Order for Free?
Assuming any complicated system of interconnected parts can be broken down into both deterministic and stochastic interactions, is there a way to quantify and qualify all salient aspects of the system? Salient is a tricky word, because it implies some sort of functionality or significance, which are only meaningful in the context of a human defined ontology. It seems that there should a way to define such systems, and search for patterns in the genesis of other patterns.
For example, consider the following system of ODE's:
x' = x^2 - y^2
y' = x^2-x+xy
where x' and y' are dx/dt and dy/dt respectively. The Jacobian matrix of this system looks as follows:
A =[ (2x) (-2y)
(2x-1 +y) (x) ]
The fixed points in this system occur where the nullclines intersect, or in other words where
0 = x^2-y^2
0 = x^2-x+ xy
intersect. This occurs at the points (1/2,1/2) and (-1/2,-1/2) in the x,y plane. The Jacobian evaluated at these points is A(1/2,1/2) = [1,-1;1/2,1/2] and A(-1/2,-1/2) = [-1,1;-5/2,-1/2]. For (1/2,1/2) the trace is 3/2 and the determinant is 1. For (-1/2,-1/2) the trace is -3/2 and the determinant is 3. This suggests that the first fixed point is unstable and the second is stable, with the dynamics of the two points being spirals (since the discriminant is negative in both cases). Hence the long-term behavior of the system will eventually approach (-1/2,-1/2) as long as it does not start exactly at (1/2,1/2) with no noise.
This tends to suggest that such a system would destroy information over time. In other words, if the initial conditions contain important information (say, the position in two dimensional space of some particle whose time evolution of its x and y coordinates is dictated by the previous system of equations) that information is eventually transformed into (1/2,1/2) after a long enough period of time. The system might transition through some interesting intermediate locations in phase space, but it will eventually end up at the stable fixed point.
Could this be an example of order for free? It is hard to say. It all comes down to what order means. If in this case order means a reduction in information, then yes, this system definitely reduces the information encoded in the initial conditions into the fixed point. But it also transforms that information into a trajectory through space-time that asymptotically reaches the fixed point, and different initial conditions could have wildly different trajectories. It would be interesting to see if the distribution of trajectories given some uncertainty in initial conditions adds any more information to the system...e.g. nearby trajectories tend to be like one another, or, on the other hand, are radically different.
Plus, adding uncertainty to the equations themselves could present some interesting results. In that case the system would never reach a total equilbrium, but would probably oscillate around a stochastic equilibrium.
Hopefully in my next posts I will have some more interesting thoughts with respect to more complicated structures...here are some rough ideas:
-multiple fixed points (or steady-states)
-how to perturb a system to transfer between fixed points (or steady-states)
-in the case of polynomial ODE's, the importance of the order of the equations
-connectivity (for systems with greater number of nodes)
-systems where each element has multiple dimensions (e.g. interactions between particles in space, each particle has three dimensions which describe its position)
-the importance of stochasticity and types of stochasticity (PDE, Monte Carlo, etc...)
-how does information (and perhaps entropy) play a role...and most importantly...
-how can a computer recognize a novel and useful phenomenon and classify it as something new and significant?
-and then, how can that computer take that information and apply it to both information it already has available as well as new information presented to it?
For example, consider the following system of ODE's:
x' = x^2 - y^2
y' = x^2-x+xy
where x' and y' are dx/dt and dy/dt respectively. The Jacobian matrix of this system looks as follows:
A =[ (2x) (-2y)
(2x-1 +y) (x) ]
The fixed points in this system occur where the nullclines intersect, or in other words where
0 = x^2-y^2
0 = x^2-x+ xy
intersect. This occurs at the points (1/2,1/2) and (-1/2,-1/2) in the x,y plane. The Jacobian evaluated at these points is A(1/2,1/2) = [1,-1;1/2,1/2] and A(-1/2,-1/2) = [-1,1;-5/2,-1/2]. For (1/2,1/2) the trace is 3/2 and the determinant is 1. For (-1/2,-1/2) the trace is -3/2 and the determinant is 3. This suggests that the first fixed point is unstable and the second is stable, with the dynamics of the two points being spirals (since the discriminant is negative in both cases). Hence the long-term behavior of the system will eventually approach (-1/2,-1/2) as long as it does not start exactly at (1/2,1/2) with no noise.
This tends to suggest that such a system would destroy information over time. In other words, if the initial conditions contain important information (say, the position in two dimensional space of some particle whose time evolution of its x and y coordinates is dictated by the previous system of equations) that information is eventually transformed into (1/2,1/2) after a long enough period of time. The system might transition through some interesting intermediate locations in phase space, but it will eventually end up at the stable fixed point.
Could this be an example of order for free? It is hard to say. It all comes down to what order means. If in this case order means a reduction in information, then yes, this system definitely reduces the information encoded in the initial conditions into the fixed point. But it also transforms that information into a trajectory through space-time that asymptotically reaches the fixed point, and different initial conditions could have wildly different trajectories. It would be interesting to see if the distribution of trajectories given some uncertainty in initial conditions adds any more information to the system...e.g. nearby trajectories tend to be like one another, or, on the other hand, are radically different.
Plus, adding uncertainty to the equations themselves could present some interesting results. In that case the system would never reach a total equilbrium, but would probably oscillate around a stochastic equilibrium.
Hopefully in my next posts I will have some more interesting thoughts with respect to more complicated structures...here are some rough ideas:
-multiple fixed points (or steady-states)
-how to perturb a system to transfer between fixed points (or steady-states)
-in the case of polynomial ODE's, the importance of the order of the equations
-connectivity (for systems with greater number of nodes)
-systems where each element has multiple dimensions (e.g. interactions between particles in space, each particle has three dimensions which describe its position)
-the importance of stochasticity and types of stochasticity (PDE, Monte Carlo, etc...)
-how does information (and perhaps entropy) play a role...and most importantly...
-how can a computer recognize a novel and useful phenomenon and classify it as something new and significant?
-and then, how can that computer take that information and apply it to both information it already has available as well as new information presented to it?
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