Blogg :: heliga tidsdimensionen

Blogg

The Alchemic Ether Model: An Organic Conception of Physical Space

02.05.2014 11:15

alchemic-ether.pdf (273,8 kB)

The Alchemic Ether Model: An Organic Conception of Physical Space

Paul A. LaViolette

May 1978, Portland State University

The following are excerpts taken from an unpublished paper by the above title

written 7 years prior to the first journal publication of subquantum kinetics.

Many of the excerpted portions focus on a discussion of how chemical reactiondiffusion

systems as well as the alchemic ether approach to physics (subquantum

kinetics) fulfill the organic conception of space that Alfred North Whitehead was

proposing. This paper is interesting from the standpoint of both the early

development of subquantum kinetics and the philosophical context for the theory.

In this paper, LaViolette begins to speak of "displaced media concentrations,"

values taken relative to the ambient steady state concentration. In later editions he

identifies these as the counterparts of energy potentials in physics. He also

begins to speak of the media being composed of "etheric units". This paper was

written about three months prior to a major development in the theory, made in

August 1978, when he discovered the importance of the reverse reaction X ← G,

which was the final step in the development of Model G.

. . . Comprehension of the alchemic ether model requires a significant reorientation from

Cartesian conceptions of space. The organic, processual concept of space expounded by Alfred

North Whitehead(7, 18) provides this sought for reorientation and so may usefully serve as a

philosophical underpinning to the alchemic ether model. Thus, in the way of an introduction we

will begin with a discussion of some of Whitehead's ideas. For example, Whitehead's criticism of

the seventeenth century, Cartesian conception of space is reviewed here (a perspective which

still hangs on in contemporary "linear" field theory models). To compliment this we will discuss

the trimolecular (Brusselator) chemical reaction model as a preparation to the discussion of the

alchemic ether model and also as a means of illustrating Whitehead's organic principles. The

Whiteheadian conception of space will then be correlated with the concept of physical space

presented in the alchemic ether model.*

. . . It is understandable why physical science has retained the linear assumption; linear

equations have been successful in representing a substantial range of phenomena, and more

importantly, they are relatively easy to solve.** The significance of this last point cannot he

overly stressed, for the necessity of achieving mathematical workability and the availability of

_____________________

* The alchemic ether model was originally developed without knowledge of Whitehead's works.

However, parallels with Whitehead's notions are understandable -- while Whitehead's concepts were

inspired from biology, the alchemic ether model was inspired from recent studies of chemical

reaction patterning which, of course, are akin to phenomena found in biochemistry.(6, 9)

** On the other hand, nonlinear equations, which may provide a better grounding for field theory

have, until recently remained insoluble, and it is not surprising that they are highly uncommon in

conventional field theories. However, in the last two decades the availability of high speed

computers has made it possible to model certain classes of nonlinear systems. Thus we may be on

the brink of a new mathematical revolution.

mathematical tools since the time of Galileo has profoundly influenced the direction which

physics has taken, the concepts of reality it has adopted and the kind of experiments it has seen

fit to conduct. Modern physics has adopted mathematics as the primary language in which

theoretical models are expressed. But, the limitations of these mathematical languages influence

the kinds of concepts which may be communicated and experimentally checked. The concepts

which achieve workability make up the reality framework in which the physicist operates and

determine to a large extent the kinds of experiments he performs. As Ludwig von Bertalanffy has

said:(22)

"Linguistic, and cultural categories in general, will not change the potentialities of sensory

experience. They will, however, change apperception, i.e., which features of experienced

reality are focused and emphasized, and which are underplayed."

Along similar lines, Joseph Pearce states:(23)

"Even our most critical, analytical, scientific, or 'detached' looking is a verification search,

sifting through possibilities for a synthesis that will strengthen the hypotheses that

generate the search... When the scientists look at the forest, they look for additions to

their garden, and they look with a gardener's eye."

The linear hypothesis has become deeply ingrained in the present framework of physical field

theory, and consequently, physicists have tended to view their forest with "linear eyes". In

losing sight of the contingent character of our mental categories, Whitehead would say we are

committing the fallacy of misplaced concreteness, mistaking the abstract for the concrete.

