Search

The use of joint distribution functions for noncommuting observables in quantum thermodynamics is investigated in the light of L. Cohen's proof that such distributions are not determined by the quantum state. Cohen's proof is irrelevant to uses of the functions that do not depend on interpreting them as distributions. An example of this, from quantum Onsager theory, is discussed. Other uses presuppose that correlations betweenp andq values depend at least on the state. But correlations may be fixed by the state even though the distribution varies from one ensemble to another represented by that state. Taking covariance as a measure of correlation, it is shown that the different commonly used joint distributions yield the same correlations for a given state. A general characterization is given for a family of distributions with this same covariance.

Monte Carlo data have been generated for a simple model fluid consisting of hard spheres with an attractive triangular well potential. The ranges spanned by the temperature and density are as follows. [Formula: see text] and [Formula: see text]. The machine data have been compared to the modern perturbation theories of (i) Barker, Henderson, and Smith and (ii) Weeks, Chandler, and Andersen. Comparison with the machine data shows that the latter theory is successful in the high density region only, but over a wide range of temperature. The Barker–Henderson approach is best in the low density region but the use of the superposition approximation limits the utility of this theory at high densities.

Birth‐and‐death type master equations for linear chemical reactions in closed and open systems are solved to discuss the properties of fluctuations. For nonlinear chemical reactions the generating functions of the probability of number fluctuation are usually second order differential equations, and a method for obtaining the moments of fluctuations is presented with applications to simple examples. The deterministic kinetic equation is valid for large systems and the second moments are the same as those of Poisson distributions in open systems treated here, while for the higher moments this is not always the case.

Results are presented for the density of states of binary alloys from a calculation using the matrix coherent potential approximation. A form of the density of states with realistic features was obtained even though the quasicrystalline assumption for the off-diagonal elements was not employed. These features arise solely from the local information about small clusters of atoms. However, it is felt from these results that a proper density of state can only be calculated from information of larger clusters.

The mathematical techniques of the previous paper are used to calculate the fluctuations of the order parameter for large, finite Kac ring models and their subsystems. It is shown that the fluctuations have a characteristic frequency even in the thermodynamic limit. This indicates that a prepared state far from equilibrium does not behave like a large fluctuation. A simplified version of the Fourier spectrum of the fluctuations is also presented which contains all the general features of finite Kac models and is more amenable to analysis.

Relationships between positional variances and covariances that refer to different choices of origin are presented, and the situation is considered in which the uncertainties in the positioning of atoms are direction-independent and the only covariance terms are those that arise from polar aspects and nonorthogonality of the coordinate axes and from symmetry. Covariance terms can often be made small by placing the origin at a suitable centroid.

Summary We show in this paper how the unified formalism obtained in a previous communication 1 embraces several situation discussed in the literature thus far. Several new results are also given. The special cases range from the classical results of Warburg 3 , Randles-Ershler 2 , Frumkin-Melik-Gaikazyan 5 , Lorenz 7 and others to more recent ones of Delahay et al . 11 on the faradaic-non-faradaic coupling. No conscious distinction between the polarizable and the non-polarizable electrode and no undue preoccupation with the mass transfer problem are required. The straightforward methodology given makes it possible to write down immediately the various formulae by forming the suitable matrix representations for the variables and the coefficients. Multi-step electron transfer processes are especially easy to handle with this formalism. Interesting schemes for which results have been obtained as illustrations include the e.e. and e.c.e. mechanisms under varied mass transfer conditions and with adsorption coupling. The non-equilibrium desad case and even processes such as two-dimensional association do not cause any undue difficulty, and it is indicated how to obtain impedance/admittance expressions for such involved reaction schemes and mass-transfer mechanisms.

A new formulation involving fulfillment of all the Kolmogorov axioms is suggested for acomplete probability theory. This proves to be not a purely mathematical discipline. Probability theory deals with abstract objects—images of various classes of concrete objects—whereas experimental statistics deals with concrete objects alone. Both have to be taken into account. Quantum physics and classical statistical physics prove to be different aspects ofone probabilistic physics. The connection of quantum mechanics with classical statistical mechanics is examined and the origin of the Schrodinger equation is elucidated. Attention is given to the true meaning of the wave-corpuscle duality, and the incompleteness of nonrelativistic quantum mechanics is explained.

The paper is concerned with the Khintchine analytical theory of stationary point processes, and in particular with the distributions and joint distributions of intervals between events following analytically defined arbitrary events. Some earlier results of Khintchine and others are strengthened, in the sense that the revised assumptions for them are simpler and may more easily be verified in applications to particular processes; for instance, the stationarity required for the intensity parameter is shown to be that the probability of zero events in a specified interval should only depend on the interval through its length. The Khintchine theory of stationary point processes is widened to include results on the joint distribution of intervals between events, and formulae are derived which enable the joint distributions to be determined for any particular process. Questions of stationarity of the interevent intervals are discussed, and a direction for their resolution is outlined.

