Probability in classical physics

Question

I have read lots of thing on probability in QM and the different ways of intending it. Now, I am wondering how physicists intend probability in classical physics. To be more specific, I have read some articles about the fact that probability in classical physics is seen by physicists as bayesian probability. I am not sure if the majority of physicists agree with this idea. How probability is generally intended in classical physics? And are there other consistent interpretation of probability in classical physics than subjective degrees of belief? Please let me know about them. Moreover, please suggest me if there are any articles that talk about this theme. Thanks.

Answer 1

Probability theory is a mathematical discipline used in physics. As such it is the same in quantum or classical mechanics, just like matrix algebra, differential equations, etc.

Having said that, it is necessary to point out that classical and quantum physics use probability theory differently or in different situations, which apparently sometimes generates a confusion that probability is somehow specific a part of quantum physics. Thus, classical mechanics is completely deterministic, whereas quantum mechanics is inherently probabilistic. The probability theory used in quantum mechanics is however the same as the one used in statistical physics, for description of Brownian motion, or inn measurement theory. The key difference in using the probability theory is that classical approaches usually aim at constructing equations for the probability itself (e.g, Fokker-Planck equation), whereas in quantum physics the equations are written for the wave function ("probability amplitude") or the density matrix.

An rather separate issue is the frequentist and Bayesian interpretations of the probability theory. While it is intuitively clear what probability means, it is hard to define rigorously. Thus, frequentists define probability as a limit (frequency) of an even occuring in an infinite number of trials (this is the belief typically held in physics, e.g., preparing an infinite number of atoms in the same state in quantum mechanics and then measuring them). Bayesians define probability as a degree of belief, updated during experiments via the Bayes formula. The frequentists frown at the Bayesians belif as unscientific, while the Bayesians retort that existence of an infinite number of trials is a belief in itself... there are books written on the subject.

Importantly, the two interpretations are identical in their mathematical formalism, although some methods correspond in spirit to one or other interpretation, and often have Bayesian or other characteristic words in their name. These names are however historic - occasionally a method frequentist in its spirit will be called "Bayesian" end vice versa (It reminds me of a well-known physics adagio that The standard model is a theory, while the string theory is only a model.)

Answer 2

For better understanding of yours, the implementation of Probability, both in QM and CM can be presented as following:

In QM, the probability of any attribute can be used for two different aspects:

1. For determining the distribution of constituents, within a system, &
2. For determining individual attributes of the constituents within a system (like, position, momentum, energy etc. ) (Note that: In QM, where position, momentum, and so other attributes follow Principle of Uncertainty, the show their Eigenvalues in a given instance, and the implementation of probability here is pretty much different, than that in representing any distribution)

However, in Classical Mechanics, the probability is mostly implemented in determining a given distribution of any material, especially in the context of Statistical Mechanics, and that is due to the random motion, like Brownian Motion of atoms and molecules within a matter, and this is same in QM.

But, a classical particle isn't uncertain for any of the attributes, as it provides a deterministic knowledge about the system. (Which does not provide the perfect information about a system and for that we need QM.)

Hope it helps!

Answer 3

Classical probability treats probability as objective. The probability of event A is the number of times the event occurs out of an infinite number of trials. The classical probability is fixed (one value) but in reality we never truly know it since we cannot conduct an infinite number of trials. Basic courses in probability assume the probability is known. Classical statistical inference provides confidence intervals for the probability based on the results of a sample. Based on the sample, an interval is constructed that bounds the fixed probability. Subjective approaches, such as Bayesian and belief/plausibility, do not assume the probability is constant but assume it changes based on our state of knowledge. Instead of treating the probability is fixed it is assumed the probability itself is a random variable with an uncertainty distribution. The random variable is the classical objective probability and its uncertainty is considered as a subjective probability; so we have a subjective probability distribution for an objective probability. In this sense, we use the name probability in two ways: objective and subjective which is very confusing to a general audience. It has been suggested we denote the classical probability as "frequency" since it is a ratio of events to trials, and reserve the name "probability" for the subjective probability. Then we have a probability distribution reflecting our uncertainty in the frequency; search online for "On Risk" by Kaplan et al for further information. The Bayesian approach assigns a prior (subjective) probability based on the state of knowledge and updates the prior to a posterior probability distribution as new information is obtained. (Belief/plausibility is based on focal elements assigned to subevents of the sample space based on the state of knowledge.) NUREG/CR-6823 "Handbook of Parameter Estimation for Probabilistic Risk Assessment", available online, provides a good basic discussion of all this. Also, the introductory sections in the textbook Bayesian Reliability Analysis by Martz and Waller address this in a clear manner.

Answer 4

If I have only partial knowledge of a system in classical mechanics I can describe it using propability.

For example if I have a gas molecule in a box and I dont know where they are, I can assume that they are on every point in space with an equal propability. If I open the box and look at it this propabalistic state "collapses" and I know exactly where the molecule is. I can say the same about the momentum of the molecule.

In quantum mechanics I can do the same. However according to borns rule I will find the particle in an eigenstate of the position operator after this observation. So far this is the same as in classical physics, because this eigenstate is the delta distribution of the position. Which is a fancy way of saying I know exacly where the particle is.

However if I want to know the momentum as well, I run into a problem. Because the momentum and poistion operators do not commute and thus dont share an eigenbasis I can only know one of the two things at a time. Because the wave function collapses into an eigensate of the corresponding operator.

This is fundamentally different from classical mechanics. In classical mechanics it doesn't matter, if I have knowledge of the position of a particle when I measure its momentum. In quantum mechanics however, it does.

We can see the transition from quantum to classical by looking at the commutator in the limit $\hbar \to 0$ because then $$ [x,p] = i\hbar \to 0 $$ and we can thus see that in classical mechancis the two observables do in fact commute.