Is Brownian motion a deterministic system? I.e the motion of all particles are completely determined or is there an innate indeterminism like quantum systems?
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$\begingroup$ No proof or reference, but I suspect that you are looking at "sensitive dependence on initial conditions" (i.e. chaos), which when combined with the Heisenberg Uncertainty principles leads to necessary unpredictability. $\endgroup$– dmckee --- ex-moderator kittenJun 2, 2013 at 1:32
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1$\begingroup$ Yes, you can derive (in a suitable limit) the stochastic description of Brownian motion from the purely deterministic dynamics of hard spheres, see the following recent and quite remarkable paper: arxiv.org/abs/1305.3397 . $\endgroup$– Yvan VelenikJun 2, 2013 at 8:13
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$\begingroup$ Related (mathematically deterministic theories and Langevin dynamics) physics.stackexchange.com/a/762425/226902 $\endgroup$– QuilloMay 13 at 14:58
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1 Answer
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If you describe the combined system of the molecules of the liquid and the Brownian particle and you know the mechanism of the collisions and all initial conditions, then it is deterministic.
If you want to describe only the Brownian particle, then you would do so by a stochastic processes (called Brownian motion or the Wiener process) and it would be non-deterministic (i.e. random or stochastic).
Does this answer your question?
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$\begingroup$ Yes , but then would that mean that there are indeterminism in classical physics as well as quantum physics ? $\endgroup$– JasmineJun 1, 2013 at 21:53
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1$\begingroup$ Yes, but generally it is viewed as a consequence of lack of information, e.g., the initial conditions are know only approximately or only a part of the system is known. $\endgroup$– UwFJun 1, 2013 at 21:57
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$\begingroup$ This is the traditional answer of the 19th century, but it doesn't really account for the combination of quantum mechanical uncertainty and the fact that sufficiently complicated systems can diverge exponentially starting from arbitrarily small perturbations. The unpredictability is built in at the ground level; there is not--even in principle--a way to treat the universe as a deterministic system. $\endgroup$ Jun 2, 2013 at 1:34
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$\begingroup$ so you are saying that brownian motion is indeterministic because of only lake of knowledge of initial conditions ? $\endgroup$– JasmineJun 2, 2013 at 7:04
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2$\begingroup$ But since we believe that our world is quantum, a more realistic model would probably give additional randomness that doesn't just come from lack of information... When you describe the collisions on a quantum level, then - even if you know perfectly well the initial states of all molecules - you can only predict the probilities with which the Brownian particle will be in a certain place, when you measure it. $\endgroup$– UwFJun 2, 2013 at 9:12