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?
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$\begingroup$ Related: physics.stackexchange.com/q/739865/35952 $\endgroup$– user35952Commented May 1, 2023 at 12:54
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3 Answers
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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.
answered Apr 30, 2023 at 16:45
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1$\begingroup$ Eh, I hesitate to say that the random walk is a model for Brownian motion. Sure in large $N$ limit it replicates it, but typically you derive it using probability distribution functions. $\endgroup$ Commented Apr 30, 2023 at 16:55
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1$\begingroup$ @KyleKanos Good point - technically it is a Wiener process. I have amended my answer. $\endgroup$ Commented Apr 30, 2023 at 17:40
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$\begingroup$ Can you cite some sources saying it requires charge symmetry I'm finding it difficult to find sources myself , Thank you very much for the answer by the way $\endgroup$– RazzCommented May 1, 2023 at 3:21
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$\begingroup$ and of course the model of necessity only applies at long distances from the edges of any volume, so that collisions with the container don't influence the observations. $\endgroup$– jwentingCommented May 2, 2023 at 7:53
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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?"
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$\begingroup$ I suppose to avoid potential confusion, instead of saying "we can get excellent predictions" (referring to lottery numbers), you might say random numbers form a good/sufficient statistical model of a lottery machine. There is no method to predict the actual values of uniformly random numbers (I know that you know this but others reading your answer might do a double take) $\endgroup$ Commented May 1, 2023 at 7:40
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1$\begingroup$ "For example, the motions of the lottery balls blown around by the air machine are totally deterministic, governed by the laws of Newtonian mechanics." Along the lines of your caveat, I was wondering whether it is truly the case that such systems are truly deterministic. In reality, all measurements have error and we use statistics to work around that. We say it's due to the fact there many factors that aren't known, but is that provable? Why wouldn't quantum effects e.g.: at surface when balls collide create small scale randomness that makes the system non-deterministic? $\endgroup$ Commented May 1, 2023 at 15:48
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$\begingroup$ Jimmy, I'm not aware of any research that has identified sources of QM randomness in Newtonian macroscopic systems, but that's not to say it doesn't exist. But it is the same as when we flip a coin 1000 times. In each event, if given the initial conditions (geometry, rotation speed, distance to the ground, elastic properties of the ground and coin, etc) we can predict with 100% certainty that it will be heads (or tails, as the case may be). But those initial conditions vary in a way that's practically unknowable, and so the outcome imitates a random variable to a very good approximation. $\endgroup$– RC_23Commented May 2, 2023 at 0:24
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$\begingroup$ The lottery ball example seems a poor fit. Blown air flows turbulently (chaotically) introducing what is in principle and practice unpredictable behavior. $\endgroup$ Commented May 2, 2023 at 17:37
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2$\begingroup$ A chaotic system is not the same as an undeterministic system. A turbulent flow is completely predictable in principle, if we have fine enough spatial and temporal resolution to account for all the flow structure. The turbulent mixing of the lottery balls is unpredictable in practice because of its complexity, and we don't have the computation power to model it at fine enough resolution. In contrast, a spin up/spin down measurement of a single electron is incredibly simple, but the result is unpredictable in advance because of true randomness. $\endgroup$– RC_23Commented May 2, 2023 at 18:05
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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?
