Computer Simulation of Quantum Mechanics - Porthouse
The name of this file is VORTEX.TXT

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                   VORTEX DYNAMICS and WAVE-PARTICLE DUALITY

   In the file BOHR.TXT it is described how the Copenhagen Interpretation 
denies the possibility of a general-purpose computer simulation of quantum 
mechanical wave-particle duality, much as the laws of thermodynamics deny 
the possibility of a perpetual motion machine. One may of course argue along 
the lines of "Niels Bohr never said that", but what one is then left with is 
too anaemic to be worthy of further consideration.

   Such a general-purpose simulation might typically be written in Modula-2
and would have three principal modules. One of these would deal with the
generation of random numbers, and might be so contrived that the user had no
idea of how it operated. Perhaps it would dial the Internet from time to 
time to sample some of the traffic, so as to reseed itself. The second 
module would deal with the numerical solution of the appropriate 
differential equations, accessing the random number module when appropriate. 
This module would also be permanently invisible to the user. The third 
module would deal with the specification of boundary conditions, and input 
and output, and the user would be able to modify the source code of this as 
required. This three-module arrangement makes it impossible to cheat by 
passing off a cartoon as a simulation. See RAINYDAY.TXT on this website for 
an explanation of the difference between a cartoon and a simulation. It 
suffices to say that we will be playing fair.

   Here we will present an example of a computer simulation, based upon
two-dimensional point-vortex dynamics, which arguably is an example of wave-
particle duality. Perhaps this presentation will be tongue-in-cheek, but it
ought to shake the confidence of believers in the Copenhagen Interpretation
just enough. We will see.

   Alexandre Chorin proposed to model complex fluid flows such as the Von
Karman vortex street using the method of two-dimensional point-vortices. The
vortices all convect each other. In addition, they are given some Brownian
motion to represent viscosity. A cloud of point-vortices in Brownian motion
can be expected to diffuse or spread out with time, which is just the effect
that viscosity would also have. In addition, if we take a vortex sheet and
give the vortices some random motion, then we can expect to see a display of
the Kelvin-Helmholtz instability. Further, if we look at the flow downstream
of a circular cylinder from an initial condition of potential flow, then
initially we can expect to see a flow separation. This will be symmetric at
first, but then an asymmetry will appear. Soon a vortex street will form.
All this can be seen in the programs KHI.BAS and KARMAN.BAS to be found on 
this website. Chorin's method is able to deal with all this behaviour in a 
straightforward way, and it is natural to ask if something similar may be 
attempted in quantum mechanics?

   Now, the kinematic viscosity and the strength of an individual point-
vortex have the same dimensions (length squared/time). Suppose we introduce
an additional rule which says that the vortex strength is quantised, and the
value of the quantum is equal to the kinematic viscosity. In the computer
simulation, if say, we wish to create 0.123 of a quantum of vorticity in
order to satisfy the no-slip condition, what we do instead is to create one
whole quantum with 0.123 probability. This is done by taking our standard
random number generator which generates numbers in the range 0 to 1, and
testing to see if the number generated on this occasion is less than 0.123,
a procedure which does not cause embarrassment if we change the number of
computational nodes.  At low flow speeds, only a few vortices will be 
created, and these will drift along the surface of the cylinder without much
happening. At higher flow speeds, many more vortices will be created, there
will be a crisis at some point, and then we will get a flow separation and a
vortex street.

   This is quite like a description of superfluidity. If we take some liquid
helium (strictly the commonest isotope helium-4), then a cylinder dragged
through this liquid at very low speed will not leave any wake behind it
because the helium behaves as an inviscid superfluid. At higher speeds, a
wake is formed. Effectively, the kinetic energy of the cylinder goes into
warming up the liquid helium until superfluidity breaks down.

   This author is no expert on superfluidity, but is really interested in
something else. Quantisation of vorticity is a wavelike phenomenon. The 
vortex is quantised so that an integer number of standing waves can be 
fitted on any circle drawn around it. Brownian motion of vorticity is a 
particle-like phenomenon. The point-vortex itself is both a particle in the 
Newtonian or Boscovitchian sense, and by virtue of its Brownian motion it is 
also a contributor to entropy-production, and therefore a particle in the 
Prigogine sense. What we have here is a computer simulation of something 
which is both a wave and a particle. Doesn't the Copenhagen Interpretation 
forbid that?

   This offbeat example of wave-particle duality is not conclusive by any
means, but it is something to set us thinking. Our quantised vortex in
Brownian motion is exhibiting two types of behaviour which are as different
as chalk and cheese, and yet they are both quantified by the same parameter,
which is like a kinematic Planck's constant (= Planck's constant divided by
mass of helium atom?). This is truly mysterious. In WPD3.BAS, the author has
suggested that the fundamental entity is an ultraluminal oscillation driving
a tachyonic Brownian motion, this being the first attempt ever to suggest
how something can be both a wave and a particle. A shortcoming of this
explanation is that we cannot deal with spin. Instead we take for granted 
the existence of spin and the spin-statistics theorem and the Pauli 
exclusion principle, and then point out that with one fermionic field and 
one linked bosonic field, we can have a Vernam cipher in operation in our 
computer simulations in order to deal with known nonlocal behaviour. 

   Whatever the case, it is proposed that there is no Copenhagen Principle
to frustrate our efforts. The computer simulation described is a valid 
example of wave-particle duality. The simplifying feature is the 
incompressibility of the fluid, so that all signals travel at infinite speed
and we do not have relativity to worry about. Once we do have relativity in
the picture, then although we can send nonlocal communications via a Vernam
cipher, we must still worry about the question of the relativity of
simultaneity. The answer to that one is to say that in phenomena such as the
formation of the Von Karman vortex street, the same cause (one vortex 
breaking away first) is always followed by the same effect (the other vortex
breaking away second), and therefore the concept of cause-and-effect is a
tautology (echoing J.S. Mill). Cause-and-effect could then just as well be
effect-and-cause, and so we don't have to worry about the relativity of
simultaneity. This argument is not sophistry for this rare type of 
phenomenon, but it does take a while to get used to it.

   There is of course no distinction to be made between bosonic and 
fermionic vortices. At this stage, we are concerned with proposing a 
counter-example to the Copenhagen Interpretation as an epistemological
interpretation. Questions about the distinction between bosons and fermions
are ontological questions.

   One other question should be cleared up. Either a general-purpose
computer simulation of wave-particle duality is possible, or it is not. We
are assuming the validity of the Law of the Excluded Middle. Perhaps, 
though, something like our example of a computer simulation of quantised 
vorticity is that overlooked middle possibility?  Well, this author won't be 
satisfied until he has produced an orthodox general-purpose computer 
simulation, and nothing less will do. He will certainly be crowing if he 
produces that simulation. There is nothing to boast about as yet. A 
simulation of quantised vorticity is merely an intermediary step. 


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