The relativistic statistical theory and Kaniadakis entropy: an approach through a molecular chaos hypothesis

We have investigated the proof of the H theorem within a manifestly covariant approach by considering the relativistic statistical theory developed in [G. Kaniadakis, Phys. Rev. E 66, 056125 (2002); G. Kaniadakis, Phys. Rev. E 72, 036108 (2005)]. As it happens in the nonrelativistic limit, the molecular chaos hypothesis is slightly extended within the Kaniadakis formalism. It is shown that the collisional equilibrium states (null entropy source term) are described by a κ power law generalization of the exponential Juttner distribution, e.g., , with θ=α(x)+β_μp^μ, where α(x) is a scalar, β_μ is a four-vector, and p^μ is the four-momentum. As a simple example, we calculate the relativistic κ power law for a dilute charged gas under the action of an electromagnetic field F^μν. All standard results are readly recovered in the particular limit κ→0.     

Entropy Production and Information Gain¶in Axiom-A Systems

Anosov systems are mathematical models for chaotic systems in statistical mechanics and fluid dynamics. Most of these systems enjoy the property of positive entropy production. We introduce the concept of specific information gain (or specific relative entropy) h(μ₊,μ₋) in Anosov systems and prove that it is identical to the entropy production rate e _p (μ₊) defined by Ruelle and Gallavotti in Anosov systems. From this point of view, the entropy production rate e _p (μ₊ characterizes the degree of macroscopic irreversibility of the system. Received: 2 August 1999 / Accepted: 14 April 2000     

Neutron halos in hypernuclei

Properties of single- Λ and double- Λ hypernuclei for even-N Ca isotopes ranging from the proton dripline to the neutron dripline are studied using the relativistic continuum Hartree-Bogoliubov theory with a zero-range pairing interaction. Compared with ordinary nuclei, the addition of one or two Λ-hyperons lowers the Fermi level. The predicted neutron dripline nuclei are, respectively, ⁷⁵ _ΛCa and ⁷⁶ _2ΛCa, as the additional attractive force provided by the Λ-N interaction shifts nuclei from outside to inside the dripline. Therefore, the last bound hypernuclei have two more neutrons than the corresponding ordinary nuclei. Based on the analysis of two-neutron separation energies, neutron single-particle energy levels, the contribution of continuum and nucleon density distribution, giant halo phenomena due to the pairing correlation, and the contribution from the continuum are suggested to exist in Ca hypernuclei similar to those that appear in ordinary Ca isotopes. Received: 21 October 2002 / Accepted: 11 January 2003 / Published online: 8 April 2003     

Weak chaos and metastability in a symplectic system of many long-range-coupled standard maps

We introduce, and numerically study, a system of N symplectically and globally coupled standard maps localized in a d=1 lattice array. The global coupling is modulated through a factor r^-α, being r the distance between maps. Thus, interactions are long-range (nonintegrable) when 0≤α≤1, and short-range (integrable) when α>1. We verify that the largest Lyapunov exponent λ_M scales as λ_M ∝ N^-κ(α), where κ(α) is positive when interactions are long-range, yielding weak chaos in the thermodynamic limit N↦∞ (hence λ_M→0). In the short-range case, κ(α) appears to vanish, and the behaviour corresponds to strong chaos. We show that, for certain values of the control parameters of the system, long-lasting metastable states can be present. Their duration t_c scales as t_c ∝N^β(α), where β(α) appears to be numerically in agreement with the following behavior: β>0 for 0 ≤α< 1, and zero for α≥1. These results are consistent with features typically found in nonextensive statistical mechanics. Moreover, they exhibit strong similarity between the present discrete-time system, and the α-XY Hamiltonian ferromagnetic model.     

