Statistical mechanics of lattice systems

We study the thermodynamic limit for a classical system of particles on a lattice and prove the existence of infinite volume correlation functions for a large set of potentials and temperatures.On partial leave from the University of Aix-Marseille.     

Statistical mechanics of a Bethe lattice

The statistical mechanics of a spin-one Bethe lattice is discussed. The exact expression for Curie temperature is derived using a Bethe-Peierls approximations. The effect of the inclusion of a single-spin anisotropy to each Ising model of the lattice is also studied. It is seen that under certain situations a first-order phase transition appears.     

Statistical mechanics of a one-dimensional lattice gas

We study the statistical mechanics of an infinite one-dimensional classical lattice gas. Extending a result ofvan Hove we show that, for a large class of interactions, such a system has no phase transition. The equilibrium state of the system is represented by a measure which is invariant under the effect of lattice translations. The dynamical system defined by this invariant measure is shown to be aK-system.     

Gauge theory on a random lattice

A general formulation of gauge theory on a random lattice is developed and the strong coupling limit of the Wilson string tension worked out. The confining force found in this strong coupling limit is identical to that predicted by the relativistic string model. In particular, the force between two color-triplet charges is a constant for large separation and the tube of electric flux joining the charges fluctuates, giving it a net thickness proportional to the logarithm of its length.     

Statistical mechanics of random paths on disordered lattices

The dependence of the universality class on the statistical weight of unrestricted random paths is explicitly shown both for deterministic and statistical fractals such as the incipient infinite percolation cluster. Equally weighted paths (ideal chain) and kinetically generated paths (random walks) belong, in general, to different universality classes. For deterministic fractals exact renormalization group techniques are used. Asymptotic behaviors for the end-to-end distance ranging from power to logarithmic (localization) laws are observed for the ideal chain. In all these cases, random walks in the presence of nonperfect traps are shown to be in the same universality class of the ideal chain. Logarithmic behavior is reflected insingular renormalization group recursions. For the disordered case, numerical transfer matrix techniques are exploited on percolation clusters in two and three dimensions. The two-point correlation function scales with critical exponents not obeying standard scaling relations. The distribution of the number of chains and the number of chains returning to the starting point are found to be well approximated by a log-normal distribution. The logmoment of the number of chains is found to have an essential type of singularity consistent with the log-normal distribution. A non-self-averaging behavior is argued to occur on the basis of the results.     

Statistical mechanics of quantum lattice systems without translation invariance

The well-known results concerning the equilibrium of a translation invariant quantum lattice system — existence of the pressure and of the time automorphisms, variational principle for the pressure — are generalized to a large class of quantum lattice systems with potentials not exhibiting covariance under the group of lattice translations.     

Statistical mechanics results in the P(φ)2 quantum field theory

We apply ideas and methods from classical statistical mechanics to study the P(φ)₂ self-coupled two-dimensional Boson field theory in the Euclidean region. In particular, we consider correlation inequalities of Griffiths type; the thermodynamic limit for the pressure, the average interaction and the entropy; and the equilibrium equations for states associated with a given interaction.     

Bohmian mechanics and quantum field theory

We discuss a recently proposed extension of Bohmian mechanics to quantum field theory. For more or less any regularized quantum field theory there is a corresponding theory of particle motion, which, in particular, ascribes trajectories to the electrons or whatever sort of particles the quantum field theory is about. Corresponding to the nonconservation of the particle number operator in the quantum field theory, the theory describes explicit creation and annihilation events: the world lines for the particles can begin and end.     

Statistical mechanics of quenched inhomogeneous systems with random disorder

We propose a method to treat quenched disordered systems within the frame of an inhomogeneous ensemble of configurations. To describe a definite configuration, we make use of occupation numbers. By choosing an appropriate representation of these occupation numbers, we are able to deal with an inhomogeneous ensemble by elementary techniques as for instance used for uncorrelated occupation numbers in the frame of a translation invariant ensemble.     

Statistical mechanics of a one-dimensional lattice gas with exponential-polynomial interactions

Some properties of the transfer-matrix for a one-dimensional classical lattice-gas with exponential-polynomial pair interactions are studied using Hilbert space techniques.     

Statistical theory of the diffusion of a passive tracer in a random velocity field

A passive tracer on the surface of an incompressible liquid behaves like a tracer in two-dimensional compressible flows, whose characteristic feature is the formation of cluster structures, i.e., compact regions of increased density surrounded by vast low-density regions. The cluster formation dynamics are studied, and statistical spatiotemporal characteristics of the density fields, which faithfully reflect the properties of the cluster structures, are calculated. Zh. éksp. Teor. Fiz. 111, 1297–1313 (April 1997)