quantum entanglement and information transfer

I have been reading about the EPR paradox (Einstein, Podolsky and Rosen) and the implications of quantum entanglement. You can think of entanglement like the drawing of one of two cards. The two cards are entangled in the sense that if I draw a red card while we know the other is blue. Then by checking the card I have drawn, you will know exactly what card is left, the blue card. This concept is the same when talking about two objects in entangled quantum states. If I am on the moon with a particle which may be in state 1 or 2, and you are on the earth with a different particle in state 3 or 4, but our pair of particles may only be in an exclusive combo state of either 1 and 3 or 2 and 4, then if you measure the state of your particle, I need only to ask you what your result is to determine the state of mine. HOWEVER, information can only travel at most the speed of light. SO, if we measure our particles simultaneously, then we should be able to have the possibility of obtaining a result which does not match either of the two possible states I listed before (1 and 3 or 2 and 4), since information of your particle and my particle will not be able to reach the other in time to "let the other particle know" that something has changed or that a measurement has occurred. I am not convinced that entanglement can exist at such distances. Furthermore, how does the effect of measurement influence entanglement (by measuring particles or systems, we effectively put our system into a certain state...from which it may then evolve according to the state we measure it in).

critical states

Do we live in a perpetual or constant world that is in the critical state? Is there any difference between "action at a distance" and microscopic forces influencing macroscopic properties and behavior?

If I have a ferromagnetic material near its Curie temperature and change one electron's spin direction, then it should have an effect on another electron's spin at any range from it. Certainly, the force that the change in spin of the electron I choose first influences to some degree its neighboring electrons, then they influence their neighbors and so on until they find electrons at any distance to influence a complete change in direction. The influential role of the electron I have chosen is merely a fluke of probabilities in my eyes while certain in the eyes of nature.

However, is this any different than if I were to blow a feather off a table. My lungs create a pressure change which causes a chain reaction of colliding air molecules which in the end, and along the direction from my mouth to the feather, is just events that can be described microscopically in order to affect the feather from a distance. To provide another example of microscopic local influences causing global reactions, consider social networks.

My undergraduate research advisor, Dr. Ojakangas, would tell his students about some physical law or theory, then asked, "Do you buy that? Because that's all I'm selling!" Now, when I tell some story or give a lecture, I might ask whomever I am talking to, "You buy that? That's what I'm selling!"...or something to that effect. The point is this: after Ojakangas fed us (his Mechanics II students) that line the first time, he told us that he had heard a professor of his say that (I think at CalTech). He apparently liked it, so he made a similar comment to us. I like it as well, so I make the comment to whomever cares to listen to me on occasion. I imagine that by this point, others either those who have heard Ojakangas' professor at CalTech, Ojakangas, or myself say this line have or will also say this to others. Other people who have no idea where the source of the silly line came from. This is like action at a distance. (I'm not going to claim that the source is even Ojakangas' professor; that's just as far as I know who came up with what!)

An even bigger social network analogy is that of the internet. Before the internet one person with a video of something ridiculous would only be able to show the people they knew and not too many others. Now, that video can go viral and effect millions of people who have absolutely no direct connection to that person. The internet has provided a way to make a correlation distance between people in the world near its maximum possible value, just as in the critical state of electrons in a ferromagnetic material near the Curie temperature.

Does this all mean that we live in a constant state of criticality? Where the butterfly effect really changes everything? If not everything, does it at least make great dents in the previous order that existed? Has there ever been order? I suppose when talking about correlation distances of one object or idea influencing another at a distance, we must consider correlation times. For our brief time on this earth, most of us probably won't cause the global changes in our lifetimes, but our actions now may influence the next generations in ways we would not expect. Stories your parents may have told you about their times in life may influence the way you conduct yourself. Your actions may then influence others which may lead to global implications later, like the leaders of nations deciding between good and evil.

My bet is that we live in a constant state of criticality. I think that every action now influences the current order to be reordered. Whether there is an end to the criticality, I doubt it exists. I cannot think of anything on any scale in which a scale's microscopic forces do not propagate to influence objects at a distance. However, the time scale in which to consider universal objects may need to be characteristically near infinite. Keep in mind that pockets of objects that do not seem to be influenced are part of the property which determines criticality. Nothing is globally special, no matter the amount of detail you consider.

species & fractals, linear v nonlinear

So I've been thinking the past few days about speciation and the splitting that occurs. I wonder if phylogenetically, speciation can be fractal under certain conditions.

A model which I'm imagining could at a very concrete and highly unrealistic state begin with a species splitting into three branches. Each of these three branches would then split into three branches, and so on. I doubt it would be too difficult to show or at least understand that this should inevitably lead to a fractal.

In order to arrange some realistic aspect to the model, I would then allow some or all of the three branches to not form. These instances would model extinction of species in a way. Instead of forming, then dying out, the branch just doesn't form. The rate at which this occurs would depend on how many branches are currently able to form (once a branch has divided or failed to branch at all, it would no longer be counted), then a random number of branches that would form will not. I would think this sort of model would retain fractal behavior, but certainly will not look as "nice" as without extinctions.

