Probability and astronomy fuse to provide interesting observations of how dynamics initial steady states are. We know from applied physics that the properties surrounding escape velocity at a certain distance, R, from a gravitation force (in this case a unit mass acceleration) are related to the orbital speed that the object must uniformly maintain. This gravitational pull provides unique dynamics when modeled through probability techniques as it is unlike most absolute and fixed probability biases. And it is also unlike proportional biases that we introduced in the gambler's ruin and survival blog note. This is because the centrifugal acceleration is inversely proportional to just the radial distance (this is where the variable symbol R comes from), from the gravitational center. We'll pick this apart later below to better understand the interconnections between the physical dynamics and the probability disturbances on the velocity direction.
First though, we see that there is increased propensity for events where one breaks above escape velocity, to continue to forever escape. This is similar to the fractal sets where we showed escapes from multidimensional domains, except here we are only looking at distance (and it's time derivatives) as a unidimensional variable. At the same time, we see that increased probabilistic dispersion about the velocity, starting tangential to the orbit, and any event situation where one breaks below the escape velocity, almost assuredly leads to a quick implosion into the center (e.g., a zero height level).
How sensitive are the results? Well the vector equations are easy to solve in either the two dimensional orbital case, or the one dimensional escape velocity case. On one side of the expression, we have gravitational acceleration times mass, divided by distance. In our blog piece here, we will set mass equal to one so that we only focus on the distance component (and generally speaking we can consider distance a proxy for anything from the growing U.S. national debt, to something intangible such as our comfort with a movie that we are watching). On the other side of the expression, we have the familiar 1/2 times mass, times the square of escape velocity. See the formulae below to describe our efforts and simplifications thusfar:
G*M*m(1)/R = ½*m(1)*v^2
G*M/R = ½*v^2
And this is related to centripetal force, which acts perpendicular to the orbital path, and is equal to the left side of the equation above, divided a second time by R.
We learn from the simulations that we will do soon, that the concept of central mean reversion fails. For example, when we roll a couple 6-sided dices and add the numbers on faces, we generally get 7 more than any other number. This is the central tendency. In applications of escape velocity, we generally do not get this central tendency. For example, the outcome of tug-of-war competitions are rarely deadlocked and result in a tie. Generally there is large movement in the middle of the rope and an ultimate victor. Or a novice rollerblader in San Francisco pushed into motion is unlikely to come to natural stop on his or her own, and is likely to either speed down for blocks and crash, or simply grab an object quickly such as a street pole or a meter. And what about the the polarized impressions we have of the investment value of things we have seen this year such as Bitcoins and Twitter? Where our unique cultural biases eventually comes into our calculation of the utility trade-off, between price momentum and investment cash flow (e.g., dividends or interest). These escape velocity examples show the complicated degree to which probability influences how unstable a "steady state" can be.
The simulations, which you can try here, shows how much effort must be put into not escaping at each turn, and how much dispersion in movement we need to have, just so that we pass through a steady state and implode (e.g., never escape). Whether in physical orbits or lifestyle choices we make with one another here on Earth, we see there is tremendous sensitivity to continuously guiding a process to its initial stationary state; else we can easily lose that steady state.
Instructions are provided on the top of this simulations webpage. And two extreme sample outputs of this are shown below, just for illustration. In each case we have 900 samples, from 30 simulations of 30 time trials. The top illustration here projects a 50% multiple on the coefficient of variation (CV) against the escape velocity, and 75% bias towards escaping at each incremental time. The bottom illustration projects a higher 150% CV multiple, and a lower 25% bias incrementally towards escaping (i.e., 75% bias toward imploding). Notice not much in the way of a steady state level, or R at one, staying that way.
On the simulation (the link again is here), you can model the appropriate variation from the options given, to appreciate the nuances embedded in situations that you currently face.
First though, we see that there is increased propensity for events where one breaks above escape velocity, to continue to forever escape. This is similar to the fractal sets where we showed escapes from multidimensional domains, except here we are only looking at distance (and it's time derivatives) as a unidimensional variable. At the same time, we see that increased probabilistic dispersion about the velocity, starting tangential to the orbit, and any event situation where one breaks below the escape velocity, almost assuredly leads to a quick implosion into the center (e.g., a zero height level).
How sensitive are the results? Well the vector equations are easy to solve in either the two dimensional orbital case, or the one dimensional escape velocity case. On one side of the expression, we have gravitational acceleration times mass, divided by distance. In our blog piece here, we will set mass equal to one so that we only focus on the distance component (and generally speaking we can consider distance a proxy for anything from the growing U.S. national debt, to something intangible such as our comfort with a movie that we are watching). On the other side of the expression, we have the familiar 1/2 times mass, times the square of escape velocity. See the formulae below to describe our efforts and simplifications thusfar:
G*M*m(1)/R = ½*m(1)*v^2
G*M/R = ½*v^2
And this is related to centripetal force, which acts perpendicular to the orbital path, and is equal to the left side of the equation above, divided a second time by R.
We learn from the simulations that we will do soon, that the concept of central mean reversion fails. For example, when we roll a couple 6-sided dices and add the numbers on faces, we generally get 7 more than any other number. This is the central tendency. In applications of escape velocity, we generally do not get this central tendency. For example, the outcome of tug-of-war competitions are rarely deadlocked and result in a tie. Generally there is large movement in the middle of the rope and an ultimate victor. Or a novice rollerblader in San Francisco pushed into motion is unlikely to come to natural stop on his or her own, and is likely to either speed down for blocks and crash, or simply grab an object quickly such as a street pole or a meter. And what about the the polarized impressions we have of the investment value of things we have seen this year such as Bitcoins and Twitter? Where our unique cultural biases eventually comes into our calculation of the utility trade-off, between price momentum and investment cash flow (e.g., dividends or interest). These escape velocity examples show the complicated degree to which probability influences how unstable a "steady state" can be.
The simulations, which you can try here, shows how much effort must be put into not escaping at each turn, and how much dispersion in movement we need to have, just so that we pass through a steady state and implode (e.g., never escape). Whether in physical orbits or lifestyle choices we make with one another here on Earth, we see there is tremendous sensitivity to continuously guiding a process to its initial stationary state; else we can easily lose that steady state.
Instructions are provided on the top of this simulations webpage. And two extreme sample outputs of this are shown below, just for illustration. In each case we have 900 samples, from 30 simulations of 30 time trials. The top illustration here projects a 50% multiple on the coefficient of variation (CV) against the escape velocity, and 75% bias towards escaping at each incremental time. The bottom illustration projects a higher 150% CV multiple, and a lower 25% bias incrementally towards escaping (i.e., 75% bias toward imploding). Notice not much in the way of a steady state level, or R at one, staying that way.
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