What is probability and what is then statistical mechanics?

Domagoj Kuić
University of Split, Faculty of Science, Department of Physics


From the point of view of predictive statistical mechanics, with the exception of quantum mechanical probabilities, there is no reason to consider any probability distribution as the only correct distribution. Such a view is in a marked contrast to the interpretation that defines probabilites only in terms of frequencies as the objective property of the observed system. In the frequentist interpretation the probabilities are experimentaly verifiable, and consequently, the foundational problem of statistical mechanics would be to derive them and to justify them in the sense of frequencies. Jaynes presented the opposite view, that if we the choose to represent the degree of our knowledge about the individual system, then there can not be anything physically real in the frequencies of the ensemble of a large number of systems, nor there is any sense in asking which ensemble is the only correct one. What we call different ensembles in reality corresponds to the different degrees of knowledge about the individual system, or about certain physical situation. In the argumentation of this viewpoint, Jaynes referred to the statement by Gibbs, according to which the ensembles are chosen to illustrate the probabilites of events in the real word.

The simplest interpretation of Gibbs formalism follows from the fact that by maximizing the information entropy, which is also known as uncertainty, subject to given constraints, one predicts just the macroscopic behaviour that can happen in the greatest number of ways compatible with those constraints. Without going deeper into the problem of interpretation of probabilites, which is even more pronounced in the case of non-equilibrium states, it is important that the distributions that follow from the application of the principle of maximum information entropy depend only on the available information. If one referres only to predictions, from the same viewpoint one can speak about the objectivity only in the extent in which the incompleteness of information is taken into account. Consistent with this way of thinking, by applying the principle of maximum information entropy, we come to the relevant statistical distributions.