Since field theory deals with phenomena which are observationally far removed from the

human level, we can never hope to directly measure the behavior of fields (like we can chemical

compounds) to determine the extent to which the linear assumption is valid. We can only hope

to draw inferences from indirect observations and as we have seen, physicists have chosen to

draw these inferences in terms of linear models. It must be conceded that linear equations, which

have been useful in modeling many mechanical phenomena, have served well as a framework in

physics for integrating experimental observation. Yet, physics, especially field theory, seems to

be alone among the sciences where use of the linearity assumption (simple location) has been

tolerated. Other sciences (i.e., biology, psychology, sociology, etc.) have at some time or other,

discovered that the linearity assumption was grossly inadequate, and that instead, phenomena

were more appropriately described by nonlinear equations and interactive concepts. That is, it is

found that the constituents of natural systems generally do not act independently of one another,

but are behaviorally interrelated.

Still though, the assumption of linearity does not hold a monopoly in the physical sciences.

There are several physical phenomena which have been more appropriately described by

nonlinear mathematics. Most of these deal with nonequilibrium systems and involve either flow

processes (physical translocation) coupled with dissipative processes, or, on the other hand,

purely dissipative processes. The former includes thermal convection, tornadoes, and weather

patterns, while the latter includes transmutational phenomena such as chemical or nuclear

reactions and ecological predation interactions. Moreover, certain nonequilibrium chemical

reaction systems exhibit some interesting properties which may provide useful insights for

understanding microphysical phenomena. One such system, which has received extensive

mathematical treatment is known as the trimolecular model (or the Brusselator).

The Trimolecular Model

The trimolecular model is a chemical reaction scheme developed by the Brussels group;(4, 24,

25) see below:

Then the paper discusses the Brusselator reaction-diffusion system. This is followed by a

section (given below) which discusses attempts that had been made to model the chemical

reaction-diffusion wave phenomenon with linear wave equations and the implications of

microphysics. proposing that our linear equations in physics may be approximations

describing nonlinear processes taking place in a subquantum ether.

A General Representation of Reaction-Diffusion Phenomena

Much work has been done recently in attempting to derive general mathematical

representations for reaction systems such as the trimolecular model and others.(28 - 30) These

approaches study various mathematical solutions of a generalized reaction system having i

species. Each species is considered to have a concentration ci, a net rate of chemical production

Ri = Ri (c1, c2 ... cn), and a net diffusive flux Ji. Wave solutions have been worked out for 1 and 2

dimensional geometries.

The point of departure for studying such wave phenomena has been to describe the character

of the underlying reaction medium. Generally, one begins by writing the following conservation

equation which must hold for all chemical species in all regions of the medium:

∂ci/∂t = Ri + Ji (5)

This basically states that the rate of increase in the concentration of a species i in a localized

region of space (say dV) is due to the net rate of production of specie i in dV by chemical

reactions plus the net rate of diffusion of specie i into dV from other regions of the medium.

If we assume that diffusion in the system can be described by a version of Fick's law that

accounts for coupled diffusion, and by making certain approximations we can express the net

diffusive flux in the system as:(28)

(6)

The Dij are diffusion coefficients giving the magnitude of the flux of the ith component caused by

a gradient in the concentration of the jth component and the Laplacian operator here measures the

departure of the concentration profile from linearity, or the "bumpiness" of the concentration

field.(28)

At steady state S the concentrations of the species remain constant in time, hence, ∂cis/∂t = 0.

If in addition this state i is uniform, there will be no gradients so Ji = 0 and therefore substitution

in equation (5) yields Ri

s = 0. We may represent excursions ψi from the steady state in terms of

chemical concentrations as ψi, = ci - cis. We will refer to ψi as the displaced concentration. For

small amplitude excursions from the steady state, we may approximate equation (5) by a linear

version:

(7)

where Kij

s stands for the chemical reaction rate constants of processes that follow first-order

kinetics in the concentration perturbations.(28) In matrix notation this may be represented as a

standard linear partial differential matrix equation:

(8)

Solutions to this equation may take the form of either stationary or propagating waves. If we

consider a one dimensional reaction system, one solution to equation (8) is the function

representing simple harmonic waves propagating in the r-direction:

(9)

This equation is restricted by the condition that the wave-number k and the frequency factor ω

both must satisfy the determinantal equation:

det ([K] - k2 [D] + iω[I]) = 0 (10)

where [I] is the identity matrix, k = 1/λ = kR + ki and ω = 2Pf.(28) In the special case of marginal

stability, we have k = kR such that equation (9) dictates undamped oscillations of constant

amplitude:*

(11)

We may consider the amplitude of the wave to be a function of the wave vector k, A =A(k2)

and the frequency to also depend on the wave vector in a dispersion relation ω = ω (k2).(30) The

wave velocity given by v = ω/k is therefore a function of k, v = ω (k2)/k.

