Abstract
This contribution aims at introducing the topics of this book. We start with a brief historical excursion on the developments from the law of large numbers to the central limit theorem and large deviations theory. The same topics are then presented using the language of probability theory. Finally, some applications of large deviations theory in physics are briefly discussed through examples taken from statistical mechanics, dynamical and disordered systems.
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Notes
- 1.
Who, by the way, already knew the expression for the mean square energy fluctuations.
- 2.
We note that sometimes even in macroscopic systems (e.g. granular materials) the number of effective elementary constituents (e.g. the seeds) is not astonishingly large as in gases or liquids.
- 3.
The most rudimentary form of the LLN seems to be credited to Cardano.
- 4.
It is now well known, e.g. from KAM theorem and FPU simulations, that surely in some limit ergodicity fails, however it is fair to assume that the ergodic hypothesis holds for liquids or interacting gases.
- 5.
Note that, at variance with the molecular dynamics, the Monte Carlo dynamics is somehow artificial (and not unique), therefore the dynamical properties, e.g. correlation functions, are not necessarily related to physical features.
- 6.
It is enough to consider the variables t j = lnx j and q j = lna j , and then, noting that t N = q 1 + q 2 + … + q N , one can use the LLN and obtain the result.
- 7.
For instance the discrete one-dimensional Schrödinger equation with a random potential can be written in terms of a product of 2 × 2 random matrices.
- 8.
Such a result has been obtained by Boltzmann, who firstly noted the basic role of the entropy [14].
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Cecconi, F., Cencini, M., Puglisi, A., Vergni, D., Vulpiani, A. (2014). From the Law of Large Numbers to Large Deviation Theory in Statistical Physics: An Introduction. In: Vulpiani, A., Cecconi, F., Cencini, M., Puglisi, A., Vergni, D. (eds) Large Deviations in Physics. Lecture Notes in Physics, vol 885. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54251-0_1
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