J. E. Moyal

Quantum mechanics as a statistical theory

An attempt is made to interpret quantum mechanics as a statistical theory, or more exactly as a form of non-deterministic statistical dynamics. The paper falls into three parts. In the first, the distribution functions of the complete set of dynamical variables specifying a mechanical system (phase-space distributions), which are fundamental in any form of statistical dynamics, are expressed in terms of the wave vectors of quantum theory. This is shown to be equivalent to specifying a theory of functions of non-commuting operators, and may hence be considered as an interpretation of quantum kinematics. In the second part, the laws governing the transformation with time of these phase-space distributions are derived from the equations of motion of quantum dynamics and found to be of the required form for a dynamical stochastic process. It is shown that these phase-space transformation equations can be used as an alternative to the Schrodinger equation in the solution of quantum mechanical problems, such as the evolution with time of wave packets, collision problems and the calculation of transition probabilities in perturbed systems; an approximation method is derived for this purpose. The third part, quantum statistics, deals with the phase-space distribution of members of large assemblies, with a view to applications of quantum mechanics to kinetic theories of matter. Finally, the limitations of the theory, its uniqueness and the possibilities of experimental verification are discussed.

The general theory of stochastic population processes

Abstract : Contents: Point processes Counting processes Generating functionals Stochastic population processes Sigma-finite population processes Cluster process Markov population processes Multiplicative population processes

The spectra of turbulence in a compressible fluid; eddy turbulence and random noise

The state of a real fluid is completely specified by its velocity, density, pressure and temperature fields. When the fluid is in turbulent flow, all these quantities fluctuate in a disordered manner. The method of space Fourier spectra is used to show that these field variables separate into two physically distinct groups, one corresponding to fluctuating acoustical waves, or random noise , and the other to fluctuating vorticity, or eddy turbulence . The corresponding decomposition of the spectral and correlation tensors in a homogeneous field of turbulence is given. The noise Fourier components are shown to be coupled to the eddy Fourier components only through the non-linear inertia terms in the dynamical equations of the fluid; whereas the former propagate as acoustical waves, the wave character of the latter is due entirely to the mean motion of the fluid. The measurement of the noise component, its attenuation through absorption by walls and its effects on the eddy component are discussed. Finally, the dynamical equations for the eddy component of the velocity spectral tensor in a homogeneous field of turbulence are compared with the corresponding equations for an incompressible fluid.

Discontinuous Markoff processes

SummaryThe present paper is devoted to the theory of discontinuous Markoff processes, that is processes where the transitions between states take place either by “jumps” of some specified kind, or by other means. States are taken as pointx in an abstract space;phases are points (x, t) in the product state×time space; sets of states are denoted byX, sets of phases byS.It is shown in § 2 that such a process is specified bytwo functions:the probabilityχ0 (X, t|x0,t0) of a transitionx0→X without “jumps” in the time interval [t0,t), and the probability distribution Ψ (S|x0,t0) of the first jump time and the consequent state, given an initial phase (x0,t0). The total transition probability χ (X, t|x0,t0) is required to satisfy the integral equation(I.E.)

A birth, death, and diffusion process

\chi \left( {\left. {X,t} \right|x_0 ,t_0 } \right) = \chi _0 \left( {\left. {X,t} \right|x_0 ,t_0 } \right) + \int {\chi \left( {\left. {X,t} \right|\xi ,\tau } \right)} \psi \left( {\left. {d\xi ,d\tau } \right|x_0 ,t_0 } \right).

Exact solutions of one-dimensional scattering problems

The main problem is to study the existence and uniqueness of the solutions of I.E. which also satisfy the conditions (stated in § 1) for being transition probabilities of a Markoff process.Previous work (cf. § 4) on this subject relates to special cases, mainly to processes where transitions occuronly by jumps. In § 5, two auxiliary sets of functions are introduced: the distributions ψn(|x0,t0) of thenth jump time and consequent state (which form a Markoff chain), and the transition probabilitiesχ0 (X, t|x0,t0)

Mean Ergodic Theorems in Quantum Mechanics

It is shown that the basic results concerning single-state multiplicative chains can be extended to general multiplicative population chains, where the probability distribution of the issue at each generation depends on the state x of the parent.

Causality, Determinism and Probability

This paper treats of a model of a population whose individuals diffuse on a line according to the usual Brownian motion scheme (with constant diffusion coefficient σ22) and at the same time undergo births and deaths with constant rates λ and μ respectively. The “backward” equation governing the transition probability of such a process and an iterative expression for its solution are given. The main results concern the asymptotic spatial distribution of the population conditional on its size as time increases: if the process starts with a single individual at the origin, and if at time t the population has exactly n members in positions y1, …, yn, then the mean position y = ∑1nyin has asymptotically a Gaussian distribution with zero mean and standard deviation σ √t, while the spatial dispersion s2 = ∑1n (yi − y)2 about y has a limiting non-degenerate distribution. Thus for fixed n the mean position gets more diffuse but the dispersion does not grow without limit.