# Statistical Mechanics vs Statistics

Just how much of a representation of statistics do we get in a statistical mechanics curriculum. What are some of the useful facets of stat not in stat mech/quantum mech that physicists should really consider taking a closer look at?

I ask this question in a position of mild ignorance from general statistics and with interest in learning what untaught aspects of it can come in handy for physicists in the cases of

a)material science\superconductivity and

b)quantum information.

(Feel free to add any other cool anecdotes of less-used statistical methods in physics if you have them.)

• My personal impression: More actual statistics is usually done in (introductory courses for) the lab curriculum (central limit theorem, sometimes hypothesis tests, linear regression, ...) and in lectures on stochastic processes. The statistical mechanics lectures are usually pretty unconcerned with general statistics. – Sebastian Riese May 13 '15 at 9:55
• First of all, the terminology statistical mechanics is rather unfortunate, and it would have been much better to call this theory probabilistic mechanics, for example. Indeed, the concepts and techniques of probability theory play an essential role in statistical mechanics (even if they are, quite wrongly, somewhat hidden in many books and courses). Statistics, however, is mostly irrelevant to statistical mechanics. – Yvan Velenik May 13 '15 at 11:19

First let me repeat what Yvan Velenik says in the comment: The terminology is somewhat unfortunate, because you don't need that much statistics, rather you'll need some probability theory.

To elaborate, quoting Wikipedia,

Statistics is the study of the collection, analysis, interpretation, presentation, and organization of data. [...] Statistics deals with all aspects of data including the planning of data collection in terms of the design of surveys and experiments.

So statistics is primarily about data and its analysis. So, when you do an experiment, you'll have a hypothesis and collect data. Then, you'll use a lot of techniques of statistics to evaluate them such as error analysis, confidence tests (such as $\chi^2$-tests), etc.

None of this is relevant in statistical physics, because this refers to a purely theoretical model. Statistical mechanics studies the collective (read: average) behaviour of a mechanical systems whose microscopic behaviour is unkown in detail. This is not data. What is actually, useful however, is probability theory.

So let me be so free as to rephrase the question:

What untaught aspects of probability theory can come in handy for physicists?

And we limit our field to material science and/or quantum information. Since I'm a quantum information guy, I'll mostly talk about that.

Of course, certain basic elements have to be known: Central limit theorems, basic expecation values and the idea of probability densities is already inherent in the formulation of quantum mechanics, but every physicist has probably heard about them.

Another, much richer, field for probability theory would be considering noise models and doing stuff with noise. "Noise" is referred to as an influence that cannot be described fully, so you introduce random variables or even stochastic processes (usually very simple ones). As a random example (pun intended), consider this paper by Bravyi and König about disorder assisted error correction for majorana fermions. It can help here to have basic understanding of what random variables can do.

I chose this also because Anderson localization is a well-studied phenomenon in mathematical physics that can occur in material science. This involves certain stochastic processes, so a knowledge thereof can be interesting. Likewise, probability theorists like to study Brownian motion (which is a special case of a stochastic process) and percolation models which can be applied to material science questions.

Another very important area of probability theory that you might encounter in quantum information is Markov theory. There is the notion of a quantum Markov chain, which is studied a lot in the literature. In addition, similar definitions of "Markovianity" exist for arbitrary channels and quantum semigroups (channels of type $e^{\mathcal{L}t}$ where $\mathcal{L}$ is a Liouvillian of some open quantum system) and their study profits from knowing the classical literature.

I could very much go on in this style - the question is actually really "too broad" - but let me only mention two last aspects, which you would encounter in classical probability courses: Martingales and Random Matrices. Both objects are used once in a while in quantum information for various purposes. Random Matrices are often used to model noise, for martingales, let me guide you to a paper that studies them for state estimation processes.

Finally, let me just note that all the examples above mostly consider only introductory-level knowledge of the topics, but that does not mean that one could not apply more fancy stuff, but that it has not been done yet.