Another way to make the model more realistic is to allow a variable number of branches, say between 0 and 5 or so, or whatever current estimates on speciation might suggest. This part may destroy the fractal like behavior, but as I have seen in Barnsley's fern, this variable extension of the phylogeny may be fine.

I'm sure there are plenty of other ideas which may be implemented, but I think what I have listed should be reasonable for a simple model. The real question would then be, how closely does an averaging of many simulations of the model reflect the phylogeny of today's species within all higher levels of taxonomy?

Another thing I've been thinking about is linear versus nonlinear systems. Primarily, I've been thinking about why some nonlinear systems cannot be linearized, especially globally. Locally, around fixed points, nonlinear systems may be linearized by evaluating the Jacobian matrix at the fixed points. However, what restricts us from taking the nonlinear terms and calling them a new term by a change of variables? Certainly, most, if not all, cases will result in a system with more dimensions (each independent variable gets its own dimension). However, if nonlinear terms are linearly independent among the linear terms, then could a new system be generated in order to make the system easier to solve? Perhaps this is all bogus because the nonlinear terms may be formed by the linear terms, which would then be a case of ALL nonlinear terms being linearly dependent which would then disallow one to make a change of variables of the nonlinear terms in order to present a new dimension to the problem.

Another idea related to this is to explore what is needed in order to execute linearization of a system- as in constructing the Jacobian matrix evaluated at the fixed points of a system. Following along with the steps indicated by Strogatz in his book, Nonlinear Dynamics and Chaos, I found (and am assuming) that the only requirements needed for a two dimensional system with arbitrary coupling are that the transformation functions used in the change of variables need to be differentiable and have an inverse which is also differentiable.

For example, let the derivatives of x and y be x' = f(x,y) and y' = g(x,y), and let the change of variables, u and v, be u = F(x,x*) and v = G(y,y*), where x* and y* are fixed points. Rewriting functions for x and y gives: x = Finv(u,x*) and y = Ginv(v,y*), where Finv and Ginv are the inverse functions for F and G. Differentiating u and v gives: u' = F' + u' F and v' = G' + v' G. Rearranging for Finv and Ginv, then differentiating gives: x' = Finv' + u' Finv = Finv' + u' x and y' = Ginv' + v' Ginv = Ginv' + v' y. By substitution of these new x' and y' equations with the originals gives: f = Finv' + u' x and g = Ginv' + v' y. Solving these for u' and v' and then expanding them as Taylor series evaluated at the fixed points should give appropriate Jacobian matrices from which an analysis of the eigen values will determine what sort of behavior can be expected.

Accumulation

So on my flight from Salt Lake City to St. Louis (originally from Portland, OR, the connecting flight was from Salt Lake City), I was day dreaming a bit. While listening to On the Origin of Species by Charles Darwin, read by Richard Dawkins, I was watching the clouds as we sped by them. The audio book had little influence in my thoughts about accumulation at the time...every once in a while, it seemed more like chatter background for me to think about other topics. However, while watching the clouds, I began to think up a simple model for accumulation. Since I know little about how clouds really form, my mind was free to dream up something whether it be accurate or not. I'll put the comment documentation for my Matlab instance of this model below:

% Model description:

% On a rectangular space, objects move parallel to the length of the

% space. The object's speed depends upon how many units of the objects

% share the same space; this may be thought of as a density dependent

% velocity. Object units may not diverge from others once they occupy the

% same space, and therefore result in the creation of larger objects.

% Geometric spacing and packing is not taken into account in the first

% instance of this model. However, limitations on the number of objects

% occupying a space and the distribution of those excess object units may

% provide a three-dimensional conceptualization of a "super" object drift.

% -Object creation: Objects will be randomly generated based on a

% proportion of the size of the space and/or the number of units in play.

% Once created, the objects will move and may form larger objects.

% Generation of larger objects may be restricted in some instances of this

% model, resulting in "packing" limitations.

% -Velocity determination: There will be a limiting velocity that will

% dictate the lowest rate of movement. The speed will increase for those

% objects with less units.

% -Accumulation restrictions: Once a maximum accumulation of an object in

% a single space is reached, some number of objects in that space may be

% removed.

% -Geometric conception: Distribution of the units may be to place excess

% units coming into a space into an adjacent space which has the least

% number of units. This may be done by searching first the immediate sides

% of the filled space, then the space directly following the filled space.

% Radiating outward in this manner if all immediate spaces are filled may

% result in a wall or object front.

% -Predictions: In the case of no geometric consideration, there should be

% pockets of great accumulation of the digital object units. Given a long

% enough space in which to drift, these pockets should grow very large and

% therefore move at or near the minimal speed allowed. A randomly

% distributed set of these pockets will exist within the space since they

% cannot merge with each other upon reaching the minimal speed. A complete

% occupation of the space may be possible, but very unlikely.