Galton-Watson processes where the reproduction of individuals in different generations can have different distributions retain many characteristic features of classical processes.

Using energy conservation sum rules, we derive bounds on multiplicity distributions in terms of one-particle inclusive spectra.

Plasma physicists are becoming increasingly aware of the importance of obtaining an understanding of basic plasma phenomena which cannot be explained in terms of any linear theory of small-amplitude disturbances, and which, collectively, are described as turbulent or large-amplitude phenomena. The first theoretical attempts at describing finite-amplitude effects were made over a decade ago, in the various formulations with the generic title 'quasi-linear theory'.

Analysing the histogram of anode pulse amplitudes allows a discussion of the hypotheses that have been proposed to account for the statistical processes of secondary multiplication in a photomultiplier (PM 56 DVP): a Poisson process, as was suggested by Lombard and Martin (1961), or a Polya process, as suggested by Prescott (1966). Comparison of experimental single-electron pulse spectra with calculated theoretical spectra is first performed by means of moment analysis, from which fractional variance, skewness (a measure of the asymmetry of the distribution), and excess of kurtosis (a measure of the sharpness of the peak and the extent of the tails, as compared with those of a Gaussian distribution), are deduced; then, by directly comparing the shapes of the spectra and minimizing the area between them. On the whole, this study on a focussed dynodes PM makes it clear that though a Polya process is a better approach to reality than a Poisson process, the gain distribution which it leads to does not correspond to the experimental spectrum. A better accord is obtained on assuming a dynode distribution including, in a first approximation, two Poisson distributions of distinct mean value and expressing an inhomogeneity of the secondary multiplication gain value different from that imagined by Prescott.

The three examples of ideal gas, square well, and harmonic oscillator particles illustrate the equivalence of the constant-energy and constant-temperature descriptions of statistical mechanics if the number of degrees of freedom is large. Significant differences between these descriptions appear, however, in the opposite extreme, i.e., the few-particle limit. The analysis presented here supports the conclusion that three degrees of freedom are necessary to localize a particle in the oscillator potential. This result is discussed in connection with the stability (finite rms particle displacement) of three-dimensional harmonic crystals.

This paper studies the statistical performances of multi-break circuit-breakers. For the thermal reignition regime, a formal proof is given of the fact that under certain conditions (explicitly derived) the overall performance of a multi-break breaker is better than that expected from a proportional extrapolation of the results from a single break. For the dielectric regime, it is shown that the hitherto published formulas are conservative in the sense that they overestimate failure probabilities. Using these formulas, graphs are given which compare the failure probabilities of a four-break breaker in the dielectric regime with that of a single break under certain given fractions of the stress on the complete pole. Some general results of a statistical nature relating complete circuit breaker failure probabilities in the dielectric regime to a single break failure probability are also given.

A study is made of a lattice fluid with interactions up to ninth neighbours. A simple approximation is used for the entropy and the internal energy is constructed in terms of a small number of basic configurations. The minimization of the free energy is performed using a numerical procedure described in a previous publication. The tail of the intermolecular potential is sufficiently extended to produce a full phase diagram showing both gas-liquid and liquid-solid transitions. The liquid-solid transition is second order, because of the particular lattice used, but otherwise the phase diagram is satisfactory. A direct application to monolayers is presented and a discussion of a possible biological application is given. The transition to a continuum model and methods of achieving more quantitatively realistic results are discussed.

A multiperipheral cluster model in which the clusters are independently emitted is investigated using the formalism of generating functionals and exact relations between overall correlations and cluster decay spectra are obtained. These relations and the general concept of clusters are subsequently discussed in regard to experimental data.

This paper reviews the practical problems associated with staging large scale Monte Carlo neutronics problems on an electronic computer. In advance of solving the problem the basic nuclear data require to be first assessed, then processed in such a way as to allow the subsequent Monte Carlo calculations to proceed flexibly. Thsi aspect of solving Monte Carlo neutronics problems, seldom addressed in the literature, is reviewed in Section 2. After defining different types of neutronics calculations, each of which presents its own individual problems, two case histories are described in Section 4. A final summary describes our present attitudes towards the subject.

A matrix-flushing method for quantitative multicomponent analysis by X-ray diffraction is reported. It is simpler and faster than, yet as reliable as, the conventional internal-standard method. In this new method, the calibration-curve procedure is shunted; a more fundamental `matrix-flushing' concept is introduced. The matrix-flushing theory gives an exact relationship between intensity and concentration free from matrix effect. Contrary to most theoretical methods the working equation is very simple, no complicated calculations are involved. The matrix-flushing theory and the analytical procedure are presented. Eight illustrative examples are drawn to demonstrate how this theory is applied to multicomponent analysis and amorphous-content determination. A novel `auto-flushing' phenomenon of binary systems was observed which appears to make the analysis of any binary system a simple matter.