Ortoleva and Ross(30) take an approach similar to that of Gmitro and Scriven(28) and derive

for small amplitude perturbations in a two dimensional system solutions representing circular

standing waves and rotating waves; see Figure 9. The solid lines n Figure 9 indicate nodal curves

of constant concentration while the (±) symbols indicate deviations above and below the mean

nodal value. In the case of a rotating wave, nodal axis AB would rotate in the plane of the paper.

______________________

* On the other hand, a region of stability will dictate damped oscillations while a region of instability

predicts exponentially growing oscillations possibly leading to a stable limit cycle.

Figure 9. Concentration pattern in a generalized two dimensional reactiondiffusion

system representing standing or rotating waves.

It should be noted that representations of small amplitude waves such as equation (11)

provide no direct information about the reaction diffusion processes taking place "under the

surface". They simply model how the value of Y, changes above and below the zero point as a

function of distance and time. Such equations are linear approximations of deviations from a

steady state cis. The reaction processes maintaining this steady state, however, are nonlinear.

We may put this in perspective by plotting both ψ and c for a particular species i at a particular

location r0 as a time; see Figure 10. Whereas the behavior of ψi (t) is given by a linear equation of

the form of equation (11) (where r = r0), the behavior of ci(t) is given by a nonlinear equation of a

form similar to equation (5).

Now imagine for the moment that we are observing small amplitude traveling waves of the

sort shown in Figure 4 (see next page) and that these are produced in a chemical reaction of the

Figure 10. A temporal plot of a chemical wave comparing absolute concentration ci to

displaced concentration ψi.

Figure 4. Concentrations of X and Y vs. time for a small amplitude deviation from

the steady state in the Trimolecular model (Brusselator) with A = 1 and B = 2).

type shown in scheme (1) operating at a point of marginal stability P. Also, imagine that we are

observationally myopic, that all we are capable of detecting is the superficial character of this

chemical wave, i.e., its space-time dependent concentration intensities. We would then be in a

situation which approximates the predicament of the physicist who with his sensitive

instruments is capable only of measuring electric and magnetic field intensities of an

electromagnetic wave above and below the zero norm. Under these conditions it is quite likely

that we would model the observed chemical wave with a linear equation like equation (11) and

possibly even make the mistake of assuming that the existence of the wave was not dependent on

the action of any underlying continuum. Since the spatial medium through which the chemical

wave was traveling would remain invisible to our detection, we might go so far as to suppose it to

be inert, simply a mechanical carrier or a volume called "space" possessing certain mathematical

properties.

However, the truth is that we are able to perceive the nature of this underlying substance and

know it to be reacting according to scheme (1). Also, we know that the proposed linear wave

equation is only a small amplitude approximation, a modeling of an epiphenomenon. A more

accurate representation of the wave, would be given by equations (4), but of course, such a

representation is mathematically more cumbersome.

The linear wave assumption, arrived at by modeling our myopic perception of the chemical

wave, carries with it the tacit assumption that different spatial regions of the "carrier medium"

bear only positional relationship to one another, and apart from that they exist independent of

one another. However, given that we can perceive the reaction intricacies of the chemical

medium, we find this view to be naive; regions separated in space are not isolated from one

another, but are interwoven into an organic whole. That is, the chemical concentrations observed

within a given volume of medium, dV, will depend both on the production within that volume

due to internal reactions , and on the net transport of chemical molecules to or from that volume

due to diffusion, see equation (5). These two factors constitute, so to speak, intrinsic and

extrinsic processes with respect to dV. Since any change in concentration (say in X or Y)

communicated to dV from its environment via diffusion will interact nonlinearly with the

chemical medium in dV and affect the level of chemical concentrations in dV, adjacent volumes of

medium must be considered, not as being independent of one another, but as being an inseparable

whole. Since what is true of volume dV is true of all volumes of the medium, we may say that

the entire volume must be treated in organic unity.

