Download about entropy in psoup

PSoup
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NTF About Entropy in PSoup
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Garvin H Boyle
Dated: 080424
Revised: 150108
ABOUT ENTROPY IN PSOUP
A comparison between a PSoup system and a thermodynamic system.
IN THERMODYNAMICS
A thermodynamic system is a container
filled with an ideal gas. The container
insulates the gas from outside
influences. It provides a nonaccelerating physical frame of
reference for the location of all
particles of the gas, and it allows no
heat or sound or other kind of energy to
pass from the gas to the external
environment, or from the environment
to the gas.
In a thermodynamic system there is a
very large number of particles. All are
spherical and have the same mass and
size, and therefore the same density
and the same inertia.
The state of each particle is
characterized by six variables, which
are the x, y and z location coordinates
and the vx, vy and vz velocity
components. The location coordinates
are limited in value to the interior of
the container. The velocity
components are not limited.
IN PSOUP
In PSoup a system is a bowl of PSoup
filled with mud, algae and bugs. In a
closed system, no energy can enter or
leave the system, and there are no
outside influences. This discussion
only addresses closed systems, until
otherwise stated.
In a bowl of PSoup there is a modest
number of bugs. All have the same
ability to mutate at the rate of one
standard mutation per generation.
The state of each bug is characterized
by a number of variables, which are x
and y location, age, energy, and the
Palmiter genotype.
The Palmiter genotype of a bug is
characterized by eight genes. The
values of the genes are positive powers
of 2, but in practice are limited to the
range which can be modeled on a
computer.
The motion of each particle in the
The behaviour of each bug is
container is determined by the laws of determined by the characteristics of the
physics, such as conservation of energy bowl (size, wrap), the rules of
and momentum.
interaction (collisions
allowed/disallowed), and the rules
PSoup
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NTF About Entropy in PSoup
encoded in RAT, RET, DAT, DET,
EPM and EPB.
The motion of each particle is
The motion of each bug is pseudodeterministic. Due to the very large
random, being driven by a pseudonumber of particles, statistical methods random number generator, and
are used to discuss average behaviour
moderated (biased) by the effects of the
and/or probability distributions.
Palmiter genes. Statistical methods are
used to discuss search patterns.
The state-space of a particle is a 6-D
The state space of a bug is a 12-D
space. This space has limited extent in space. This space has limited extent in
three location dimensions, and
four dimensions, and quasi-unlimited
unlimited extent in the other three
positive extent (overlooking the
velocity dimensions.
computer limitation) in the other eight.
A particle can occupy any position in
A bug can occupy any position in the
its state space as time progresses.
first four dimensions of its state space,
but has a fixed Palmiter genotype
which does not change over time.
All dimensions are relevant to the
All dimensions are relevant to the
moment-by-moment trajectory of the
second-by-second trajectory of the bug
particle through its state space, but only through its state space, but only the
the velocity-related dimensions are
Palmiter genes are relevant to the longrelevant to the large-scale
term patterns of evolution. [Is this
characteristics of the thermodynamic
true?] These patterns of evolution are
system; characteristics such as
moderated by relative efficiencies (as
temperature, pressure, internal energy. affected by bowl configuration).
These characteristics are moderated by
speed.
Restrict our interest to the 8-D space of
Restrict our interest to the 3-D space of Palmiter genes. Call it genospace.
velocities. Call it p-space.
Statistical techniques involving the law Statistical techniques involving the law
of large numbers (and reversion to the of large numbers are applicable
mean) are applicable because each
because each bug experiences at least
particle experiences a very large
800 (=RAT) random interactions
number of collisions each second. We between its Palmiter genotype and the
can therefore consider the particle
environment in each generation.
trajectory to be random in nature, but
biased.
A collection of ideal gas molecules A population of bugs would appear as a
would appear as a cloud of realdiscrete-valued cloud of points in the state
PSoup
valued points in the state-space of a
single particle, i.e. in p-space.
