PSoup 1 NTF About Entropy in PSoup

NOTE TO FILE

Garvin H Boyle

Dated: 080424

Revised: 150108

ABOUT ENTROPY IN PSOUP

A comparison between a PSoup system and a thermodynamic system.

IN THERMODYNAMICS / IN PSOUP
A thermodynamic system is a container filled with an ideal gas. The container insulates the gas from outside influences. It provides a non-accelerating 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 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 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. / 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 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. / 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 container is determined by the laws of physics, such as conservation of energy and momentum. / The behaviour of each bug is determined by the characteristics of the bowl (size, wrap), the rules of interaction (collisions allowed/disallowed), and the rules encoded in RAT, RET, DAT, DET, EPM and EPB.
The motion of each particle is deterministic. Due to the very large number of particles, statistical methods are used to discuss average behaviour and/or probability distributions. / The motion of each bug is pseudo-random, being driven by a pseudo-random number generator, and moderated (biased) by the effects of the Palmiter genes. Statistical methods are used to discuss search patterns.
The state-space of a particle is a 6-D space. This space has limited extent in three location dimensions, and unlimited extent in the other three velocity dimensions. / The state space of a bug is a 12-D space. This space has limited extent in four dimensions, and quasi-unlimited positive extent (overlooking the computer limitation) in the other eight.
A particle can occupy any position in its state space as time progresses. / A bug can occupy any position in the 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 moment-by-moment trajectory of the particle through its state space, but only the velocity-related dimensions are relevant to the large-scale characteristics of the thermodynamic system; characteristics such as temperature, pressure, internal energy. These characteristics are moderated by speed.
Restrict our interest to the 3-D space of velocities. Call it p-space. / All dimensions are relevant to the second-by-second trajectory of the bug through its state space, but only the Palmiter genes are relevant to the long-term patterns of evolution. [Is this true?] These patterns of evolution are moderated by relative efficiencies (as affected by bowl configuration).
Restrict our interest to the 8-D space of Palmiter genes. Call it genospace.
Statistical techniques involving the law of large numbers (and reversion to the mean) are applicable because each particle experiences a very large number of collisions each second. We can therefore consider the particle trajectory to be random in nature, but biased. / Statistical techniques involving the law of large numbers are applicable because each bug experiences at least 800 (=RAT) random interactions between its Palmiter genotype and the environment in each generation.
A collection of ideal gas molecules would appear as a cloud of real-valued points in the state-space of a single particle, i.e. in p-space. / A population of bugs would appear as a discrete-valued cloud of points in the state space of a single bug, i.e. in genospace.
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 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 container does not value speed along one dimension above the other dimensions. The cloud is symmetrical about the origin. / 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 thermodynamic system in a state of equilibrium has a cloud of real-valued 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. / 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 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 expected distribution about the origin with radius ‘d’. / 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 ideal phenotype.
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. / Is there an ‘entropy-analogy’ state variable in PSoup?
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. / 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.
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. / Any valid ‘entropy analogy’ should then vary inversely as the density of the cloud 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 Kelvin. / I have no idea what the units of measure would be for any ‘entropy-analogy’ in PSoup. Something to think about.
For a given value of E (energy) there are many shapes of clouds that are possible, but only one is consistent with the Maxwell speed distribution. That is the signature of an equilibrium state. / For a given bowl configuration there are many shapes of clouds possible, but only one is centred on the ideal phenotype, and has the right size.
Thermodynamic systems do not suffer the problem described for bowls of PSoup. / In a bowl of PSoup which lacks a C1 penalty, the magnitudes of the strengths of genes can increase without bounds. The population tends to occupy the same volume in genospace (I believe) but will move further and further 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?
If a dynamic system is in disequilibrium, then the distribution of points in p-space is not Maxwell distribution conformant. I.e. the distribution does not conform to the expected values, statistically.
The mathematical phenomenon of ‘reversion to the mean’ induces conformity.
This is the phenomenon that drives the increase of entropy. / What causes motion in genospace?
Search pattern differences and efficiency gradients cause conformity with the ideal phenotype. Those furthest from the ideal phenotype will suffer natural selection due to reduced efficiency. Efficiency gradients cause movement towards the ideal phenotype.
What causes the outward motion, away from the origin? The cloud contains one bug closest to the origin, and one which is the furthest away. Both are 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.
The state space of the particles can be partitioned into sub-spaces of equal size that tile the space. If the sub-spaces are indexed by i, and there are K sub-spaces, then we can count the number of particles 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. / The state space of the bugs can be partitioned into sub-spaces of equal size that tile the space. If the sub-spaces 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.
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. / 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.