Introduction: Diffusive Motion and Where It Leads

William G. Faris

1.1 DIFFUSION

The purpose of this introductory chapter is to point out the unity in the following chapters. At first this might seem a difficult enterprise. The authors of these chapters treat diffusion theory, quantum mechanics, and quantum field theory, as well as stochastic mechanics, a variant of quantum mechanics based on diffusion ideas. The contributions also include an infinitesimal approach to diffusion and related probability topics, an approach that is radically elementary in the sense that it relies only on simple logical principles. There is further discussion of foundational problems, and there is a final essay on the mathematics of music. What could these have in common, other than that they are in some way connected to the work of Edward Nelson?

In fact, there are important links between these topics, with the apparent exception of the chapter on music. However, the chapter on music is so illuminating, at least to those with some acquaintance with classical music, that it alone may attract many people to this collection. In fact, there is an unexpected connection to the other topics, as will become apparent in the following more detailed discussion.

The plan is to begin with diffusion and then see where this leads.

In ordinary free motion distance is proportional to time:

Δx = vΔt (1.1)

This is sometimes called ballistic motion. Another kind of motion is diffusive motion. The characteristic feature of diffusion is that the motion is random, and distance is proportional to the square root of time:

Δx = ±σ [square root of Δt] (1.2)

As a consequence diffusive motion is irregular and inefficient. The mathematics of diffusive motion in explained in sections 1.1–1.3 of this chapter.

There is a close but subtle relation between diffusion and quantum theory. The characteristic indication of quantum phenomena is the occurrence of the Planck constant [??] in the description. This constant has the dimensions M L2/T of angular momentum. The relation to diffusion derives from

σ2 = [??]/m, (1.3)

where m is the mass of the particle in the quantum system. The diffusion constant σ2 has the appropriate dimensions L2/T for a diffusion; that is, it characterizes a kind of motion where distance squared is proportional to time.

In quantum mechanics it is customary to define the dynamics by quantities expressed in energy units, that is, with dimensions M L2/T2. The determination of the time dynamics involves a division by [??], which changes the units to inverse time units 1/T. In the following exposition energy quantities, such as the potential energy function V(x), will be in inverse time units. This should make the comparison with diffusion theory more transparent.

One connection between quantum theory and diffusion is the relationship between real time in one theory and imaginary time in the other theory. This connection is precise and useful, both in the quantum mechanics of nonrelativistic particles and in quantum field theory. This connection is explored in sections 1.4–1.7.

The marriage of quantum theory and the special relativity theory of Einstein and Minkowski is through quantum field theory. In relativity theory a mass m has an associated momentum me and an associated energy mc2 These define in turn a spatial decay rate

mL = mc/[??] (1.4)

and a time decay rate

mT = mc2/[??]. (1.5)

These set the distance and time scales for quantum fluctuations in relativistic field theory. This theory is related to diffusion in an infinite-dimensional space of Euclidean fields. Some features of this story are explained in sections 1.8–1.10 of this introduction and in the later chapters by Leonard Gross, Barry Simon, and David Brydges.

The passage from real time to imaginary time is convenient but artificial. However, in the domain of nonrelativistic quantum mechanics of particles there is a closer connection between diffusion theory and quantum theory. In stochastic mechanics the real time of quantum mechanics is also the real time of diffusion, and in fact quantum mechanics itself is formulated as conservative diffusion. This subject is sketched in sections 1.11–1.12 of this introduction and in the chapters by Eric Carlen and Cédric Villani.

The conceptual importance of diffusion leads naturally to a closer look at mathematical foundations. In the calculus of Newton and Leibniz, motion on short time and distance scales looks like ballistic motion. This is not true for diffusive motion. On short time and distance scales it looks like the Wiener process, that is, like the Einstein model of Brownian motion. In fact, there are two kinds of calculus for the two kinds of motion, the calculus of Newton and Leibniz for ballistic motion and the calculus of Ito for diffusive motion. The calculus of Newton and Leibniz in its modern form makes use of the concept of limit, and the calculus of Ito relies on limits and on the measure theory framework for probability. However, there is another calculus that can describe either kind of motion and is quite elementary. This is the infinitesimal calculus of Abraham Robinson, where one interprets Δt and Δx as infinitesimal real numbers. It may be that this calculus is particularly suitable for diffusive motion. This idea provides the theme in sections 1.13–1.14 of this introduction and leads to the later contributions by Greg Lawler and Sam Buss. The concluding section 1.15 connects earlier themes with a variation on musical composition, presented in the final chapter by Julian Hook.

1.2 THE WIENER WALK

The Wiener walk is a mathematical object that is transitional between random walk and the Wiener process. Here is the construction of the appropriate simple symmetric random walk. Let [xi]1, ..., [xi]n be a finite sequence of independent random variables, each having the values ±1 with equal probability. One way to construct such random variables is to take the set {-1, 1}n of all sequences [xi] of n values ± 1 and give it...