Tuesday, March 01, 2016

The Mathematics of the Spongeam.

The Spomgeam: An object full of holes
The following is a mathematical impressionist’s sketch of evolution and in particular of that enigmatic object I keep talking about, the “spongeam”; an object that is a necessary condition for evolution. As usual I make known my caution about claiming the existence of this object (See here for example), but in this post I am proceeding under the assumption that it exists.

When it comes to writing an all embracing mathematical equation for an object like evolution we are likely to remain in the land of make believe for a long while; the mathematics, at least my mathematics, is never going to be anywhere near complex enough. Moreover, there is one aspect of this affair I have found difficult to render and that is what is very probably the non-linear nature of evolution. Non-linearity arises because organic forms influence and create environments and therefore this in turn will affect evolution, very likely with non-linear feedback. (More about this later) 

But not to be out done….. as the saying goes: If you can’t solve a difficult problem try solving a simple problem like it. Well that’s what I’m doing here; in fact I’ve simplified things to the point of it really only being a metaphor that helps us to picture evolution in a very abstract way. So what’s below can’t be taken too seriously; it’s still very much a toy town model and is likely to remain so. Even so, I find the model below does help me think about conventional evolution abstractedly and perhaps may even pave the way for more sophisticated metaphors sharpening the focus further.

My starting assumption is that when living configurations breed their offspring vary randomly and by this I understand that each generation shifts a little in configuration space in a way that at least approximates to random walk. In using this mathematical abstraction we abstract away all the exact details of genetics and inheritance; all we assume is that a breeding organism is surrounded by a probabilistic halo in configuration space, a halo which means it can only generate offspring that on average are within a relatively short distance of its own form. One important reason for abstracting away all the conventional microbiological mechanisms of variation is that it is entirely conceivable these details are contingencies which are not a necessary condition for more general models of evolution.

If we start by assuming that a complex of causes result in variation and imply at least a pseudo random walk, then it follows that the distributions of offspring in configuration space will approximately obey diffusional laws. So let’s start with the multidimensional diffusion equation:

Equation 1

The symbol Y represents the distribution of living things in configuration space. Because configuration space has n-dimensions I’ve used the “house”2 symbol rather than the “Del”2 symbol of three dimensional diffusion. The diffusion coefficient resolves into three factors e, w and v. These symbols represent the typical step distance, the step probability and step velocity of random walk respectively (See my book Gravity and Quantum Non-Linearity for a proof of this). I’ve also thrown in the variable A to absorb any other factors we might think necessary.

In this model we imagine that living things replicate themselves with slight differences due to a multiplicity of causes (or perhaps  even acausal changes), a difference typified by the value e with a probability of w. I’m making another simplifying assumption here by assuming  w is not positionally or directionally dependent; that is, the probability of an organism generating any variant is the same for all variants within a typical distance of e. The rate at which an organism generates offspring is measured by the step velocity, v. In ordinary random walk we usually imagine a single agent walking around and generating a distribution graph that is in fact a probability curve, but in the random walk we are dealing with here the agent “walks” by replicating incremental variants in the surrounding space; in fact the parent doesn’t step at all but stays at the original position: it’s the production of offspring which does the stepping. 

So, from this model it is clear that if Y is integrated over configuration space the resultant value changes in time as it represents a changing population number; this contrasts with ordinary random walk where the distribution normalizes to unity when integrated. But we must also  remember that organisms die.  There is therefore a term missing from the right-hand side of equation 1. Viz:

Equation 2

The second term on the RHS caters for the combined effect of organism death and reproduction. The term reflects the assumption that in a population both the death rate and birth rate are proportional to population which in turn is proportional to Y. Hence the net result is the difference between a birth rate proportionality constant R and the death rate proportionality constant D. Simplifying by putting V = R – D gives:

Equation 3
Although this equation is relatively simple it hides many potential complexities in its variables. Leaving aside possible variation in e, w and v which could in fact make them vector fields, much complexity can potentially reside in the scalar field variable V, so let’s see what we can get out of V alone.

If an organism varies randomly then the destructive nature of randomness will ensure that the (overwhelming?) majority of variations are likely to be disadvantaged when it comes to reproduction and survival. Hence we expect most random variants to display a high value of D, so much so that we expect them to lead into (eventual) extinction (Hence the black holes in the picture of the spongeam). However, evolution, as we well know, depends on the conjecture that there are some walks where populations are maintained or even grow; it is along these pathways, if they exist, that it becomes possible for a population to “evolve”; that is, diffuse into new regions. These pathways must be both broad enough and traverse great enough distances in configuration space if they are to facilitate macro-evolution. It is this network of paths in configuration space that constitutes the spongeam.

Equation 3 has a form that is clearly related to the Schroedinger equation; in fact one only need insert “i” into the diffusion coefficient to get an equation with quantum behavior. In the Schrodinger equation analogue V constitutes the potential field and in equation 3 V would have a similar role. In particular, where V has a gradient it acts as kind of force which tends to drive the random walk in a particular direction. Thus I interpret the locations in configuration space where V has a gradient to be those places where a kind of “natural selection” is operating. But let’s note that natural selection is not a necessary condition for the general kind of evolution equation 3 allows. Provided V is positive in sufficiently extensive regions the diffusion of random walk gives rise to evolution even when V has no gradient. 

Each living configuration can only survive and replicate if it has the right construction and since it is the laws of physics which ultimately determine whether a configuration survives or dies then it follows that the value of V is implicit in the physical regime. Thus, if the spongeam exists to provide evolution with the requisite information then it follows that this information is implicit in physics, perhaps even physics we have yet to understand.  Hence conventional evolution depends on frontloaded physical information and this naturally raises questions about the origins of that information.

Non-Linearity: Non linearity arises in equation 3 because V is influenced by the very existence of life; that is, life is itself a significant part of the environment in which configurations survive and replicate. Mathematically this means that V is likely to be a nonlinear function of Y. Moreover, V is not going to be just a function of the local value of Y, but also the value of V elsewhere in configuration space. This is because widely diverse living configurations influence the environment of one another.  It is this complex feed-back feature, very dependent on the twists and turns of the history of life, which make it difficult to write an all embracing succinct evolution equation.

Let’s face it, equation 3 is even worse than the Drake equation: Too many unknowns make it all but insoluble, but at least it gives us a conversation piece....that is, something to talk about.

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