Continuum models of two-dimensional random droplets

We study the statistical mechanical properties of a two-dimensional assembly of free particles coupled to a mechanical reservoir. The particles-reservoir interaction is modelised by an Hamiltonian depending on the convex hull of the particles only. We concentrate on models whose energy is the sum of an area-term, a perimeter term and possibly a term preventing the particles occupying the interior of the convex hull. The range of coupling constants insuring a thermodynamic behaviour, as well as the associated free energy per particle are exactly determined.Work partially supported by the Swiss National Foundation for Scientific Research.     

Algebraic approach to Kaluza-Klein models

We propose a new method for the construction of injections into classical Lie algebras. This method is very useful for the analysis of vacuum solutions in Kaluza-Klein models. With the help of the proposed procedure, one can obtain all the known solutions as well as a new one explicitly constructed in this letter.     

Continuum symmetries of lattice models with staggered fermions

The validity of the flavour interpretation of staggered fermions is discussed in terms of the discrete symmetries of the interaction terms. Some aspects of the embedding of these symmetries in the symmetry group of the continuum limit are clarified. An explicit calculation, at first non-trivial order in 1/N, of the four-point function for a latticized Gross-Neveu model yields the same result in the continuum limit as the continuum theory for 2N fermions. A proof is then given that flavour and C, P, and T symmetries are restored in the continuum limit of 2-point correlation functions, for interactions, including the case of 4-dimensional QCD, which respect the discrete symmetries of the free action.     

Continuum limit of lattice models built on quantum groups

We study the continuum limit of certain 2D lattice models built on quantum groups. Under the assumption that they renormalize to a Coulomb gas, we determine the central charge and critical exponents.     

A new approach to nonrenormalizable models

Nonrenormalizable quantum field theories require counter-terms; and based on the hard-core interpretation of such interactions, it is initially argued, contrary to the standard view, that counter-terms suggested by renormalized perturbation theory are in fact inappropriate for this purpose. Guided by the potential underlying causes of triviality of such models, as obtained by alternative analyses, we focus attention on the ground-state distribution function, and suggest a formulation of such distributions that exhibits nontriviality from the start. Primary discussion is focused on self-interacting scalar fields. Conditions for bounds on general correlation functions are derived, and there is some discussion of the issues involved with the continuum limit.     

Non-perturbative renormalization-group approach to lattice models

The non-perturbative renormalization-group approach is extended to lattice models, considering as an example a φ⁴ theory defined on a d-dimensional hypercubic lattice. Within a simple approximation for the effective action, we solve the flow equations and obtain the renormalized dispersion epsilon(q) over the whole Brillouin zone of the reciprocal lattice. In the long-distance limit, where the lattice does not matter any more, we reproduce the usual flow equations of the continuum model. We show how the numerical solution of the flow equations can be simplified by expanding the dispersion in a finite number of circular harmonics.     

Top-down approach to unified supergravity models

We introduce a new approach for studying unified supergravity models. In this approach all the parameters of the grand unified theory (GUT) are fixed by imposing the corresponding number of low energy observables. This determines the remaining particle spectrum whose dependence on the low energy observables can now be investigated. We also include some SUSY threshold corrections that have previously been neglected. In particular the SUSY threshold corrections to the fermion masses can have a significant impact on the Yukawa coupling unification.     

Field theoretical approach to spin models

We developed a systematic non-perturbative method base on Dyson–Schwinger theory and the Φ-derivable theory for Ising model at broken phase. Based on these methods, we obtain critical temperature and spin spin correlation beyond mean field theory. The spectrum of Green function obtained from our methods become gapless at critical point, so the susceptibility become divergent at Tc. The critical temperature of Ising model obtained from this method is fairly good in comparison with other non-cluster methods. It is straightforward to extend this method to more complicate spin models for example with continue symmetry.     

Algebraic approach to anomalous sigma models

Anomalous σ-models in 1+1 and 1+3 dimensions are analysed using purely algebraic methods. We find, in the 1+1 dimensional example, a current algebra containing an arbitrary parameter which is compatible with the Wess-Zumino anomaly and anon-vanishing curvature. The consistency of the algebra is checked by means of a consistency condition and the Jacobi identity. In the 1+3 dimensional case, however, the conventional (anomalous) current algebra with a vanishing curvature is reproduced.     

Derivation of non-local macroscopic traffic equations and consistent traffic pressures from microscopic car-following models

This contribution compares several different approaches allowing one to derive macroscopic traffic equation directly from microscopic car-following models. While it is shown that some conventional approaches lead to theoretical problems, it is proposed to use an approach reminding of smoothed particle hydrodynamics to avoid gradient expansions. The derivation circumvents approximations and, therefore, demonstrates the large range of validity of macroscopic traffic equations, without the need of averaging over many vehicles. It also gives an expression for the “traffic pressure”, which generalizes previously used formulas. Furthermore, the method avoids theoretical inconsistencies of macroscopic traffic models, which have been criticized in the past by Daganzo and others.     

Numerical linked-cluster approach to quantum lattice models

We present a novel algorithm that allows one to obtain temperature dependent properties of quantum lattice models in the thermodynamic limit from exact diagonalization of small clusters. Our numerical linked-cluster approach provides a systematic framework to assess finite-size effects and is valid for any quantum lattice model. Unlike high temperature expansions, which have a finite radius of convergence in inverse temperature, these calculations are accurate at all temperatures provided the range of correlations is finite. We illustrate the power of our approach studying spin models on kagomé, triangular, and square lattices.     

A new approach to modified gravity models

We investigate f (R)-gravity models performing the ADM-slicing of standard General Relativity. We extract the static, spherically-symmetric vacuum solutions in the general case, which correspond to either Schwarzschild de-Sitter or Schwarzschild anti-de-Sitter ones. Additionally, we study the cosmological evolution of a homogeneous and isotropic universe, which is governed by an algebraic and not a differential equation. We show that the universe admits solutions corresponding to acceleration at late cosmological epochs, without the need of fine-tuning the model-parameters or the initial conditions.     

Continuum approach to phonon gas and shape changes of second sound via shock waves theory

Summary A continuum approach, based on the principles of modern extended thermodynamics, describing the model of a phonon gas is performed. The main difference with the ideal phonon gas theory consists in the presence of athermal inertia. We apply the shock wave theory and discuss the selection rules for physical shocks (theLax conditions and theentropy growth). In this way the existence of two new kinds of shocks (hot andcold shocks) in rigid heat conductors at low temperature is pointed out. In particular a critical temperature, characteristic of each material, changing the structure of the previous types of shocks is analytically deduced. This characteristic temperature permits also to explain the modification of the received second sound wave form with respect to the initial wave profile. Finally, the results are applied to the case of high-purity crystals (NaF, Bi,³He and⁴He) and compared with experimental results.     

An approach to one-dimensional elliptic quasi-exactly solvable models

One-dimensional Jacobian elliptic quasi-exactly solvable second-order differential equations are obtained by introducing the generalized third master functions. It is shown that the solutions of these differential equations are generating functions for a new set of polynomials in terms of energy with factorization property. The roots of these polynomials are the same as the eigenvalues of the differential equations. Some one-dimensional elliptic quasi-exactly quantum solvable models are obtained from these differential equations.