So, anyway, where were we? Oh, that's right: we were in the middle of you guys telling me how natural selection (not a force) is a process found in nature like photosynthesis and what not.
Now, math is not my stong suit (just ask Telerion, he'll tell you), so you'll have to bear with me on this part. Is not the preferred means of describing the processes of nature differential equations?
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Originally posted by FreakyKBHThere's nothing fundamental about differential equations, necessarily (in the sense that there not being a DE describing natural selection does not mean that it can't be a natural process). They are a mathematical construct, but are useful in natural sceince because the natural scientist abstracts some property of the natural process and compares its behaviour with that of a solution to an initial value/boundary value problem, where the boundary conditions have been chosen as part of the process of abstracting properties of the natural process.
So, anyway, where were we? Oh, that's right: we were in the middle of you guys telling me how natural selection (not a force) is a process found in nature like photosynthesis and what not.
Now, math is not my stong suit (just ask Telerion, he'll tell you), so you'll have to bear with me on this part. Is not the preferred means of describing the processes of nature differential equations?
For example, I could consider the vibrations in the head of a circular drum when I hit it. In this case, I want to find a function h of the radial distance from the centre of the drum, the angular distance around the drum, and the time since hitting -- h is the height of the drum (measured from the height of a nonvibrating drum) at any given point in space and time. I thus start abstracting things from the physical, drum-hitting situation (ie, I abstract things about the natural process of vibration) to build a mathematical model which will predict the shape and size of the drum's vibrations. Maybe it will explain why, harmonically, drums sound like ass.
The first thing I assume is a boundary condition saying the drum stays put together -- ie h = 0 for all radial distances greater than or equal to the radius of the drum. Then I assume that no matter the t, h is always finite. Furthermore, I have to hit it, so perhaps when t = 0, h is some given function of radial distance and angle defined by the hitting. In fact, hitting seems radially symmetrical about whatever point I hit it, so this initial condition needn't depend on the angle. Finally, I might stipulate that h is solution of the wave equation. Given all of the conditions, I can solve this and no everything about the vibrations, given these assumptions.
It's a powerful technique; it tells us, in this instance, why drums are harmonically unpleasant, for instance. However, the understanding of natural processes we gain is only as good as our assumptions. For instance, solving the BVP I set up above will tell us that the vibrations never dampen, because the wave equation includes no dampening term. It's still a good model, for limited amounts of time, but it illustrates the fact that maths in natural science is not like maths alone because it in no way perfectly describes the phenomena in question, and setting up and solving differential equations is only as useful as the assumptions about the natural process which they represent.
Not all natural processes lend themselves to good modelling by DEs. Some natural processes are basically discrete, and this can lead to very complicated behaviour that a DE would miss (compare the logistic differential equation to its discrete counterpart, and their respective uses in population models). Therefore, asking what DE describes natural selection is not necessarily very sensible, especially since DEs (like you just saw with the wave equation) model specific phenomena under narrow ranges of conditions. Natural selection is a very broad and varied process, and requires much more complex modelling (potentially involving DEs or systems of them).
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Originally posted by FreakyKBHNo. Natural processes are given us by nature. DEs come from, depending on your stance in an argument irrelevant here, either some Platonic realm of consequences of mathematicalaxioms or from the human imagination. It happens to be that solutions to some DEs match some of the behaviour of quantities in nature which change as a result of natural processes. Scientists use this correspondence to make easy-to-manipulate mathematical simplifications of the processes, in order to predict and possibly find new properties of the process by observing properties of the model.
Are natural processes otherwise defined by DE's?
(For example, the wave equation in one dimension is a model of, say, a vibrating string. It gives rise to the notion of harmonics, even though the use of the model does not depend on harmonics.)
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Originally posted by FreakyKBHNot all of them, and imperfectly. I should also stress that a perfect model is almost never a desirable one; a model which is as simple as possible and preserves the interesting features of the phenomenon is desirable.
Let me rephrase that. Can the known natural processes be described utilizing DE's?
If something can be described using a DE, it is not necessarily a natural process. If something cannot be described using a DE, it is not necessarily not a natural process.
There are also natural processes for which DEs have not been used, but for which DEs may be useful in some way, presumably.
Originally posted by FreakyKBHhere's a decent starting point to begin the explaination of abiogenesis:
Let me rephrase that. Can the known natural processes be described utilizing DE's?
diffusion equation
A partial differential equation that models the statistics or distribution of many particles undergoing Brownian motion, or the diffusion of one fluid in another fluid, or the diffusion of heat
thanks for reminding me.
Originally posted by frogstompResults, right? I was referencing action.
here's a decent starting point to begin the explaination of abiogenesis:
diffusion equation
A partial differential equation that models the statistics or distribution of many particles undergoing Brownian motion, or the diffusion of one fluid in another fluid, or the diffusion of heat
thanks for reminding me.