Formation of Cs molecules via Feshbach resonances stabilized by spontaneous emission: theoretical treatment with the Fourier grid method

This paper proposes a general method to investigate Feshbach resonances in atomic collisions similar to Cs(6 s ) + Cs(6 p ) in the thermal or cold regime. In order to compute the predissociation widths of the C ¹ Π _u (6 s + 5 d ) bound vibrational levels of Cs₂, coupled both with the (2) ³ Σ ⁺ _u (6 s + 6 p ) continuum and with the (2) ³ Π _u (6 s + 5 d ) vibrational series, a Fourier grid method is implemented, with an optical potential. A convenient way of optimizing the latter is proposed. A large number of resonances are found and calculations of their cross-sections for stabilization into ground state molecules show that the rate may be important. This confirms the interpretation of Lintz and Bouchiat [Phys. Rev. Lett. 80, 2570 (1998)] who observed dimer formation in cell experiments. Possible generalization to the cold regime relies on the possibility to tune the position of a resonance to coincide with the maximum of the collisional energy distribution. Received 14 February 2002 Published online 28 June 2002     

Quantum Motion on 2D Surfaces of Spherical Topology

When the motion of a particle is constrained on the two-dimensional surface, excess terms exist in usual kinetic energy 1/(2μ) ∑ p _i ² with hermitian form of Cartesian momentum p _i (i = 1,2,3), and the operator ordering should be taken into account in the kinetic energy which turns out to be 1/(2μ) ∑ (1/f _i )p _i f _i p _i where the functions f _i are dummy factors in classical mechanics and nontrivial in quantum mechanics. In this article, the explicit forms of the dummy functions f _i for quantum motion on some 2D surfaces of revolution of spherical topology are given. PACS numbers: 03.65.-w Quantum mechanics, 04.60.Ds Canonical quantization.     

Derivatives on the isotropic tensor functions

The derivative of the isotropic tensor function plays an important part in continuum mechanics and computational mechanics, and also it is still an opening problem. By means of a scalar response function f(Λ_i, I ₁, I ₂) and solving a tensor equation, this problem is well studied. A compact explicit expression for the derivative of the isotropic tensor function is presented, which is valid for both distinct and repeated eigenvalue cases. Throughout the analysis, the formulation holds for general isotropic tensor functions without need to solve eigenvector problems or determine coefficients. On the theoretical side, a very simple solution of a tensor equation is obtained. As an application to continuum mechanics, a base-free expression for the Hill’s strain rate is given, which is more compact than the existent results. Finally, with an example we compute the derivative of an exponent tensor function. And the efficiency of the present formulations is demonstrated. We dedicate this work with deep respect and admiration to the memory of Prof. Gao Yuchen.     

Decay asymmetry in non-mesonic weak decay of light Λ -hypernuclei

The proton decay asymmetry, α^NM _p , of the polarized Λ -hypernuclei, ⁵ _ΛHe , ¹² _ΛC and ¹¹ _ΛB , has been investigated to understand the reaction mechanism of the non-mesonic weak-decay process. These Λ -hypernuclei were produced in the highest statistics ever via the (π⁺, K ⁺) reaction at 1.05GeV/c by using the SKS spectrometer at KEK 12GeV PS. The results show that the α^NM _p are very small for these s -shell and p -shell hypernuclei.     

A Statistical Mechanics Approach for a Rigidity Problem

We focus the problem of establishing when a statistical mechanics system is determined by its free energy. A lattice system, modelled by a directed and weighted graph (whose vertices are the spins and its adjacency matrix M will be given by the system transition rules), is considered. For a matrix A(q), depending on the system interactions, with entries which are in the ring Z[a ^q :a∈R ⁺] and such that A(0) equals the integral matrix M, the system free energy β _A (q) will be defined as the spectral radius of A(q). This kind of free energy will be related with that normally introduced in Statistical Mechanics as proportional to the logarithm of the partition function. Then we analyze under what conditions the following statement could be valid: if two systems have respectively matrices A,B and β _A = β _B then the matrices are equivalent in some sense. Issues of this nature receive the name of rigidity problems. Our scheme, for finite interactions, closely follows that developed, within a dynamical context, by Pollicott and Weiss but now emphasizing their statistical mechanics aspects and including a classification for Gibbs states associated to matrices A(q). Since this procedure is not applicable for infinite range interactions, we discuss a way to obtain also some rigidity results for long range potentials.     

