Narrow Results

In this paper, we aim to extend to interacting massive and massless fermionic theories the recent perturbative construction of equilibrium states developed within the framework of perturbative algebraic quantum field theory on Lorentzian spacetime. We analyze the case of interactions which depend on time by a smooth switch-on function and on space by a suitably bounded function that multiplies an interaction Lagrangian density constructed with the field of the theory. The construction is achieved by first considering the case of compact support and, in a second step, by removing the space cutoff with a suitable limit (adiabatic limit). As an application, we consider a Dirac field interacting with a classical stationary background electromagnetic potential, and we compute at first perturbative order (linear response) the expectation value of the conserved current on the equilibrium state for the interacting theory. The resulting expectation value is written as a convolution, in the space coordinates, between the electromagnetic potential and an integral kernel which, at vanishing conjugate momentum, gives the inverse of the square Debye screening length at finite temperature. The corresponding Debye screening effect is visible in the backreaction treated semiclassically of this current on the classical background electromagnetic potential sourced by a classical external current.

A gauge-invariant C*-system is obtained as the fixed point subalgebra of the infinite tensor product of full matrix algebras under the tensor product unitary action of a compact group. In this paper, thermodynamics is studied in such systems and the chemical potential theory developed by Araki, Haag, Kastler and Takesaki is used. As a generalization of quantum spin system, the equivalence of the KMS condition, the Gibbs condition and the variational principle is shown for translation-invariant states. The entropy density of extremal equilibrium states is also investigated in relation to macroscopic uniformity.

The thermodynamic impact of the Coulomb repulsion on s-wave superconductors is analyzed via a rigorous study of equilibrium and ground states of the strong coupling BCS-Hubbard Hamiltonian. We show that the one-site electron repulsion can favor superconductivity at fixed chemical potential by increasing the critical temperature and/or the Cooper pair condensate density. If the one-site repulsion is not too large, a first or a second order superconducting phase transition can appear at low temperatures. The Meißner effect is shown to be rather generic but coexistence of superconducting and ferromagnetic phases is also shown to be feasible, for instance, near half-filling and at strong repulsion. Our proof of a superconductor-Mott insulator phase transition implies a rigorous explanation of the necessity of doping insulators to create superconductors. These mathematical results are consequences of "quantum large deviation" arguments combined with an adaptation of the proof of Størmer's theorem [1] to even states on the CAR algebra.

We investigate a system of partial differential equations modeling ambipolar plasmas. The ambipolar — or zero current — model is obtained from general plasmas equations in the limit of vanishing Debye length. In this model, the electric field is expressed as a linear combination of macroscopic variable gradients. We establish that the governing equations can be written as a symmetric form by using entropic variables. The corresponding dissipation matrices satisfy the null space invariant property and the system of partial differential equations can be written as a normal form, i.e. in the form of a symmetric hyperbolic–parabolic composite system. By properly modifying the chemistry source terms and/or the diffusion matrices, asymptotic stability of equilibrium states is established and decay estimates are obtained. We also establish the continuous dependence of global solutions with respect to vanishing electron mass.

Motivated by recent strain-limiting models for solids and biological fibers, we introduce the first intrinsic set of nonlinear constitutive relations, between the geometrically exact strains and the components of the contact force and contact couple, describing a uniform, hyperelastic, strain-limiting special Cosserat rod. After discussing some attractive features of the constitutive relations (orientation preservation, transverse symmetry and monotonicity), we exhibit several explicit equilibrium states under either an isolated end thrust or an isolated end couple. In particular, certain equilibrium states exhibit Poynting like effects, and we show that under mild assumptions on the material parameters, the model predicts an explicit tensile shearing bifurcation: a straight rod under a large enough tensile end thrust parallel to its center line can shear.

The transmission of Plasmodium falciparum and Plasmodium vivax malaria in a mixed population of Thais and migrant Burmese living along the Thai-Myanmar border is studied through a mathematical model. The population is separated into two groups: Thai and Burmese. Each population in turn is divided into susceptible, infected, recovered and in case of vivax infection, a dormant subclass. The model is then modified to allow for some of the Burmese (given as a fraction P) to be infectious when they enter into Thailand. The behaviour of the modified model is obtained using a standard dynamical analysis. A new basic reproduction number is obtained. Numerical simulations of the modified model show that when P ≠ 0 and the same set of parameter values used in the initial model are used, the Thai population will be in the epidemic state. In other words, the repeated introduction of infectious Burmese (no matter how small of a number) will result in a malaria epidemic among the Thais irregardless of the public health practice undertaken by the Thai government. In the presence of the infected Burmese, the Thai government would have to increase the facilitites to treat the people who are infected by the malaria.

In the proposed study, a nonlinear model is developed to explore the interactive dynamics between cattle and invertebrates when they coexist in a grassland system and compete with one another for the same resource — the grass biomass. The constructed model is theoretically investigated using the qualitative theory of differential equations to show the system’s rich dynamical properties, which are crucial for maintaining the ecosystem’s balance in grasslands. The qualitative findings show that, depending on the parameter combinations, the system not only displays stability of many equilibrium states but also experiences transcritical and Hopf bifurcations. The model results support the idea that inter-specific competition between cattle and invertebrates does not always produce regular dynamic patterns but may also produce periodic and destabilizing patterns. The model’s outputs may assist in striking a balance between pasture and natural grass biomass in grassland with the invertebrates.

We prove that multimodal maps with an absolutely continuous invariant measure have exponential return time statistics around almost every point. We also show a "polynomial Gibbs property" for these systems, and that the convergence to the entropy in the Ornstein–Weiss formula has normal fluctuations. These results are also proved for equilibrium states of some Hölder potentials.

We prove uniqueness of equilibrium states for subshifts corresponding to intermediate beta transformations with $\beta >2$ having the property that the orbit of 0 is bounded away from 1.

Two non-interacting systems immersed in a common bath and evolving with a Markovian, completely positive dynamics can become initially entangled via a purely noisy mechanism. Remarkably, for certain phenomenologically relevant environments, the quantum correlations can persist even in the asymptotic long-time regime.

This talk contains a review of some results about homogeneous boson models, which are a special case of the general variational problem of statistical mechanics that can be solved in terms of quasi-free states. We apply these results to the model of the Mean-Field Boson Gas with Bardeen-Cooper-Schrieffer (BCS) interaction.

We survey some results on non-uniform hyperbolicity, geometric pressure and equilibrium states in one-dimensional real and complex dynamics. We present some relations with Hausdorff dimension and measures with refined gauge functions of limit sets for geometric coding trees for rational functions on the Riemann sphere. We discuss fluctuations of iterated sums of the potential −t log |f′| and of radial growth of derivative of univalent functions on the unit disc and the bound-aries of range domains preserved by a holomorphic map f repelling towards the domains.