Kramers equation for a particle interacting with a thermal heat bath and external nonlinear force field. For the force-free case, the velocity damping follows the Mittag-Leffler relaxation and the diffusion is enhanced. The equation obeys the generalized Einstein relation, and its stationary solution is the Boltzmann distribution. Our results are compared to previous results on enhanced Lévy type of diffusion derived from stochastic collision models.

Photodissociation of acetaldehyde at 308 nm via the radical CH 3 + HCO channel has been studied on the triplet state surface by DFT classical trajectory calculations using the B3LYP/6-31G* electronic structure method. A rotational analysis of product HCO predicts a non-Boltzmann distribution and that the energy is partitioned preferentially along the b and c axes rather than axis a. Product HCO is produced with no vibration excitation but the radical CH 3 is predicted to be vibrationally excited. These results corroborate the most recent LIF experiment on this reaction. When the product energies are scaled, to take into account a possible underestimation of the B3LYP/6-31G* reverse barrier height, the resulting energies are in excellent agreement with experiment.