κ-generalized statistics in personal income distribution

Starting from the generalized exponential function , with exp ₀(x)=exp (x), proposed in reference [G. Kaniadakis, Physica A 296, 405 (2001)], the survival function P_>(x)=exp _κ(-βx^α), where x∈R⁺, α,β>0, and , is considered in order to analyze the data on personal income distribution for Germany, Italy, and the United Kingdom. The above defined distribution is a continuous one-parameter deformation of the stretched exponential function P_> ⁰(x)=exp (-βx^α) to which reduces as κ approaches zero behaving in very different way in the x→0 and x→∞ regions. Its bulk is very close to the stretched exponential one, whereas its tail decays following the power-law P_>(x)∼(2βκ)^-1/κx^-α/κ. This makes the κ-generalized function particularly suitable to describe simultaneously the income distribution among both the richest part and the vast majority of the population, generally fitting different curves. An excellent agreement is found between our theoretical model and the observational data on personal income over their entire range.     

Study of inclusive strange-baryon production and search for pentaquarks in two-photon collisions at LEP

Measurements of inclusive production of the Λ, Ξ^- and Ξ^*(1530) baryons in two-photon collisions with the L3 detector at LEP are presented. The inclusive differential cross sections for Λ and Ξ^- are measured as a function of the baryon transverse momentum, p_t, and pseudo-rapidity, η. The mean number of Λ, Ξ^- and Ξ^*(1530) baryons per hadronic two-photon event is determined in the kinematic range 0.4 GeV<p_t<2.5 GeV, |η|<1.2. Overall agreement with the theoretical models and Monte Carlo predictions is observed. A search for inclusive production of the pentaquark θ⁺(1540) in two-photon collisions through the decay θ⁺→pK⁰ _S is also presented. No evidence for production of this state is found.     

Stability of incomplete entropy and incomplete expectation

Based on the principle of maximum entropy, the q-exponential distribution can be derived from several different nonextensive entropies including the incomplete entropy. It is widely used in nonextensive statistical mechanics. In the present paper, it is shown that the incomplete expectation value and incomplete entropy are stable under small deformation of the probability distribution function of the system.     

Impurity nuclear physics

Using a germanium-detector array for hypernuclear γ spectroscopy (Hyperball), we measured B(E2) of the ⁷ _ΛLi hypernucleus and observed a significant shrinkage of the ⁶Li core induced by a Λ-particle. In this way, nuclear properties can be drastically changed by introducing a Λ-particle, which can be investigated by high-resolution hypernuclear γ spectroscopy. In the future neutron-rich hypernuclei will also be studied, where interesting modifications of nuclear structure by a Λ-particle are expected. Received: 1 May 2001 / Accepted: 4 December 2001     

Generalizations of Quantum Mechanics Induced by Classical Statistical Field Theory

No Heading We show that the Dirac-von Neumann formalism for quantum mechanics can be obtained as an approximation of classical statistical field theory. This approximation is based on the Taylor expansion (up to terms of the second order) of classical physical variables – maps f : Ω → R, where Ω is the infinite-dimensional Hilbert space. The space of classical statistical states consists of Gaussian measures ρ on Ω having zero mean value and dispersion σ²(ρ) ≈ h. This viewpoint to the conventional quantum formalism gives the possibility to create generalized quantum formalisms based on expansions of classical physical variables in the Taylor series up to terms of nth order and considering statistical states ρ having dispersion σ²(ρ) = hⁿ (for n = 2 we obtain the conventional quantum formalism).     

Structure of neutron-rich Λ hypernuclei

Properties of light neutron-rich Λ hypernuclei (¹⁶ _ΛC, ¹² _ΛBe, and ¹¹ _ΛLi) are calculated within the Skyrme-Hartree-Fock approach. Interplay between hypernuclear interaction features and properties of these hypernuclei is studied. Response of weakly bound neutron states to hyperon addition depends generally on core distortion by hyperon, and it is essentially different for the different states. This response is especially sensitive to details of the ΛN interaction for 1p _1/2 states. Implications of the nuclear spin-orbit potential and nuclear incompressibility in the neutron-rich system properties are inferred. Dependence of the Λ binding energies in hypernuclei on Z at fixed A is discussed. Received: 16 December 1998     

Non-coherent scattering in subordinate lines: III. Generalized redistribution functions

