Chapter 1 - Variance and Kinetic Energy

Since Brownian motion is the description of particles colliding with each other, it is intuitive to assume that the variance and kinetic energy have some equivalence. Considering the observed particle colliding with surrounding particles, at each spatial location, the observed particle has 50/50 chance to bounce back (after collision) or continue to move deterministically (without collision) in the next time interval. In such way, variance can be linked to kinetic energy in the form of ${\sigma }^2=v^2\Delta t$. Graph below shows detailed derivation.

Another simpler way to look at it is simply comparing the definition of white noise term ${\mathrm{(dX)}}^2={\sigma }^{2}dt$ with the ${\mathrm{(dX)}}^2={\mathrm{(vdt)}}^2$. In a way, randomness reduces one $\Delta t$ such that time is measured by one unit of kinetic energy change, instead of measured by one unit of spatial change. The moment we start to equate variance with kinetic energy in this way, the following Stochastic equation start to show problems.
$$dx=\mu dt+\sigma dW$$ It is okay to say that the particle randomly colliding with surrounding particles with 50/50 chance. We just need to assume symmetric distribution of particles. But it is not okay to say that these particles move to a different mean $\mu dt$ without the presence of any force. Newtonian physics tells us that some form of forces, pressures, or asymmetry must occur in order for these particles to move.

The prevailing formulation says that the particle magically know where the mean is and maintain symmetry around it. The author challenges that this is flawed. When dealing with physical problems like hot gas, where kinetic energy is much higher than forces pulling particles together, the force field impact is negligible, but when dealing with diffusion like comet tails, a diffusion process with potential energy adjustment is needed.

Yes, we will be describing a self-organizing system with finite entropy, just like particle balls attracted by its own particles gravitational pull. Yes, the stochastic process proposed here satisfy Newtonian rules. And there is no reason not to apply it to model electron clouds in an atom.

Home Page >> Chapter 2
Disclaimer: All contents on this website have not been peer-reviewed, and it may contain errors.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.