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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 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 2 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 3 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 4 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 5 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. 6 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.