We calculate asymptotic values of the first two moments of a planar walk in which the step lengths depend on the direction of motion. The model is suggested by experiments on the locomotion of biological cells. Internally induced persistence due to nonuniform turn angle distributions is also accounted for.

Simple models for the statistical decay of fireballs are formulated in terms of integral equations for different physical quantities like single particle distributions, multiplicity distributions, etc. The integral equations are solved assuming the statistical bootstrap for fireball states and reciprocity between the fireball’s composition and decay. The case of nonspherically symmetric fireballs is also considered. The resulting physical picture of the fireball is that of a strongly interacting, weakly correlated gas of hadrons.

Lattice statics and dynamics calculations for α-nitrogen have been done using the 6-exp atom-atom interaction to derive intermolecular forces. The three parameters of this interaction are reduced to one by constraining them so that the unit cell and sublimation energy are calculated to agree with experiment. Choice of the one parameter is then made using spectroscopic data. This 'best' potential is then used to calculate the statics and dynamics of β-nitrogen. This is stable only under pressure, and the calculation of the lattice dynamics under pressure gives a satisfactory agreement with experiment.

A theoretical analysis has been performed to study molecular and turbulent transport phenomena between subchannels of infinite bare rod arrays at laminar, transition and turbulent flow conditions. For this investigation, the theoretical approach of Ramm and Johannsen for predicting turbulent momentum and heat transfer in rod bundles has been extended to evaluate three-dimensional temperature fields. Results are presented enabling the prediction of the onset and growth of laminarization in typical subchannels of square and triangular rod arrays. These results are further applied to interpret the characteristic effects of variations in Reynolds number, Prandtl number or geometric spacing on integral exchange parameters as the thermal mixing flow rate and mixing length scale. These results are of particular significance relative to the explanation of recent data from tracer-type mixing experiments and also exhibit the importance of secondary flow effects on turbulent intersubchannel energy transport. In view of these findings, the physical relevance of current correlations derived from integral-type experiments to numerically predict exchange coefficients for use in lumped parameter subchannel analysis codes is discussed.

The energy of the breathing mode in nuclei can be calculated both from a macroscopic point of view via the nuclear compressibility and from a microscopic point of view in the TDA and RPA formalisms. We compare results of both types of calculation for 16 O using several self-consistent effective interactions and a couple of more traditional interactions motivated by nuclear structure. We comment on the degree of consistency between the results of both types of calculation, some of the technical considerations in carrying out the calculations, and some of the extremes to which the effective interaction can be pushed without changing the breathing mode energy.

The renormalization-group method for studying critical phenomena is generalized to a class of dynamical systems---the time-dependent Ginzburg-Landau models. The effects of conservation laws on the critical dynamics are investigated through the study of models with different conservation properties for the energy and the space integral of the order parameter. Dynamic critical exponents near four dimensions ($d\ensuremath{\approx}4$) are obtained from recursion relations, analogous to those of Wilson and Fisher. The physical significance of the time-dependent Ginzburg-Landau models is explored and the applicability of the results to experiments on the NMR linewidth of Fe${\mathrm{F}}_{2}$ is discussed.

We use a model due to Kac to investigate some properties of the generalized entropy proposed by the Brussels group. We show that the entropy production is positive-definite and that the entropy and entropy production per particle are finite in the limit of an infinite system. The generalized H theorem is a dynamical theorem and describes the approach to equilibrium of the whole system: generally, correlations play a role in its formulation and cannot be forgotten even if macroscopic variables satisfy the chaos condition. However, a detailed investigation of the evolution equations for the moments shows that correlations reach their equilibrium value faster than the one-particle reduced distribution function and that, asymptotically, there is a regime where one recovers Boltzmann's results.

Diffusion coefficients are computed for a typical lattice cell of the zero-power plutonium reactor experiments using the methods of Benoist and Bonalumi. It is noted that the diffusion coefficients...

It is proved that the free energy per spin in the thermodynamic limit of an Ising model on a lattice with coordination numberz approaches the classical Curie-Weiss free energy in the limitz→∞. The infinite spacial dimension limit of nearest neighbour lattice models is a special case of this result.

