# Randomness of submolecular phenomena

Why do models of submolecular phenomena involving randomness work? Do these phenomena appear random to other submolecular particles?

For example, people can use Einstein-Smoluchowski to characterize ion diffusion through solutions: this model is based on the assumption of a random walk. Do ions in a solution view the movements of other ions as random?

• Are you looking for a theoretical proof (if so, look up Mori-Zwanzig theory), or a qualitative explanation? – alarge Dec 6 '14 at 22:43
• Both would be appreciated. – vrume21 Dec 7 '14 at 21:10

## 1 Answer

yes, the equations of motion with collisions behave chaotically and from an observers point of view behave randomly. What can be shown is that there is some structure in that chaos, for instance, the distribution of energies among the molecules follow the Maxwell-Boltzmann distribution. This probability distribution indicates which speeds are more likely: a particle will have a speed selected randomly from the distribution, and is more likely to be within one range of speeds than another. The distribution depends on the temperature of the system and the mass of the particle.

• The answer does not answer the question of "why", but rather blindly uses statements of very limited scope (equilibrium thermodynamics) that at best might be applied to explaining some of the consequences, but as I understand the question, certainly not the underlying reasons. This is particularly unfortunate as the question is asking about the processes themselves, modeled with Langevin equations etc. which are manifestly non-equilibrium. What is essential here are correlations and time scales, and the answer in its present state does not even mention these. – alarge Dec 6 '14 at 23:14
• @alarge I answered these "Do ions in a solution view the movements of other ions as random?" "Do these phenomena appear random to other submolecular particles?", if you think there are questions that I did not noticed, I encourage you to write an answer to them. – Wolphram jonny Dec 6 '14 at 23:20
• the why too, it is because the solutions ar chaotic. – Wolphram jonny Dec 6 '14 at 23:21
• Q: "Why do models of submolecular phenomena involving randomness work?" A: "Because the phenomena are chaotic". This is hardly a satisfying explanation even if you meant chaotic to be interpreted in the technical sense of the word (rather than as a synonym for random as one might easily read it). Furthermore, the arguments of your answer apply equally well for solids (they, too, obey the Maxwell-Boltzmann velocity distribution). Given this, are you suggesting that the movements of the atoms of a solid appear random to one another? – alarge Dec 6 '14 at 23:42
• No, you reversed the logic, I pointed out that the chaotic (in the formal sense) solutions look random, but even if so they still have some structure, like the Maxwell-Boltzmann velocity distribution, instead of being uniform. And that is important when you model a random walk. – Wolphram jonny Dec 6 '14 at 23:45