Narrow Results

The Hamiltonian of mean force is a widely used concept to describe the modification of the usual canonical Gibbs state for a quantum system whose coupling strength with the thermal bath is non-negligible. Here, we perturbatively derive general approximate expressions for the Hamiltonians of mean force in the weak-coupling approximation and in the high-temperature one. We numerically analyze the accuracy of the corresponding expressions and show that the precision of the Bloch–Redfield equantum master equation can be improved if we replace the original system Hamiltonian by the Hamiltonian of mean force.

We consider models of classical statistical mechanics satisfying natural stability conditions: a finite spin space, translation-periodic finite potential of finite range, a finite number of ground states meeting Peierls or Gertzik–Pirogov–Sinai condition. The Pirogov–Sinai theory describes the phase diagrams of these models at low temperature regimes. By using the method of doubling and mixing of partition functions we give an alternative elementary proof of the uniqueness of limiting Gibbs states at low temperatures in ground state uniqueness region.

We consider the one-dimensional ferromagnetic Ising model with very long range interaction under weak and sparse biased external field and prove that at sufficiently low temperatures, the model has a unique limiting Gibbs state.

The absence of phase transitions in one-dimensional Widom–Rowlinson model with long-range interaction is established in the non-symmetric case when different particles have different activity parameters.

A zero-temperature phase-diagram of the one-dimensional ferromagnetic Ising model is investigated. It is shown that at zero temperature spins of any compact collection of lattice points with identically oriented external field are identically oriented.

In this paper, we consider a one-dimensional long range Widom–Rowlinson model when particle activity parameters are periodic and biased. We show that if the interaction is sufficiently large versus particle activities then the model does not exhibit a phase transition at low temperatures.

Asymptotic behavior of spectral distribution of the adjacency operator on the Johnson graph with respect to the Gibbs state is discussed in infinite volume and zero temperature limit. The limit picture is drawn on the one-mode interacting Fock space associated with Meixner polynomials.

Some a priori bounds for measures on configuration spaces are considered. We establish relations between them and consequences for corresponding measures (such as support properties etc.). Applications to Gibbs measures are discussed.

For a topologically transitive subshift of finite type defined by a symmetric transition matrix, we introduce a temperature-based problem related to the usual thermodynamic formalism. This problem is described by an operator acting on Hölder continuous observables which is actually superlinear with respect to the max-plus algebra. We thus show that, for each fixed absolute temperature, such an operator admits a unique eigenfunction and a unique eigenvalue. We also study the convergence as the temperature goes to zero and we relate the limit objects to an ergodic version of Kantorovich transshipment problem.

Let $M$ be a compact metric space and $X=M^{\mathbb{N}}$, we consider a set of admissible sequences $X_{A,I}\subset X$ determined by a continuous admissibility function $A:M\times M\to ℝ$ and a compact set $I\subset ℝ$. Given a Lipschitz continuous potential $\phi :X_{A,I}\to ℝ$, we prove uniqueness of the Gibbs state ${\mu }_{\phi }$ and we show that it is a Gibbs–Bowen measure and satisfies a central limit theorem.

In this paper, we provide an insight into the emergence of power-law and two-phase behavior in the financial market fluctuations by defining an analytical model for time evolution of stock share prices. The defined model can exhibit bimodal behavior in the supply-demand structure of the market. Moreover, it differs from existing Ising-type models. It turns out that the constructed model is a solution of a thermodynamic limit of a Gibbs probability measure when the number of investors and the number of stock shares approaches the infinity. The energy functional of the Gibbs probability measure is derived from the Nash equilibrium of the underlying game.

The diffusion approximation for a Boltzmann–Poisson system is studied. Nonlinear relaxation type collision operator is considered. A relative entropy is used to prove useful $L^2$-estimates for the weak solutions of the scaled Boltzmann equation (coupled to Poisson) and to prove the convergence of the solution toward the solution of a nonlinear diffusion equation coupled to Poisson. In one dimension, a hybrid Hilbert expansion and the contraction property of the operator allow to exhibit a convergence rate.