The speeds of the particles in an
ideal gas are distributed according
to the Maxwell-Boltzman
distribution, with an average speed
of ‘d’. Therefore the cloud would
have maximum density at a point
‘d’ units (measured using
Pythagorean distance function) from
the origin, forming a hollow sphere.
The value of the parameter ‘d’ is
determined by the energy and
volume.
The container does not value speed
along one dimension above the
other dimensions. The cloud is
symmetrical about the origin.
What does equilibrium look like? A
thermodynamic system in a state of
equilibrium has a cloud of realvalued points, in the 3-D p-space,
that is a hollow symmetrically
formed sphere of radius ‘d’ centred
at the origin, which can be derived
from knowledge of the volume and
energy of the system.
While the individual particles move
about in the cloud along a
continuous trajectory as time goes
by, the shape and density
distribution of the cloud does not
change, except for momentary
fluctuations.
A system which is far from
equilibrium, i.e. a system that does
not have the characteristic
distribution in state space, will alter
itself, will reshape itself, to have the
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NTF About Entropy in PSoup
space of a single bug, i.e. in genospace.
The genotypes in a bowl of PSoup at
equilibrium have an unknown distribution,
but they are tightly grouped near their
average genotype.
If we project the points which represent
the 8-D genotype onto the 7-D plane of
the unit phenotypes, the cloud should have
a maximum density near the ideal
phenotype. The ideal phenotype will be
determined by the bowl’s configuration.
The bowl may (does) value gene strength
along one dimension above the other
dimensions. The cloud is localized about
an arbitrary average point.
What does equilibrium look like? A bowl
of PSoup in a state of equilibrium has a
cloud of discrete-valued points, in the 7-D
phenospace, that is (symmetrically?)
formed about the ideal phenotype, which
can not (yet) be derived theoretically from
the bowl’s configuration.
While the descendants of each individual
bug move about in the cloud along a
discrete-valued, and possibly bifurcated,
trajectory as generations go by, the shape
and density distribution of the cloud does
not change, except for momentary
fluctuations.
A population which is far from
equilibrium, i.e. a population that does not
have the characteristic distribution in state
space, will alter itself, will reshape itself,
to have the expected distribution about the
PSoup
expected distribution about the
origin with radius ‘d’.
Entropy is a state variable in
thermodynamics.
That is to say, the difference in the
entropy S between any two
equilibrium states of the
thermodynamic system, call them a
and b, is the same, no matter what
paths you follow to get from a to b.
In terms of p-space, suppose we
have a state of equilibrium
determined by energy Ea. A stable
cloud has formed with radius da.
An outside agency causes
disequilibrium by injecting an
amount of energy such that the total
energy is now Eb. The cloud moves
to reform itself with a radius db. No
matter what trajectory the particles
of the cloud take to reform the
cloud, the change in entropy is the
same.
Entropy increases as the system
approaches state b. The speeds still
follow the M-B distribution, with
some close to zero and some very
high values, so the effect is to
reduce the density of points in state
space.
Therefore, entropy varies inversely
as the density of points in state
space.
The entropy of a thermodynamic
system in equilibrium is defined as
Q/T where Q is the internal energy
and T is the temperature. The units
of measure are Joules/degree
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NTF About Entropy in PSoup
ideal phenotype.
Is there an ‘entropy-analogy’ state variable
in PSoup?
In terms of phenospace, suppose we have
a state of equilibrium determined by bowl
Ba. A stable cloud has formed about the
ideal phenotype pa. An outside agency
causes disequilibrium by altering the bowl
configuration such that we now have
configuration Bb. The cloud moves to
reform itself about the phenotype pb. No
matter what trajectory the genetic lines of
the bugs follow to reform the cloud, the
change in this ‘entropy-analogy’ must be
the same.
Any valid ‘entropy analogy’ should then
vary inversely as the density of the cloud
in state space.
I have no idea what the units of measure
would be for any ‘entropy-analogy’ in
PSoup. Something to think about.
