Gibbs Measures and Phase Transitions in Statistical Mechanics
Summary
Gibbs measures form the mathematical backbone of equilibrium statistical mechanics, providing a probability distribution over microscopic configurations of a system that is proportional to the exponential of minus the energy of each configuration. In a lattice or graph setting, the Hamiltonian encodes interactions—nearest-neighbour, next-nearest-neighbour or long-range—and the inverse temperature parameter regulates fluctuations. Infinite-volume Gibbs measures emerge in the thermodynamic limit and are characterised by consistency conditions (DLR equations) linking local conditional probabilities to global behaviour. Phase transitions occur when multiple Gibbs measures coexist under identical external conditions, signalling spontaneous symmetry breaking or the emergence of ordered phases such as ferromagnetic, antiferromagnetic or modulated states. Critical phenomena appear at the boundaries of these regimes, where correlation lengths diverge and universal scaling laws apply. Rigorous analyses span spin systems on regular lattices, trees and hierarchical structures, as well as gradient fields and continuum models. These results have profound implications for condensed-matter physics, information theory, combinatorial optimisation and the theory of complex networks, illuminating both foundational mathematical questions and practical applications in material science and beyond.
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Technical terms
Gibbs measure: A probability distribution over configurations proportional to exp(−βH), satisfying consistency conditions in infinite volume.
Phase transition: A change in macroscopic behaviour marked by the coexistence of distinct Gibbs measures under identical external parameters.
Cayley tree: An infinite, connected acyclic graph in which each vertex has the same number of neighbours, often used to study exact solutions.
Gradient Gibbs measure: A Gibbs measure defined on the discrete gradients of a field, relevant to interfaces and random conductance models.
References
- Gibbs measures of an Ising-Vannimenus Model with one-level competing interactions on 4th order Cayley tree. Heliyon (2023).
- Hierarchical Cubes: Gibbs Measures and Decay of Correlations. Journal of Statistical Physics (2024).
- Phase transitions for a class of gradient fields. Probability Theory and Related Fields (2021).
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