MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

Multi-information in the thermodynamic limit

Ionas Erb and Nihat Ay


From information theory, mutual information is known to measure stochastic interdependence of probability distributions with two subsystems. We use a generalised version of this measure: multi-information, the Kullback-Leibler distance of a distribution from its corresponding independent distribution, and give a definition within the framework of statistical mechanics. There, the theory of infinite-volume Gibbs measures allows for the description of phase coexistence: The interaction potential of a model can yield several Gibbs measures at the same time. We propose to take the least multi-information of all the translation-invariant Gibbs measures to define a quantity directly depending on the interaction potential. We show that it is enough to take this infimum over the pure, i.e. physically relevant states only. Our definition is applied to the two-dimensional Ising model and the main result is derived: In the Ising square lattice, multi-information as a function of temperature attains its isolated global maximum at the point of phase transition. There, the one-sided derivatives diverge. Finally, we also briefly discuss the behaviour for the one-dimensional Ising chain in a magnetic field.

MSC Codes:
60K35, 79XX, 60XX
mutual information, ising model, phase transitions, complexity

Related publications

2004 Repository Open Access
Ionas Erb and Nihat Ay

Multi-information in the thermodynamic limit

In: Journal of statistical physics, 115 (2004) 3-4, pp. 949-976