Correlation of clusters: Partially truncated correlation functions and their decay

In this article, we investigate partially truncated correlation functions (PTCF) of infinite continuous systems of classical point particles with pair interaction. We derive Kirkwood-Salsburg-type equations for the PTCF and write the solutions of these equations as a sum of contributions labelled by certain forests graphs, the connected components of which are tree graphs. We generalize the method introduced by R.A. Minlos and S.K. Poghosyan (1977) in the case of truncated correlations. These solutions make it possible to derive strong cluster properties for PTCF which were obtained earlier for lattice spin systems.Comment: 31 pages, 2 figures. 2nd revision. Misprints corrected and 1 figure adde

A combinatorial identity

We give an elementary proof of an interesting combinatorial identity which is of particular interest in graph theory and its applications

Correlation of clusters: Partially truncated correlation functions and their decay

In this article, we investigate partially truncated correlation functions (PTCF) of infinite continuous systems of classical point particles with pair interaction. We derive Kirkwood-Salsburg-type equations for the PTCF and write the solutions of these equations as a sum of contributions labelled by certain forests graphs, the connected components of which are tree graphs. We generalize the method introduced by R.A. Minlos and S.K. Pogosyan (1977) in the case of truncated correlations. These solutions make it possible to derive strong cluster properties for PTCF which were obtained earlier for lattice spin systems

Theory of interacting quantum fields

This monograph is devoted to the systematic and encyclopedic presentation of the foundations of quantum field theory. It represents mathematical problems of the quantum field theory with regardto the new methods of the constructive and Euclidean field theory formed for the last thirty years of the 20th century on the basis of rigorous mathematical tools of the functional analysis, the theory of operators, and the theory of generalized functions. The book is useful for young scientists who desire to understand not only the formal structure of the quantum field theory but also its basic concepts and connection with classical mechanics, relativistic classical field theory, quantum mechanics, group theory, and the theory of functional integration.This monograph is devoted to the systematic and encyclopedic presentation of the foundations of quantum field theory. It represents mathematical problems of the quantum field theory with regardto the new methods of the constructive and Euclidean field theory formed for the last thirty years of the 20th century on the basis of rigorous mathematical tools of the functional analysis, the theory of operators, and the theory of generalized functions. The book is useful for young scientists who desire to understand not only the formal structure of the quantum field theory but also its basic concept

Gibbs State Uniqueness for Anharmonic Quantum Crystal with a Nonpolynomial Double-Well Potential

45 pagesWe construct the Gibbs state for $\nu$-dimensional quantum crystal with site displacements from $\R^d$, $d\geq 1$, and with a one-site \textit{non-polynomial} double-well potential, which has \textit{harmonic} asymptotic growth at infinity. We prove the uniqueness of the corresponding {\it Euclidean Gibbs measure} (EGM) in the \textit{light-mass regime} for the crystal particles. The corresponding state is constructed via a cluster expansion technique for an arbitrary temperature $T\geq 0$. We show that for all $T\geq 0$ the Gibbs state (correlation functions) is analytic with respect to external field conjugated to displacements provided that the mass of particles $m$ is less than a certain value $m_* >0$. The high temperature regime is also discussed

On diffusion dynamics for continuous systems with singular superstable interaction

Kondratiev Y, Rebenko AL, Röckner M. On diffusion dynamics for continuous systems with singular superstable interaction. JOURNAL OF MATHEMATICAL PHYSICS. 2004;45(5):1826-1848.We consider the time evolution of states for continuous infinite particle systems which corresponds to nonequilibrium diffusion dynamics. For initial states mu(0) which are perturbations of the equilibrium we obtain a bound for finite volume nonequilibrium correlation functions and their continuity in time uniformly in volume for any finite time interval. This gives the possibility to construct the time evolution of correlation functions and corresponding states in the thermodynamic limit. (C) 2004 American Institute of Physics

Small mass behaviour in quantum lattice models

Minlos RA, Rebenko AL, Albeverio S, Kondratiev Y. Small mass behaviour in quantum lattice models. Journal of Statistical Physics . 1998;92(5-6):1153-1172

Phase transitions in continuum ferromagnets with unbounded spins

Daletskii A, Kondratiev Y, Kozitsky Y. Phase transitions in continuum ferromagnets with unbounded spins. Journal of Mathematical Physics. 2015;56(11): 113502.States of thermal equilibrium of an infinite system of interacting particles in R-d are studied. 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. It is proved that the set of such measures contains at least two elements if the activity is big enough. (C) 2015 AIP Publishing LLC