This motivating textbook gives a friendly, rigorous introduction to fundamental concepts in equilibrium statistical mechanics, covering a selection of specific models, including the Curie–Weiss and Ising models, the Gaussian free field, O(n) models, and models with Kac interactions. Using classical concepts such as Gibbs measures, pressure, free energy, and entropy, the book exposes the main features of the classical description of large systems in equilibrium, in particular the central problem of phase transitions. It treats such important topics as the Peierls argument, the Dobrushin uniqueness, Mermin–Wagner and Lee–Yang theorems, and develops from scratch such workhorses as correlation inequalities, the cluster expansion, Pirogov–Sinai Theory, and reflection positivity. Written as a self-contained course for advanced undergraduate or beginning graduate students, the detailed explanations, large collection of exercises (with solutions), and appendix of mathematical results and concepts also make it a handy reference for researchers in related areas.

Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction

Nature has a hierarchical structure, with time, length, and energy scales ranging from the submicroscopic to the supergalactic. Surprisingly, it is possible, and in many cases essential, to discuss these levels independently—quarks are irrelevant for understanding protein folding and atoms are a distraction when studying ocean currents. Nevertheless, it is a central lesson of science, very successful in the past three-hundred years, that there are no new fundamental laws, only new phenomena, as one goes up the hierarchy. Thus arrows of explanations between different levels always point from smaller to larger scales, although the origin of higher-level phenomena in the more fundamental lower-level laws is often very far from transparent. (In addition some of the dualities recently discovered in string theory suggest possible arrows from the highest to the lowest level, closing the loop.)

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Statistical Mechanics: A Selective Review of Two Central Issues

Theory of phase transitions

(2 + 1) [XTpXpH12, CTH11]. + [Zuc11b]. 0 [Fed17]. 1 [BELP15, CAS11, Cor16, Fed17, GDL10, GBL16, Hau16, JV19, KT12, KM19c, Li19, MN14b, Nak17, Pal11, Pan14, RT14, RBS16b, RY12, SS18c, Sug10, dOP18]. 1 + 1 [Sak18, CP15b]. 1/2 [MD10]. 1/f [FDR12]. 1/n [Per17]. 1/|x− y| [MSV10, MSV13]. 13 [DFL17]. 1 ≤ p ≤ ∞ [Dud13]. 1/f [HPF15]. 2 [AB19, ADS19, BF12, BNT13, DSS15, EKD12, Her13, Ily12, Lan10, Li12, Li19, LZ11, Ny13, Ost16, PSS16, ST14, Sch13b, TJ15, WPB15, dWL10]. 2 + 1 [dWL14]. 2.5 [BC15a]. 2R [WLEC17]. 2× 2 [CLTT13]. 3 [BCF19, BLS17, ESPP14, Kar18, SH16, SWKS14, dCCS19]. 3/2 [DK10]. 38 [Cam13]. 4 [BBS14, Zha14]. 4× 4 [LN19a]. 5/2 [DK10, EKD12]. 6 [EC11]. 8 [Zha14]. 90◦ [YM11]. 3 [Afz12]. 1−x [EFO11]. 13 [CDCL18]. 2 [ML15, QR13, ST11c]. 4 [HBB10]. 6 [BCL10a, BCL10b, EFO11]. x [EFO11].

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A Complete Bibliography of the Journal of Statistical Physics: 2000{2009

The rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade. We review here the main results that have been obtained, both in two and higher dimensions. In particular, we describe how the phenomenological Wulff and Winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases. We focus on the main conceptual issues and describe the central ideas of the existing approaches.

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Rigorous probabilistic analysis of equilibrium crystal shapes

We consider particles in ℝd, d≥2, interacting via attractive pair and repulsive four-body potentials of the Kac type. Perturbing about mean-field theory, valid when the interaction range becomes infinite, we prove rigorously the existence of a liquid–gas phase transition when the interaction range is finite but long compared to the interparticle spacing for a range of temperature.

Liquid–Vapor Phase Transitions for Systems with Finite-Range Interactions

We consider extended Pirogov–Sinai models including lattice and continuum particle systems with Kac potentials. Call λ an intensive variable conjugate to an extensive quantity α appearing in the Hamiltonian via the additive term −λα. We suppose that a Pirogov–Sinai phase transition with order parameter α occurs at λ=0, and that there are two distinct classes of DLR measures, the plus and the minus DLR measures, with the expectation of α respectively positive and negative in the two classes. We then prove that λ=0 is the only point in an interval I of values of λ centered at 0 where this occurs, namely the expected value of α is positive, respectively negative, in all translational invariant DLR measures at {λ>0}⊓I and {λ<0}⊓I.

/pdf/on-the-gibbs-phase-rule-in-the-pirogov-sinai-regime-3tyqut2osr.pdf

On the Gibbs phase rule in the Pirogov-Sinai regime

We prove a convexity property of the surface tension corresponding to a non-local, anisotropic free-energy functional of van der Waals type which implies that the Wulff shape is strictly convex and smooth. We also prove that the transport coefficients of the limiting anisotropic motion by mean curvature obtained in [33] are strictly positive and equal to the stiffness parameters determined by the surface tension.

Sharp Interface Limits for Non-Local Anisotropic Interactions

We consider the Ising model on $$\mathbb Z\times \mathbb Z$$
 where on each horizontal line $$\{(x,i), x\in \mathbb Z\}$$
 the interaction is given by a ferromagnetic Kac potential with coupling strength $$J_{ \gamma }(x,y)\sim \gamma J(\gamma (x-y))$$
 at the mean field critical temperature. We then add a nearest neighbor ferromagnetic vertical interaction of strength $$\epsilon $$
 and prove that for every $$\epsilon >0$$
 the systems exhibits phase transition provided $$\gamma >0$$
 is small enough.

/pdf/layered-systems-at-the-mean-field-critical-temperature-2ysqoxf0l5.pdf

Layered Systems at the Mean Field Critical Temperature

We propose a new definition of surface tension and check it in a spin model of the Pirogov-Sinai class without symmetry. We study the model at low temperatures on the phase transitions line and prove: (i) existence of the surface tension in the thermodynamic limit, for any orientation of the surface and in all dimensions $ d \geq 2 $; (ii) the Wulff shape constructed with such a surface tension coincides with the equilibrium shape of the cluster which appears when fixing the total spin magnetization (Wulff problem).

https://link.springer.com/content/pdf/10.1007/s00023-003-0149-1.pdf

Errico Presutti

Papers

Liquid–Vapor Phase Transitions for Systems with Finite-Range Interactions

On the Gibbs phase rule in the Pirogov-Sinai regime

Sharp Interface Limits for Non-Local Anisotropic Interactions

Layered Systems at the Mean Field Critical Temperature

Surface Tension and Wulff Shape for a Lattice Model without Spin Flip Symmetry