Scope

Preceding chapters have dealt with relatively elementary applications of the EPI principle. These are problems whose answers are well-established within known experimental limits. Historically, such problems constituted ‘first tests’ of the principle. As we found, EPI agreed with the established answers, albeit with several extensions and refinements. The next step in the evolution of the theory is to apply it ‘in the large’, i.e., to more difficult problem areas, those of active, current research by the physics and engineering communities.

Two such are the phenomena of (i) quantum gravity and (ii) turbulence at low Mach number. We show, next, the current status of EPI work on these problems. Aspects of the approaches that are, as yet, uncertain are pointed out in the developments. It is hoped that, with time, rigorously correct EPI approaches to the problems will ensue. It is expected that final versions of the theory will closely resemble the first ‘tries’ given below.

Quantum gravity

Introduction

The central concept of gravitational theory is the metric tensor gμν(x). (For a review, see Exercise 6.2.5 and the material preceding it.) This defines the local distortion of space at a four-coordinate x. The Einstein field Eq. (6.68) permits the metric tensor to be computed from knowledge of the gravitational source ― the stress energy tensor Tμν(Sec. 6.3.4). This is a deterministic view of gravity. That is, for a given stress energy tensor and initial conditions, a given metric tensor results. This view holds over macroscopic distances x.

Motivation

As is well-known, gravitational effects are caused by a local distortion of space and time. The Einstein field equation relates this distortion to the local density of momentum and energy. This phenomenon seems, outwardly, quite different from the electromagnetic one just covered. However, luckily, this is not the case. As viewed by EPI they are very nearly the same.

This chapter on gravity is placed right after the one on electromagnetic theory because the two phenomena derive by practically identical steps. Each mathematical operation of the EPI derivation of the electromagnetic wave equation has a 1:1 counterpart in derivation of the weak-field Einstein field equation. For clarity, we will point out the correspondences as they occur. The main mathematical difference is one of dimensionality: quantities in the electromagnetic derivation are vectors, i.e., singly subscripted, whereas corresponding quantities in the gravitational derivation are doubly subscripted tensors. This does not amount to much added difficulty, however.

A further similarity between the two problems is in the simplicity of the final step: in Chap. 5 this was to show that Maxwell's equations follow from the wave equation. Likewise, here, we will proceed from the weak-field approximation to the general-field result. In fact, the argumentation in this final step is even simpler here than it was in getting Maxwell's equations.

Gravitational theory uses tensor quantities and manipulations. We introduce these in the section to follow, restricting attention to those that are necessary to the EPI derivation that follows. A fuller education in tensor algebra may be found, e.g., in Misner et al. (1973) or Landau and Lifshitz (1951).

Classical measurement of four-vectors

In the preceding chapter, we found that the accuracy in an estimate of a single parameter θ is determined by an information I that has some useful physical properties. It provides new definitions of disorder, time and temperature, and a variational approach to finding a single-component PDF law p(x) of a single variable x. However, many physical phenomena are describable only by multiple- component PDFs, as in quantum mechanics, and for vector variables x since worldviews are usually four-dimensional (as required by covariance). Our aim in this chapter, then, is to form a new, scalar information I that is appropriate to this multi-component, vector scenario. The information should be intrinsic to the phenomenon under measurement and not depend, e.g., upon exterior effects such as the noise of the measuring device.

The ‘intrinsic’ measurement scenario

In Bayesian statistics, a prior scenario is often used to define an otherwise unknown prior probability law; see, e.g., Good (1976), Jaynes (1985) or Frieden (1991). This is a model scenario that permits the prior probability law to be computed on the basis of some ideal conditions, such as independence of data, and/or ‘maximum ignorance’ (see below), etc. We will use the concept to define our unknown information expression.

Goals

Classical statistical physics is usually stated in the non-relativistic limit, and so we restrict ourselves to this limit in the analyses to follow. However, as usual, we initiate the analysis on a covariant basis.

The overall aim of this chapter is to show that many classical distributions of statistical physics, defining both equilibrium and non-equilibrium scenarios, follow from a covariant EPI approach. Such equilibrium PDFs as the Boltzmann law on energy and the Maxwell–Boltzmann law on velocity will be derived. Non-equilibrium Hermite–Gaussian PDFs on velocity will also be found. Finally, some recently discovered inequalities linking entropy and Fisher information will be derived.

Covariant EPI problem

Physical scenario

Let a gas be composed of a large number M of identical molecules of mass m within a container. The temperature of the gas is kept at a constant value T. The molecules are randomly moving and mutually interacting through forces due to potentials. The particles randomly collide with themselves and the container walls. All such collisions are assumed to be perfectly elastic. The mean velocity of any particle over many samples is zero.

Measurements E and μ are made of the energy and momentum, respectively, of a randomly selected particle. This defines an (energy,momentum) joint event, and is a Fourier-space counterpart to the (space, time) event characterizing the quantum mechanical analysis in Chap. 4. The measurements perturb the amplitude functions q(E, μ) of the problem, initiating the EPI process.

Aim of the book

The overall aim of this book is to develop a theory of measurement that incorporates the observer into the phenomenon under measurement. By this theory, the observer becomes both a collector of data and an activator of the physical phenomenon that gives rise to the data. These ideas have probably been best stated by J. A. Wheeler (1990), (1994):

The measurement theory that will be presented is largely, in fact, a quantification of these ideas. However, the reader might be surprised to find that the ‘information’ that is used is not the usual Shannon or Boltzmann entropy measures, but one that is relatively unknown to physicists, that of R. A. Fisher.

