Chapter 3 - Variance + Arrival = Kinetic + Potential Energy

In previous chapter, we define the arrival rate as velocity gradient, or the kinetic energy with respect to time, i.e., $dK/dt=-\lambda K$. It is telling us that the kinetic energy change per unit time from force field is $\lambda K$. In a traditional Heat diffusion PDE, the term $u$ is the kinetic energy function. Since the maximum velocity is observed at the center (per Newtonian rule) and Dirac Delta function $\delta$ is the unit kinetic energy function, the governing Reaction-Diffusion PDE can be expressed as follows -- (a) Diffusion term (${\sigma }^2=v^{2}dt$) + (b) Potential energy adjustments from where the velocity is observed.

$$\frac{\mathrm{\partial <!--\; \partial \; -->}u}{\mathrm{\partial <!--\; \partial \; -->}t}=\frac{{\sigma }^2}{2}\frac{{\mathrm{\partial <!--\; \partial \; -->}}^{2}u}{\mathrm{\partial <!--\; \partial \; -->}{x}^2}+\mathrm{\lambda <!--\; \lambda \; -->}(\delta -u)$$ The equivalence (jump-to-coordination origin) stochastic process is:
$$dX=\sigma d\xi -XdJ$$ Or, more generically, the diffusion term should be the expected local kinetic diffusion (in the particle ball system, it is in the form of Hook's Law):
$${(dx)}^2=({\sigma }^2-\lambda x^2)dt={\sigma }_x^{2}dt$$
This chapter is intended to be very short and only lay out the key formulations in the particle ball system to assist the readers to go through more detailed derivations in the Paper. In essence, the derivation assumes that force is applied on a probabilistic basis, instead of tracking each individual particle. This is rather unconventional but necessary in order to reduce the complexity of the formulation to describe the collective particles movements. The conclusion is rather simple --- diffusion should be based on local kinetic energy, adjusted by energy potential field. And most importantly, this is a stochastic process satisfying Newtonian energy conservation constraints.

Once we follow the rule of energy conservation, various shape of particle ball system can be formed, as long as we define the potential energy adjustment terms. For example, for a system with Dual minimum potential energy center, the Dirac delta term can be replaced with two terms each with 50/50 probability to model a dual core system such as below:

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