This is essentially what Whitehead had in mind when he spoke of "prehensive unification" of

things being "together in space, and together in time even if they be not contemporaneous". The

diffusion of chemicals from one region of space to another and their interaction with chemicals

present in that region would constitute what Whitehead called a prehensive event.

I would suggest that the term "empathy" might also be suitable since this word carries the

meaning of both spatial communication and affection. Thus, if the reactions at a point in space A

are affected by the reactions at a distant point B, point A can be said to be empathizing with

point B. If the empathy is mutual, we can say that points A and B are sympathizing with one

another.

Prehension, or "empathy," is seen to be the basis for the propagation of chemical waves

through the reaction volume. To get a rough idea of how such wave propagation arises, consider

the trimolecular model operating at a critical point P and consider two adjacent volumes dV1 and

dV2 in a linear reaction vessel. Also, suppose that initially the concentrations of X and Y are

uniformly distributed at their steady state values. If now in region dV1 the concentration of X

increases and Y decreases in a departure from the steady state, this deviation will be

communicated to region dV2 through diffusion. That is, component X will diffuse into dV2 and

component Y will diffuse out of dV2. Now since the reaction system is operating at point P, Dy

> Dx. So, the concentration of Y in dV2 will be the first to change. Its concentration in dV2 will

fall causing the concentration of X to fall exponentially via reaction step ii (see scheme 1). This

fall in X will reduce the rate at which Y is consumed in dV2 via reaction (ii) allowing Y to build

up in concentration via step (iii). This increase in Y will now increase X exponentially and

consequently decrease Y. This increase in X will be compounded by the contribution of X

diffusing from dV1 (delayed until now). Thus we find that the conditions of high X and low Y

previously present in dV1 have now been transmitted to region dV2 via reaction and diffusion

processes. This analysis may be extended by considering successively other volumes along the

reaction vessel. We would find that the disturbance initiated in dV1 would be transmitted along

the vessel as a wave having a particular velocity and wavelength. The concentration oscillation in

each volume would be synchronized but slightly out of phase with the oscillation in an adjacent

volume. Thus, the wave emerges as a macrolevel inhomogeneity which is propagated as a result

of microlevel molecular reaction and diffusion processes taking place throughout the medium.

The remainder of this paper will explore a new conception of space based on a nonlinear ether

reaction scheme that is similar in many respects to the trimolecular model (Brusselator). Such a

model of space embodies the essential features which Whitehead has stressed in his organic

theory of nature and offers a basis for understanding the emergence of inhomogeneities such as

photons and material particles.

The paper then goes on to describe a Brusselator-like reaction-diffusion ether scheme that

hypothesized in the subquantum kinetics approach. This approach was at this early stage of

development referred to as the alchemic ether model.

The alchemic model allows a new conception of space. The notion of simple location is

abandoned; space is now viewed as being active and organic, composed of media which are

engaged in a process of mutual transformation.* Consequently, a medium inhomogeneity

communicated from point A to point B by diffusion will affect the nature of the media

transmutations taking place at point B. With this in mind, the notion of spatial volume must be

recast from a Whiteheadian perspective:(17)

". . . the prehensive unity of the volume is not the unity of a mere logical aggregate of parts.

The parts form an ordered aggregate, in the sense that each part is something from the

standpoint of every other part, and also from the same standpoint every other part is

something in relation to it. Thus if A and B and C are volumes of space, B has an aspect

from the standpoint of A, and so has C, and so has the relationship of B and C. This aspect

of B from A is of the essence of A. The volumes of space have no independent existence.

They are only entities as within the totality; you cannot extract them from their environment

without destruction of their very essence."

Whitehead defines the "mode in which B enters into composition of A" as "the aspect of B

from A." Accordingly, the modal character of space of which he speaks is expressed by his

statement that ''the prehensive unity of A is the prehension into unity of the aspects of all other

volumes from the standpoint of A," or that "every volume mirrors in itself every other volume in

space". In a certain sense he says, then, "everything is everywhere at all times. For every

location involves an aspect of itself in every other location."

The paper then goes on to describe the alchemic ether model's application to the formation of

photons and particles. Eventually this leads into a continued discussion of Whitehead's ideas.

Figure 15. One-dimensional cross sectional view of a 3-dimensional dissipative space

structure (material particle) showing how spatial inhomogeneity is maintained through

a balance between reaction and diffusion.

