[Paper Review] Gibbs Measures on Marked Configuration Spaces: Existence and Uniqueness

Diana Conache, Alexei Daletskii+2 more|arXiv (Cornell University)|Mar 21, 2015

Theoretical and Computational Physics32 references3 citations

TL;DR

This paper establishes sufficient conditions for the existence and uniqueness of Gibbs measures on marked configuration spaces for infinite systems of particles with unbounded spins and position-dependent spin-spin interactions. By developing a non-classical stability framework that extends Ruelle's superstability estimates and applying Dobrushin's uniqueness criterion, the authors prove the existence of a unique equilibrium state under polynomial growth conditions on the spin interaction and integrability of the single-spin measure.

ABSTRACT

We study equilibrium states of an infinite system of interacting particles in a Euclidean space. The particles bear `unbounded' spins with a given symmetric a priori distribution. The interaction between the particles is pairwise and splits into position-position and spin-spin parts. The position-position part is described by a superstable potential, and the spin-spin part is attractive and of finite range. Thermodynamic states of the system are defined as tempered Gibbs measures on the space of marked configurations. We derive sufficient conditions of the existence and uniqueness of these Gibbs measures.

Motivation & Objective

• To address the lack of existence and uniqueness results for Gibbs measures in systems with unbounded spins and non-compact marks, which fall outside classical Ruelle superstability theory.
• To develop a new thermodynamic stability concept tailored to models combining continuum particle systems and unbounded spin systems.
• To prove the existence and uniqueness of equilibrium states (Gibbs measures) for marked configuration spaces with polynomially growing spin interactions.
• To extend the Dobrushin uniqueness criterion to systems with unbounded spins and long-range interactions, overcoming the limitations of prior compact spin assumptions.
• To provide a rigorous mathematical foundation for models of magnetic gases, ferrofluids, and amorphous magnets where phase transitions may occur.

Proposed method

• Formulate the system as a marked Gibbs measure on the space of marked configurations, with positions in $\mathbb{R}^d$ and spins in $\mathbb{R}^m$, governed by a Hamiltonian combining positional and spin-spin interactions.
• Introduce a modified superstability estimate that accounts for unbounded spins by controlling the growth of spin interactions via polynomial bounds and integrability of the single-spin measure $\chi$.
• Apply the Dobrushin uniqueness criterion by estimating the total variation distance between conditional distributions under different boundary conditions, using the dependence on the spin configuration and particle density.
• Derive a uniform bound on the total variation distance between Gibbs specifications using exponential moments of spin norms, relying on Young's inequality and moment generating function estimates.
• Establish a key estimate (6.29) showing that the variation distance decays exponentially with respect to the interaction strength and particle density, leading to uniqueness under a smallness condition on $\phi(z, \mathcal{J}, L)$.
• Use disintegration of the marked Poisson measure and moment generating functions to control the partition function and conditional expectations, ensuring integrability and tightness of the Gibbs measure.

Experimental results

Research questions

• RQ1 Under what conditions does a Gibbs measure exist for a system of particles with unbounded spins and position-dependent spin-spin interactions?
• RQ2 Can the classical Dobrushin uniqueness criterion be extended to systems with non-compact spin spaces and unbounded interactions?
• RQ3 How can thermodynamic stability be defined and proven for models combining continuum particle systems and unbounded spin systems?
• RQ4 What role does the integrability of the single-spin measure $\chi$ play in ensuring the existence and uniqueness of Gibbs measures?
• RQ5 Is the equilibrium state unique in the low-density regime for such marked particle systems with polynomially growing spin interactions?

Key findings

• The paper establishes a sufficient condition for the existence of Gibbs measures on marked configuration spaces for systems with unbounded spins and polynomially growing spin interactions.
• A new superstability-type estimate is derived that controls the Hamiltonian under unbounded spin configurations, enabling the application of classical Gibbs measure theory.
• The uniqueness of the Gibbs measure is proven under a smallness condition on the function $\phi(z, \mathcal{J}, L)$, which depends on particle density $z$, interaction strength $\mathcal{J}$, and spin configuration bounds $L$.
• The total variation distance between Gibbs specifications under different boundary conditions is bounded by $z\mathcal{E}_{\mathcal{J}} \exp\{2(C_{\Phi,\mathcal{J}} + z\mathcal{E}_{\mathcal{J}})\}$, ensuring uniqueness when this quantity is sufficiently small.
• The existence of a unique equilibrium state is guaranteed under the integrability of $\int_S \exp\{\mathcal{J}( |s|^t + \mathcal{N}_0 L |s|^r )\} \chi(ds)$, which ensures the finiteness of the moment generating function.
• The results extend the applicability of Gibbs measure theory to models of magnetic gases and ferrofluids, where spin variables are unbounded and interactions are non-ferromagnetic or non-quadratic.