The basic concepts and formulation of the renormalization group are explained beginning at an elementary level. Discussion is in the framework of classical statistical mechanics with emphasis on applications to the theory of critical phenomena. The details are worked out in the large n limit for 2 < d < 4, where n is the number of components of the fluctuating field of interest and d is the dimension of the thermodynamical system. In the large n limit, the infinite sum of ``tree graphs'' offers an exact and analytically tractable description of the renormalization group. It illustrates many concepts including the fixed point, the critical surface in the space of coupling parameters, and critical exponents. Most important, it illustrates the origin and the limitation of the scaling hypothesis. The critical behavior of various correlation functions and the free energy is examined. Attention is paid to terms often ignored in qualitative scaling arguments. We have attempted to make this paper self‐contained and of pedagogical value to a wide audience.

Particle production is discussed in a class of multiperipheral models. The feature common to all of the models is a boundedness of the K -particle inclusive distributions, as s →∞. It is found that in some of the models long-range correlations can occur, but only as a critical coupling phenomenon, at a certain value of the coupling constant. For stronger couplings a gas-like distribution results with short correlations, while for weaker couplings only a fixed number of particles are produced, which condense against the “walls”, or ends of phase space. The two cases resemble respectively the “multiperipheral” and “diffractive” components of usual two-component theories, and we can have many of the combined features in our unified model at the critical coupling. There are radical differences between the usual “multicomponent” theory and ours however, which are discussed.

The well-known connection between the Poisson process and empirical c.d.f.'s is exploited from a new point of view. Distributions of functions of empirical c.d.f.'s for finite sample size n are explicitly described in some new examples, and new qualitative information is obtained for some classical examples.

The quasistationary process of shock radiation recombination is described by a stationary diffusion equation which is solved under the assumption of a constant current and taking account of the exact dependence of the diffusion and drift coefficients on the number of the atomic energy level. A comparison is made with the results of other authors obtained by other methods.

Rate equations for level populations of atomic hydrogen are solved to obtain collisional-radiative recombination coefficients. Several particular collision and radiative processes entering the rate equations have been accounted for or have been suppressed in order to obtain information on how strongly the solution depends on the different assumptions. Under certain conditions the results differ by orders of magnitude. The results also vary strongly with the number of quantum levelsp entering the rate equations. The critical quantum numbern * which defines the limit for partial L.T.E. cannot be considered as the critical number of rate equations above which the population coefficients become independent ofp.

Many physical problems are described by stochastic processes with stationary increments. We present a general description of such processes. In particular we give an expression of a process in terms of its increments and we show that there are two classes of processes: diffusion and asymptotically stationary. Moreover, we show that the nth increments are given by a linear filtering of an arbitrary stationary process. STOCHASTIC PROCESSES; STATIONARY; DIFFUSION; LINEAR FILTERING; CORRELATION THEORY Stochastic processes (s.p.) are very often used as statistical models to represent random physical phenomena. In most cases the assumption of stationarity is introduced, either for mathematical simplifications or for physical reasons. Nevertheless there are some situations in which the observed phenomenon is nonstationary, even if it is driven by a stationary source. The best example is the phase ((t) of an oscillator the instantaneous frequency F(t) of which is stationary. Thus we have

A nonlinear analysis of xenon-temperature controlled nuclear reactor dynamics is presented. The set of equations in question belongs to a general class of rate equations with quadratic nonlinearities. Boundedness of the solutions is examined. The mean value of periodic solutions for the flux is shown to be always less than the equilibrium value. The Bogoliubov's method of averaging as extended by Case is applied to obtain approximate solutions. The mechanism of the existence of relaxation oscillations in the linear stability region is analyzed. Computer calculations are performed and found in good agreement with the approximate solutions obtained by means of the method of averaging.

A model for shock heating in a staged θ-pinch is constructed which combines the simple bounce and snowplow models. The main result is the comparison of the amount of shock heating obtainable for various idealized driving fields and for various degrees of mixture of bounce and snowplow effects.

Particle models for hadron interactions are discussed. The various strong interaction phenomena are considered, and those features of the data which need to be incorporated in a model for it to be successful are discussed. 12 references. (JFP)

A simple model undergoing a structural phase transition is studied using diagrammatic methods. In the limit of an infinite number of displacement components the critical dynamics are discussed for potential range sigma and dimensionality d satisfying d/2< sigma

It is known [9] that geometric or exponential decay of a Markov chain is preserved under derivation (as defined by Cohen [2]). In this paper we consider the inverse problem, i.e., does a derived Markov chain with a geometrical or exponential decay necessarily arise from a Markov chain having the same property. For a large class of Markov chains (including time reversible chains) a complete solution is found. The method used in this paper proved accurate for obtaining exact results on the value of the decay parameter. An example extending results of Miller [8] and Teugels [9] illustrates the procedure.

For particle reactions with arbitrary spins a general prescription is outlined and illustrated for the selection of sets of experiments which can completely determine the reaction amplitudes. The prescription involves the inspection of simple numerical matrices, combined with an expedient diagrammatical scheme. (auth)