During the same years that quantum mechanics was being developed by Schroedinger (1926) and others, the field of classical measurement theory was being developed by R. A. Fisher (1922) and co-workers (see Fisher Box, 1978, for a personal view of his professional life). According to classical measurement theory, the quality of any measurement(s) may be specified by a form of information that has come to be called Fisher information.Since these formative years, the two fields - quantum mechanics and classical measurement theory - have enjoyed huge success in their respective domains of application. And until recent times it has been presumed that the two fields are distinct and independent.

Introduction

In preceding chapters, EPI has been used as a computational procedure for establishing the physical laws governing various measurement scenarios. We showed, by means of the optical measurement model of Sec. 3.8, that EPI is, as well, a physical process that is initiated by a measurement. Specifically, it arises out of the interaction of the measuring instrument's probe particle with the object under measurement. This perturbs the system probability amplitudes, which perturbs the informations I and J, etc., as indicated in Figs. 3.3 and 3.4. The result is that EPI derives the phenomenon's physics as it exists at the input space to the measuring device.

We also found, in Sec. 3.8, the form of the phenomenon's physics at the output to the measuring instrument. This was given by Eq. (3.51) for the output probability amplitude function.

The analysis in Sec. 3.8 was, however, severely limited in dimensionality. A One-dimensional analysis of the measurement phenomenon was given. A full, covariant treatment would be preferable, i.e., where the space-time behavior of all probability amplitudes were determined.

Such an analysis will be given next. It constitutes a covariant quantum theory of measurement. This covariant theory will be developed from an entirely different viewpoint than that in Sec. 3.8. The latter was an analysis that focussed attention upon the probe-particle interaction and the resulting wave propagation through the instrument. The covariant approach will concentrate, instead, on the meaning of the acquired data to the observer as it reflects upon the quantum state of the measured particle.

Return once more to our paradigm of a chaotic system, the Lorenz model. The time evolution of trajectories in this system is deterministic. But the precise sequence of events – so many turns around the left wing of the attractor, followed by so many turns around the right wing, followed by another visit to the left wing, and so on – typically seems quite patternless: the visits to the two wings appear to be randomly distributed. Is there any good sense, however, in which there really is objective randomness here? What are we to make e.g. of Joseph Ford's often-quoted remark ‘chaos is merely a synonym for randomness’ (Ford 1989, 350)? If there is a notion of randomness appropriate for describing deterministic chaotic models, can it usefully be carried over to describe the world? And how can we empirically distinguish true deterministic chaos and its sort of randomness from mere random noise? These are the central questions for this chapter.

But first, a brief comment about the claim that standard chaotic models are deterministic. John Earman's remarkable Primer (Earman 1986) teaches us that many questions about determinism are trouble-some and can require considerable care to get straight. Still, the claim being made here does seem fairly unproblematic.

Earman himself starts, as is now popular, from a definition of determinism for worlds.

In Chapter 3, it was argued that worldly phenomena of the kinds typically modelled by chaotic theories cannot exemplify in their time-evolutions the infinitely intricate patterns characteristic of chaos. So even if (as argued in Chapter 4) chaotic theories can be richly predictive, it seems that they cannot be strictly true.

Is this a kind of scepticism about chaos? Not specifically. Dynamical models in classical macrophysics always postulate an infinite precision in the values of the relevant quantities, yet we typically have excellent physical reasons to suppose that these quantities cannot take infinitely precise values. So it is not only chaotic theories that idealize: most of macrophysics is in just the same boat. Does a more global scepticism threaten, then? Surely not. The natural line – implicit, I think, in the comments of working scientists if and when they address the issue – is to allow that such macrophysical theories may idealize, may fail to be true if interpreted by perhaps over-strict standards, but to insist that they can still be more or less approximately true (and can be known to be approximately true).

But what does approximate truth amount to? For reasons that will emerge, it is doubtful that there is any story to be told which is both substantive and general. So let's continue to concentrate on a restricted class of cases – classical dynamical theories that can be regimented by means of a system of linked first-order differential equations of form (D) (§1.1).

Let's take stock. We now have some idea of the kinds of behaviour in dynamical models – involving sensitive dependence on initial conditions, aperiodicity, and the like – that are thought of as typical of ‘chaos’. We have seen that models exhibiting these kinds of behaviour can at least in principle be richly predictive and that we can coherently think of them as candidates for approximate truth. We have seen too how such models can throw some explanatory light on natural phenomena (even when the models are constructed by radical idealization). So most of the main general issues about the status of chaotic dynamical models that we raised back in §1.6 have turned out to be fairly easily resolved. One major issue remains – business for Chapter 9 – concerning the claim that deterministic chaos involves a kind of randomness. But leaving that aside, chaos theory so far looks to be conceptually in good order.

But what is the state of play empirically? As remarked in §§4.6 and 7.5, the Lorenz model, our paradigm example of a model with chaotic behaviour, is in fact empirically rather unsuccessful in its intended domain. It would be good to be able to note some more robust empirical successes for chaotic models. The story, however, is a mixed one.

To repeat, it is no surprise that the Lorenz model does not work very well as an account of Rayleigh-Bénard flow, given the raft of radical simplifications involved in its construction (see §1.4).