Is Brownian motion truly random?

Question

We say that Brownian motion is caused by the random collisions of particles. But let's consider an ionized gas; in that case, there's a nonzero net charge on the atom. Doesn't this mean the electrostatic force determines the paths the gas ions take, and hence their motion is predictable rather than random? Even with molecules, there's still the Van der Waals force and isn't that what determines molecular motion? So how can Brownian motion be random if particle motion is predictable?

Answer 1

The standard mathematical model for Brownian motion is the continuous limit of a random walk, known as a Wiener process. This model may or may not be a good fit to the observed behaviour of physical particles in a particular situation. There are certainly many scenarios in which the Brownian motion model is not a good fit.

Note that one of the assumptions of the Brownian motion model is that it is modelling the behaviour of a relative massive particle surrounded by a large number of much lighter particles. As such, it is not usually applied to individual atoms. It also assumes spherical symmetry, and once you introduce electrostatic forces you have probably broken this assumption.

Answer 2

Treating something as a random variable is a tool that's used to predict the behavior of a large number of particles in bulk, when it would be practically impossible (but possible in principle) to track all of the motions individually. And the random variable model makes good predictions in these cases because statistical facts like "laws of large numbers" and "central limit theorem" mean that particles in bulk approach Gaussian distributions (or a distribution appropriate to the situation, like Maxwell velocity distribution). Just because it is modeled well by a random variable doesn't mean the basic interactions are not fully deterministic.

For example, the motions of the lottery balls blown around by the air machine are totally deterministic, governed by the laws of Newtonian mechanics. But in practice, we can get excellent predictions* by assuming, for all intents and purposes, the lottery numbers are chosen "at random."

Quantum mechanical processes, like whether a given particle will be measured "spin up" or "spin down" are the only events in nature that seem to be "truly random," based on our current understanding.

To your specific question, adding an electric field to a gas may allow to you predict the bulk motion in a different way than a neutral gas, but that doesn't change the basic determinism of the situation.**

*That is, numbers will appear with a frequency that matches extremely well with a random distribution.

**Caveat: since gas molecules are quantum mechanical objects, one could debate whether their collisions and interactions should be considered "truly random" in the same way as spin measurements or wavefunction collapse. It gets at the heart of the question "where does QM end and continuum classical mechanics begin?"

Answer 3

Random can mean many different things. Wiener process, already mentioned by @gandalf61 is a specific type of a random process, which assumes the fluctuations to be white noise, that is Gaussian noise that have the same intensity on all frequencies (which also means that it has zero correlation time.) This is a model, since any real physical noise has a finite correlation length, cannot have the same energy at all frequencies, and may deviate from a Gaussian distribution.

Whenever studying a specific phenomenon, we have to evaluate, whether this model is applicable or not. The details of interactions, discussed in the OP, influence the correlation time and limit the applicability of the model to experiments where the characteristic length scales are longer than these times. However the presence of this or that interaction does not automatically mean that the model is not applicable. Every experiment is characterized by an averaging time, since no measurement is instantaneous, thus, the collisions happening on scales shorter than this averaging time can be considered as delta-correlated.

Finally, I could like to warn against claims like "then in this model the electrostatic force creates the "path" for the gases and this can be predicted with the net electrostatic force on the molecule right ?" (emphasis mine.) The unpredictability of molecular motion is the basis of statistical physics - it is not at all clear that the behavior of systems with $N_A\sim10^{24}$ is more degrees of freedom is predictable even in principle, see, e.g., Is limited computational capacity a fundamental obstacle?