__________________________

* (from p. 7) The view that the underlying character of the universe is one of activity and change is

rooted in eastern religion and also corresponds to Shelley's portrayal of nature. According to

Whitehead: "Shelly thinks of nature as changing, dissolving, transforming as it were at a fairy's touch.

The leaves fly before the West Wind 'like ghosts from an enchanter fleeing.' In his poem The Cloud,

it is the transformations of water which excite his imagination. The subject of the poem is the

endless, eternal, elusive change of things: 'I change but I cannot die."

. . . Consider Figure 15 which shows a one-dimensional profile of this spherical propagation

[steady state dissipative space structure produced by the alchemic ether model]. It is seen that

any diffusion of medium X from region X1 into its complimentary region X1' is compensated by

an equal diffusion from X1' into X1. Consequently, the diffusive loss of medium X can only take

place in directions away from center point 0; i.e., from X1 into Y1 or from X1' into Y1'.

Consider for the moment just regions X1 and Y1. Suppose a quantity dx of medium X diffuses

from region X1 into region Y1 and an equal quantity dy of medium Y diffuses from region Y1 into

X1. As was discussed earlier with respect to the tri-molecular model (Brusselator), in regions of

high Y concentration there is an affinity for X to be preferentially converted to Y and in regions

of high X concentration an affinity for Y to be preferentially converted to X. Consequently in

region Y1, dx will be transmuted into an equivalent amount dy as symbolized by the process dx

→ dy. This transmutational input of dy into region Y1 will exactly compensate for the diffusive

loss of dy from this region provided that the rate of transmutation keeps pace with the rate of

diffusion, as will be the case when the pattern has fully developed and maintains a steady state.

Consequently, the concentration of Y in region Y1 will be preserved unchanged in the face of

diffusion. Likewise, the concentration of X in region X1 will also remain invariant, and similarly

for other regions to the left and right of point 0.

We may regard a material particle as being composed of a distribution of etheric units

spatially ordered according to whether they are type X or Y. From a thermodynamic perspective

this constitutes a state of negentropy. The fact that etheric units diffuse from regions of higher

concentration to regions of lower concentration indicates that the Second Law of Thermodynamics

is in operation. However, certain open reaction systems, such as the one proposed here,

have the ability to oppose the increase of entropy through the action of their processually

ordered transmutational (dissipative) processes. Thus, if we visualize the regions of high X and

high Y in Figure 15 as alternating regions of red and blue beads engaged in random motion, we

would observe, in conformance with the Second Law, that red beads tended on the whole to

diffuse into blue regions and blue beads into red regions. However, the original segregated order

would remain preserved, for just as fast as the red beads enter into blue regions, they are

converted into blue beads, and vice versa.

Thus, a material particle, viewed as a dissipative space structure, has the unique property

that, by virtue of its geometry and autocatalytic character, it is able to preserve itself as a stable

pattern. Consequently, as the initial quantum X, Y disturbance travels outward radially its

"past" becomes frozen; inner lying regions formerly excited into their inhomogeneous state

remain so as an enduring structure is formed. A particle is essentially the historical travel

itinerary of its parent nucleating fluctuation. Linearly propagating quanta, on the other hand, are

more "now oriented." They leave no trace behind as they travel.

Earlier (cf. p. 6) we noted that the prehensive character of the trimolecular (Brusselator)

reaction medium is the basis by which chemical reaction waves are propagated. The prehensive

character of space as depicted in the alchemic ether model, similarly may be regarded as the basis

for the spatial propagation of quanta (both linearly and radially) and for the maintenance of the

inhomogeneous steady state known as matter. As an illustration of this latter case, we may note,

by reference to Figure 15, that when a quantity dx diffuses from region X1 into Y1, region Y1

"takes account" of this environmental communication by incorporating it into its local reaction

kinetics, i.e., through the transformation dx → dy. Consequently, the structure of a particle of

matter can be said to maintain its ordered state by virtue of spatial empathy, or what may be

otherwise called nonlinear interaction.