The concept of the redistribution function, which was originally introduced in order to describe a correlation between frequencies and directions of the absorbed and emitted photon in resonance scattering, has been extended to other resonance two-photon processes including resonance Raman scattering, resonance two-photon absorption and emission, and inverse Raman scattering. We have derived, within the frame of the impact approximation, the appropriate form of the generalized redistribution function. Using a suitable formalism, the generalized redistribution function takes the same form for all types of two-photon processes and contains all the redistribution functions, considered previously, as various limiting cases. In analogy to Hummer's original scheme of redistribution functions, we have derived a similar set of generalized redistribution functions, denoted as p_i(i = I, II, III, IV, V), and we have shown that the most general case is described by a linear combination of p_III and p_V, analogously to the previous results. Explicit formulae for the velocity-averaged (laboratory-frame) generalized redistribution functions p_i(i = I, II, III, IV, V) are given and possible numerical methods for their evaluation are briefly indicated.     

Second-order moving average and scaling of stochastic time series

Long-range correlation properties of stochastic time series y(i) have been investigated by introducing the function σ² _MA = [y(i) - (i)]², where (i) is the moving average of y(i), defined as 1/n y(i - k), n the moving average window and N_max is the dimension of the stochastic series. It is shown that, using an appropriate computational procedure, the function σ _MA varies as n^H where H is the Hurst exponent of the series. A comparison of the power-law exponents obtained using respectively the function σ _MA and the Detrended Fluctuation Analysis has been also carried out. Interesting features denoting the existence of a relationship between the scaling properties of the noisy process and the moving average filtering technique have been evidenced. Received 31 December 2001     

Nonadditive entropy: The concept and its use

The thermodynamical concept of entropy was introduced by Clausius in 1865 in order to construct the exact differential dS = Q/T , where Q is the heat transfer and the absolute temperature T its integrating factor. A few years later, in the period 1872-1877, it was shown by Boltzmann that this quantity can be expressed in terms of the probabilities associated with the microscopic configurations of the system. We refer to this fundamental connection as the Boltzmann-Gibbs (BG) entropy, namely (in its discrete form) , where k is the Boltzmann constant, and {p _i} the probabilities corresponding to the W microscopic configurations (hence ∑^W _i=1 p _i = 1 . This entropic form, further discussed by Gibbs, von Neumann and Shannon, and constituting the basis of the celebrated BG statistical mechanics, is additive. Indeed, if we consider a system composed by any two probabilistically independent subsystems A and B (i.e., , we verify that . If a system is constituted by N equal elements which are either independent or quasi-independent (i.e., not too strongly correlated, in some specific nonlocal sense), this additivity guarantees S_BG to be extensive in the thermodynamical sense, i.e., that in the N ≫ 1 limit. If, on the contrary, the correlations between the N elements are strong enough, then the extensivity of S_BG is lost, being therefore incompatible with classical thermodynamics. In such a case, the many and precious relations described in textbooks of thermodynamics become invalid. Along a line which will be shown to overcome this difficulty, and which consistently enables the generalization of BG statistical mechanics, it was proposed in 1988 the entropy . In the context of cybernetics and information theory, this and similar forms have in fact been repeatedly introduced before 1988. The entropic form S_q is, for any q 1 , nonadditive. Indeed, for two probabilistically independent subsystems, it satisfies . This form will turn out to be extensive for an important class of nonlocal correlations, if q is set equal to a special value different from unity, noted q_ent (where ent stands for entropy . In other words, for such systems, we verify that , thus legitimating the use of the classical thermodynamical relations. Standard systems, for which S_BG is extensive, obviously correspond to q _ent = 1 . Quite complex systems exist in the sense that, for them, no value of q exists such that S_q is extensive. Such systems are out of the present scope: they might need forms of entropy different from S_q, or perhaps --more plainly-- they are just not susceptible at all for some sort of thermostatistical approach. Consistently with the results associated with S_q, the q -generalizations of the Central Limit Theorem and of its extended Lévy-Gnedenko form have been achieved. These recent theorems could of course be the cause of the ubiquity of q -exponentials, q -Gaussians and related mathematical forms in natural, artificial and social systems. All of the above, as well as presently available experimental, observational and computational confirmations --in high-energy physics and elsewhere-- are briefly reviewed. Finally, we address a confusion which is quite common in the literature, namely referring to distinct physical mechanisms versus distinct regimes of a single physical mechanism. This paper is part of the Topical Issue Statistical Power Law Tails in High-Energy Phenomena.