Before going into Newtonian definition, the best way to to build intuition between arrival rate and force is actually everywhere in daily life. Please see picture below:
For those who have been in a relationship with someone, it is evident that, the higher the nagging arrival rate from the partner, the less freedom we would have. It's almost as if there's a tight leash pulling us back to where the center is. Most important of all, they seems to be memory-less of previous nagging events, and keep on nagging at a constant rate. It is in fact Poisson (nagging) process. In the example of particle ball, the equivalence can be shown as:
In Newtonian expression, it is expressed as follows:
In a particle ball where each particle collides with one another, we can assume that it will go into some form of equilibrium, with two main assumptions (1) All particles have the same expected kinetic energy, and (2) expected force (pressure) will be constant in the space. 
For those who have been in a relationship with someone, it is evident that, the higher the nagging arrival rate from the partner, the less freedom we would have. It's almost as if there's a tight leash pulling us back to where the center is. Most important of all, they seems to be memory-less of previous nagging events, and keep on nagging at a constant rate. It is in fact Poisson (nagging) process. In the example of particle ball, the equivalence can be shown as:
In Newtonian expression, it is expressed as follows:
The constant force implies that velocity spatial gradient will be constant on an expected basis (). And tells us that the randomization of arrival rate should be exponentially distributed.
It is important to note that the nagging-partner analogy is not only a joke, but rather a truth to life. In the philosophy of financial market pricing, we often benchmark valuation off comparable prices. For example, when look at BBB assets, the discount yield is often based off the BBB index. The further away from market expectation, the more difficult it is to justify it.
Another example is that when looking at stock prices, financial analysts often benchmark their targeted pricing off the quarterly communication or announcements from the companies. In a way, when an event occurs, market collectively jump to the new prices from market consensuses with varying degree of opinions. The model will suggest a "probabilistic jump to new mean", as not all market participants would agree on the new prices. In a way, the distribution is like the comet tails introduced in chapter 1:
In the next chapter, we will be superimposing the "variance/kinetic energy" with "arrival rate/velocity gradient" in a form a "jump-to-force center process" to formulate the stochastic process & PDE governing equation.
Disclaimer: All contents on this website have not been peer-reviewed, and it may contain errors. It is strictly the author’s personal ideas that do not reflect any organization’s view. Furthermore, it is in disagreement with generally accepted practice of Brownian process. The proposal is original to the best of author’s knowledge. The author also reserves all the rights to the contents, including but not limited to, any copy rights, rights to modify and remove any statements, and the right to distribution. Any application of the model is used at the readers own risks.
No comments:
Post a Comment