It is interesting to note that Whitehead's organic conception of material particles in many

respects resembles the dissipative space structure view proposed here.* Whitehead visualizes a

subatomic particle, what he calls a "primate", as being associated with a pattern of

electromagnetic waves, or what he terms "vibratory organic deformations," which are radially

disposed in a stationary manner:(17)

"A primate must be associated with a definite frequency of vibratory organic deformation so

that when it goes to pieces it dissolves into light waves of the same frequency, which then

carry off all its average energy. It is quite easy (as a particular hypothesis) to imagine

stationary vibrations of the electromagnetic field of definite frequency, and directed radially

to and from a centre, which, in accordance with the accepted electromagnetic laws, would

consist of a vibratory spherical nucleus satisfying one set of conditions and a vibratory

external field satisfying another set of conditions. ...The total energy, according to one of

these ways, should satisfy the quantum condition; so that it consists of an integral number of

units or cents, which are such that the cent of energy of any primate is proportional a to its

frequency.

. . . In this particular hypothesis of vibratory primates, the Maxwellian equations are

supposed to hold throughout all space, including the interior of a proton. They express the

laws governing the vibratory production and absorption of energy. The whole process for

each primate issues in a certain average energy characteristic of the primate, and proportional

to its mass. In fact the energy is mass. There are vibratory radial streams of energy, both

without and within a primate. Within the primate, there are vibratory distributions of electric

density. On the materialistic theory such density marks the presence of material: on the

organic theory of vibration, it marks the vibratory production of energy. Such production is

restricted to the interior of the primate."

His "vibratory radial streams of energy" might be likened in the alchemic ether model to the

electric field propagating outward at the boundary of the space structure. However, these

stationary vibrations would not be "in accordance with the accepted electromagnetic laws," as he

states, since the laws of electromagnetics accepted by contemporary physics do not presuppose

that such waves are sustained by underlying nonlinear interactions.

One major discrepancy between Whitehead's view of particles and that proposed in the

alchemic ether model is that Whitehead accepts the field-source dualism. He distinguishes

between the interior and exterior of a particle, which he says each satisfy a different set of

conditions. The alchemic ether model, on the other hand, makes no distinction between inside

and outside. A particle has only an "inside" in the conventional sense, no "outside." The extent

of this inside region is determined by the particle's event horizon, i.e., the radially propagating

electromagnetic field boundary induced by the photon that created it (i.e., in the case of pair

production). Thus, the particle's radius is time dependent, expanding at the rate of 186,000 miles

per second, where r = ct.

What is usually thought of as the "inside*' of a particle, i.e., the nucleus (which is on the

order of 10-13 cm in the case of a proton) is not distinguishable in the alchemic ether model, save

for the central concentration inhomogeneity. But, this core inhomogeneity contains no units of

charge or mass. As a will be discussed in a subsequent paper, the seat of a particle's electrostatic

potential resides throughout its space structure. The attractive or repulsive action of a particle's

"electrostatic field" may be traced to the geometry of the particle and whether an elevated

concentration of X or Y resides at its center. Thus "charge" becomes viewed as a property

characteristic of the entire volume of a particle. Likewise, "mass" is also viewed as a property of

the entire particle space structure, being related to the peak intensity of the displaced medium

concentration in the particle's standing wave pattern.

The paper then goes on to discuss de Broglie's wave theory of matter and how the dissipative

structure particle predicted by the alchemic ether model (i.e., subquantum kinetics) satisfies

quantitative experimental observations of particle diffraction and correctly predicts the radii

of the Bohr orbits in the hydrogen atom. All this is essentially as is described in the

published version of subquantum kinetics.

References

(4) (reference lacking)

(6) (reference lacking)

(7) Whitehead, A. N. (reference lacking)

(9) (reference lacking)

(17) Whitehead, A. N. Science and the Modern World, New York, Macmillan, 1925, pp. 81, 168.

https://evankozierachi.com/uploads/Whitehead_-_Science_and_the_Modern_World.pdf

(18) Whitehead, A. N. (reference lacking)

(22) L. von Bertalanffy, "The Relativity of Categories," General System Theory (Braziller, New

York, 1968), p. 235.

(23) J. Pearce, The Crack in the Cosmic Egg (Pocket Books, New York, 1973), p. 4, 16.

(24) R. Lefever, J. Chem. Phys. 49, 4977 (1968).

(25) M. Herschkowitz-Kaufman and G. Nicolis, J. Chem. Phys. 56, 1890 (1972).

(28) J. Gmitro and L. Scriven, in K. Warren (ed.), Intracellular Transport (Academic , New York,

1966), p. 221.

(29) N. Kopell and L. Howard, Studies App. Math. 52, 291 (1973).

(30) P. Ortoleva and J. Ross, J. Chem. Phys. 60, 5090 (1974).