Majorization properties of generalized thermal distributions

We examine the majorization properties of general thermal-like mixed states depending on a set of parameters. Sufficient conditions which ensure the increase in mixedness, and hence of any associated entropic form, when these parameters are varied, are identified. We then discuss those exhibiting a power law distribution, showing that they can be characterized by two distinct mixing parameters, one associated with temperature and the other with the non-extensivity index q. Illustrative numerical results are also provided.     

Fock spaces with generalized statistics

Fock representations for quantum fields which obey generalized statistics are explicitly constructed. The main features of these representations are investigated.     

Quantum groups and generalized statistics in integrable models

The paper deals with the integrable massive models of quantum field theory. It is shown that generalized statistics of physical particles is closely connected with the invariance under quantum groups. This invariance provides the possibility to construct quasi-local operators (parafermions) possessing generalized statistics which interpolates the physical particles. For the particular examples of SG, RSG models and scaling 3-state Potts model the parafermions are described completely (all their matrix elements in the space of states are presented).     

A generalized thermodynamics for power-law statistics

We show that there exists a natural way to define a condition of generalized thermal equilibrium between systems governed by Tsallis thermostatistics, under the hypotheses that (i) the coupling between the systems is weak, (ii) the structure functions of the systems have a power-law dependence on the energy. It is found that the q values of two such systems at equilibrium must satisfy a relationship involving the respective numbers of degrees of freedom. The physical properties of a Tsallis distribution can be conveniently characterized by a new parameter η which can vary between 0 and +∞, these limits corresponding, respectively, to the two opposite situations of a microcanonical distribution and of a distribution with a predominant power-tail at high energies. We prove that the statistical expression of the thermodynamic functions is univocally determined by the requirements that (a) systems at thermal equilibrium have the same temperature, (b) the definitions of temperature and entropy are consistent with the second law of thermodynamics. We find that, for systems satisfying the hypotheses (i) and (ii) specified above, the thermodynamic entropy is given by Rényi entropy.     

Thermodynamic Bethe ansatz for generalized extensive statistics

We investigate properties of the entropy density related to a generalized extensive statistics and derive the thermodynamic Bethe ansatz equation for a system of relativistic particles obeying such a statistics. We investigate the conformal limit of such a system. We also derive a generalized Y-system. The Gentile intermediate statistics and the statistics of γ-ons are considered in detail. In particular, we observe that certain thermodynamic quantities for the Gentile statistics majorize those for the Haldane–Wu statistics. Specifically, for the effective central charges related to affine Toda models we obtain nontrivial inequalities in terms of dilogarithms.     

Quantum liquids of particles with generalized statistics

We propose a phenomenological approach to quantum liquids of particles obeying generalized statistics of a fermionic type, in the spirit of the Landau Fermi liquid theory. The approach is developed for fractional exclusion statistics. We discuss both equilibrium (specific heat, compressibility, and Pauli spin susceptibility) and nonequilibrium (current and thermal conductivities, thermopower) properties. Low-temperature quantities have the same temperature dependences as for the Fermi liquid, with the coefficients depending on the statistics parameter. The novel quantum liquids provide an explicit realization of systems with a non-Fermi liquid Lorentz ratio in two and more dimensions. Consistency of the theory is verified by deriving the compressibility and f-sum rules.     

On some relations in the generalized Boltzmann-Enskog model

The generalized Boltzmann-Enskog equation with long-range potential which was considered to be approximate is proved to possess exact microscopic solutions. Solutions corresponding to a small periodical disturbance of the distribution function are studied. Corrections to the sound velocity are determined.     

Boundary relations and generalized resolvents of symmetric operators

The Kreĭn-Naĭmark formula provides a parametrization of all selfadjoint exit space extensions of a (not necessarily densely defined) symmetric operator in terms of maximal dissipative (in ℂ₊) holomorphic linear relations on the parameter space (the so-called Nevanlinna families). The new notion of boundary relation makes it possible to interpret these parameter families as Weyl families of boundary relations and to establish a simple coupling method to construct generalized resolvents from given parameter families. A general version of the coupling method is introduced and the role of the boundary relations and their Weyl families in the Kreĭn-Naĭmark formula is investigated and explained. These notions lead to several new results and new types of solutions to problems involving generalized resolvents and their applications, e.g., in boundary-value problems for (ordinary and partial) differential operators. For instance, an old problem going back to M. A. Naĭmark and concerning the analytic characterization of the so-called Naĭmark extensions is solved. The present research was supported by the Academy of Finland (project no. 116842).     