PSoup
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NTF About Entropy in PSoup
Kelvin.
For a given value of E (energy)
For a given bowl configuration there are
there are many shapes of clouds that many shapes of clouds possible, but only
are possible, but only one is
one is centred on the ideal phenotype, and
consistent with the Maxwell speed
has the right size.
distribution. That is the signature of
an equilibrium state.
Thermodynamic
In a bowl of PSoup which lacks a C1 penalty, the
systems do not suffer magnitudes of the strengths of genes can increase without
the problem
bounds. The population tends to occupy the same volume
described for bowls in genospace (I believe) but will move further and further
of PSoup.
from the origin. The projection against the 7-D
phenospace will decrease in size and the density of points
will increase (in accordance with a decrease in entropy?).
Such a bowl is never in equilibrium.
The addition of the C1 penalty (large gene strengths cause
use of extra energy per move) causes the strengths to have
a maximum value, and causes the bowl to reach
equilibrium.
What causes motion in p-space?
What causes motion in genospace?
Search pattern differences and
If a dynamic system is in
efficiency gradients cause conformity
disequilibrium, then the distribution of with the ideal phenotype. Those
points in p-space is not Maxwell
furthest from the ideal phenotype will
distribution conformant. I.e. the
suffer natural selection due to reduced
distribution does not conform to the
efficiency. Efficiency gradients cause
expected values, statistically.
movement towards the ideal
phenotype.
The mathematical phenomenon of
‘reversion to the mean’ induces
What causes the outward motion, away
conformity.
from the origin? The cloud contains
one bug closest to the origin, and one
This is the phenomenon that drives the which is the furthest away. Both are
increase of entropy.
subject to random mutations away
from/towards the ideal phenotype. The
offspring of the closest will suffer the
strongest selection due to low
efficiency. Efficiency gradients cause
motion away from the origin, as well.
PSoup
The state space of the particles can be
partitioned into sub-spaces of equal
size that tile the space. If the subspaces are indexed by i, and there are K
sub-spaces, then we can count the
number of particles in each such subspace. Represent those counts by the
variables xi. Then, the K-tuple (x1, ...
,xK) is called a macro-state of the
system, which we can denote as X(x1,
... ,xK), or just X.
The thermodynamic formula for
entropy associated with macro-state X
that connects thermodynamics to
probability theory is S = k ln(Ω(X))
where k is the dimensionless
Boltzmann constant, and Ω(X) is called
the disorder parameter. Let N be the
total number of particles. Ω(X) is the
count of the number of ways that the
particles can be arranged consistent
with the definition of macro-state X.
Ω(X) can be expressed as a
combinatorial formula as (N!)/Π[xi!].
Using Stirling’s approximation for
ln(N!) this can be expressed as S = - k
Σ Pi* ln(Pi); where Pi = xi/N.
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NTF About Entropy in PSoup
The state space of the bugs can be
partitioned into sub-spaces of equal
size that tile the space. If the subspaces are indexed by i, and there are K
sub-spaces, then we can count the
number of bugs in each such sub-space.
Represent those counts by the variables
xi. Then, the K-tuple (x1, ... ,xK) is
called a macro-state of the system,
which we can denote as X(x1, ... ,xK),
or just X.
Is there a similar formula in PSoup, an
ABM, that produces a ‘state variable’
we might call PSoup Entropy. Easily
done.
The formula for entropy associated
with macro-state X in PSoup that
connects PSoup to probability theory is
S = f ln(Ω(X)) where f is some scaling
constant, and Ω(X) is the count of the
number of ways that the bugs can be
arranged consistent with the definition
of macro-state X. Let N be the total
number of bugs. Ω(X) can be
expressed as a combinatorial formula
as (N!)/Π[xi!]. Using Stirling’s
approximation for ln(N!) this can be
expressed as S = - k Σ Pi* ln(Pi); where
Pi = xi/N.