Uncertainty relations for noise and disturbance in generalized quantum measurements

Heisenberg’s uncertainty relation for measurement noise and disturbance is commonly understood to state that in any measurement the product of the position measurement noise and the momentum disturbance is not less than Planck’s constant divided by 4π. However, it has been shown in many ways that this relation holds only for a restricted class of measuring apparatuses in the most general formulation of measuring processes. Here, Heisenberg’s uncertainty relation is generalized to a relation that holds for all the possible quantum measurements, from which rigorous conditions are obtained for measuring apparatuses to satisfy Heisenberg’s relation. In particular, every apparatus with the noise and the disturbance statistically independent from the measured object is proven to satisfy Heisenberg’s relation. For this purpose, all the possible quantum measurements are characterized by naturally acceptable axioms. Then, a mathematical notion of the distance between probability operator valued measures and observables is introduced and the basic properties are explored. Based on this notion, the measurement noise and disturbance are naturally defined for any quantum measurements in a model independent formulation. Under this formulation, various relations for noise and disturbance are also derived for apparatuses with independent noise, independent disturbance, unbiased noise, and unbiased disturbance as well as noiseless apparatuses and nondisturbing apparatuses. Two models of position measurements are also discussed in the light of the new uncertainty relations to show that Heisenberg’s relation can be violated even by approximately repeatable position measurements.     

On the relations of antibunching,sub-poissonian statistics and squeezing

The squeezing phenomenon is investigated in optical parametric processes and in Raman and hyper-Raman scattering. It is related to the antibunching effect and the sub-poissonian statistics of optical fields in these processes, and it is demonstrated that squeezing accompanies antibunching very often, but not always. In some cases squeezing may occur and antibunching may not and vice versa.     

Fock representations of quantum fields with generalized statistics

We develop a rigorous framework for constructing Fock representations of quantum fields obeying generalized statistics. The main features of these representations are investigated. Various aspects of the underlying mathematical structure are illustrated by means of explicit examples.     

Some metallic properties in the framework of Tsallis generalized statistics

Some metallic quantities are calculated on the grounds of Tsallis generalized statistics: the specific heat at constant volume, c _V (T); the chemical potential, ; the Pauli paramagnetic susceptibility, and the Korringa constant, . First it is found that for a general value of q, the Sommerfeld expansion series will exhibit both, odd and even terms, contrary to what is obtained if we use the Fermi-Dirac (FD) statistics, where only even terms appear. It follows that: (i) the specific heat coefficient, , is q-dependent, but the temperature dependence of c_V remains linear, as in the FD case; (ii) the Fermi energy, , differs from the chemical potential by a linear term in T, and not quadratic, as in FD, the same happening for ; (iii) the Korringa constant is q-dependent, but not T-dependent. In the limit the results of FD statistics are recovered. Metallic thin films and multilayers exhibiting fractal surface structures are possible systems where the present results could be tested. Received 30 June 1999 and Received in final form 7 September 1999     

Transformation of recursion relations for generalized random walks

It is shown that recursion relation for the generalized random walks (GRW) or correlated random walks can be directly transformed into the recursion relation for the usual random walks. The recursion relation for the GRW is expressed by a non-linear difference equation. To transform the non-linear difference equation, the Hopf-Cole transformation is modified and expressed in a discrete form. Formal solution of the GRW is obtained in an integral representation.     

New invariant relations for the generalized two-field gyrostat

In the paper, we consider a completely integrable Hamiltonian system with three degrees of freedom found by V.V. Sokolov and A.V. Tsiganov. This system is known as the generalized two-field gyrostat. For the case of only gyroscopic forces present, we find new invariant four-dimensional submanifolds such that the induced dynamical systems are almost everywhere Hamiltonian with two degrees of freedom.