A mathematical structure for the generalization of conventional algebra

An abstract mathematical framework is presented in this paper as a unification of several deformed or generalized algebra proposed recently in the context of generalized statistical theories intended to treat certain complex thermodynamic or statistical systems. It is shown that, from a mathematical point of view, any bijective function can in principle be used to formulate an algebra in which the conventional algebraic rules are generalized.      

Generalized algebra within a nonextensive statistics

By considering generalized logarithm and exponential functions used in nonextensive statistics, the four usual algebraic operators: addition, subtraction, product and division, are generalized. The properties of the generalized operators are investigated. Some standard properties are preserved, e.g. associativity, commutativity and existence of neutral elements. On the contrary, the distributivity law and the opposite element are no more universal within the generalized algebra.     

Theoretical self-consistency in nonextensive statistical mechanics with parameter transformation

Self-consistency in nonextensive statistical mechanics is studied as a recourse to parameter transformation, where different nonextensive parameters are presented for various theoretical branches. The unification between the first and third choices of the average definition and that between the normal and escort distributions are both examined. The problem of parameter inversion in the generalized H theorem is also investigated. The inconsistency between the statistical ensemble pressure and molecular dynamics pressure can be eliminated. This work also verifies the equivalence of physical temperature and gravitational temperature in nonextensive statistical mechanics. In these parameter transformations, the Tsallis entropy form is observed to remain invariant.     

Landau damping of longitudinal oscillation in ultra- relativistic plasmas with nonextensive distribution

The generalized dispersion equation for longitudinal oscillation in an unmagnetized, collisionless, isotropic and relativistic plasma is derived in the context of nonextensive q-distribution. An analytical expression for the Landau damping is obtained in an ultra-relativistic regime, which is related to q-parameter. In the limit q → 1, the result based on the relativistic Maxwellian distribution is recovered. It is shown that the interactions between the wave and particles are stronger and the waves are more strongly damped for lower values of q-parameter. The results are explained by the increased number of superthermal particles or low velocity particles contained in the plasma with the nonextensive distribution.     

谐振势阱中旋转广义玻色系统的热力学性质

以非广延Tsallis统计理论为基础,导出了广义玻色-爱因斯坦统计分布表达式,并用其分别讨论了三维和二维谐振势阱约束的旋转广义玻色气体的热力学性质.结合系统粒子数、玻色-爱因斯坦凝聚(BEC)临界温度、基态粒子占据率和比热等物理量的解析表达式,分析了非广延参数和势阱旋转频率等因素对系统热力学性质的影响.     

非广延统计力学与完全开放系统的统计分布

简介了非广延统计力学的Tsallis统计,用其计算了理想气体;推导出了以含有非广延熵常数的Shannon熵为基础和以Tsallis熵为基础的非广延统计力学的完全开放系统的统计分布及计算热力学量的公式;讨论表明:Tsallis熵对应的统计分布及计算热力学量的公式在非广延参量q→1时,完全过渡到了Shannon熵对应的形式.     

Incomplete nonextensive statistics and the zeroth law of thermodynamics

On the basis of the entropy of incomplete statistics (IS) and the joint probability factorization condition, two controversial problems existing in IS are investigated: one is what expression of the internal energy is reasonable for a composite system and the other is whether the traditional zeroth law of thermodynamics is suitable for IS. Some new equivalent expressions of the internal energy of a composite system are derived through accurate mathematical calculation. Moreover, a self-consistent calculation is used to expound that the zeroth law of thermodynamics is also suitable for IS, but it cannot be proven theoretically. Finally, it is pointed out that the generalized zeroth law of thermodynamics for incomplete nonextensive statistics is unnecessary and the nonextensive assumptions for the composite internal energy will lead to mathematical contradiction.     

Remarks on l-conformal extension of the Newton-Hooke algebra

The l-conformal extension of the Newton-Hooke algebra proposed in [J. Negro, M.A. del Olmo, A. Rodriguez-Marco, J. Math. Phys. 38 (1997) 3810] is formulated in the basis in which the flat space limit is unambiguous. Admissible central charges are specified. The infinite-dimensional Virasoro-Kac-Moody type extension is given.     

多项式角动量代数的代数表示及实现

本文利用三参数李群求代数表示的方法求出多项式角动量代数的代数表示及其酉表示，找到一个能同时承载李代数及相对应的多项式角动量代数的基底，并在该基底下求出两种代数之间的联系，利用该联系则也可求出多项式角动量代数的代数表示.最后求出多项式角动量代数的单玻色实现及其在有限维多项式函数空间的微分实现. 关键词： 多项式角动量代数 Higgs代数 su(2)代数     

Optical Mueller matrices in terms of geometric algebra

Connection between optical Mueller matrices and geometrical (Clifford) algebra multivectors is established. It is shown that starting from 3-dimensional (3D) Cl_3,0 algebra and using isomorphism between Cl_3,0 and even Cl_3,1⁺ subalgebra one can generate canonical Mueller matrices and their combinations that describe an optical system. It appears that representation of polarization devices in terms of geometric algebra is very compact and, in contrast to Mueller matrix approach, there is no need for speculative physical restrictions. If needed, properties of media can be logically introduced into Maxwell equation in a form of Clifford algebra via constitutive relations. Since representation of polarization by Cl_3,1 algebra is Lorentz invariant it allows to include relativistic effects of moving bodies on light polarization as well. In this paper only simple examples of connection between Mueller matrices and geometric algebra multivectors is presented.     

A diagrammatic categorification of the fermion algebra

In this paper, we study the diagrammatic categorification of the fermion algebra. We construct a graphical category corresponding to the one-dimensional （1D） fermion algebra, and we investigate the properties of this category. The categorical analogues of the Fock states are some kind of 1-morphisms in our category, and the dimension of the vector space of 2-morphisms is exactly the inner product of the corresponding Fock states. All the results in our categorical framework coincide exnetlv with those in normal quantum mechanics.     

Differential geometry of the Lie algebra of the quantum plane

We present a differential calculus on the extension of the quantum plane obtained by considering that the (bosonic) generator x is invertible and by working with polynomials in ln x instead of polynomials in x. We construct the quantum Lie algebra associated with this extension and obtain its Hopf algebra structure and its dual Hopf algebra.     

Investigation of Heisenberg algebra for a modified anti-de Sitter and modified antiSnyder models

In this paper,we consider a possible modification of the de Sitter and anti-de Sitter space for the extended uncertainty principle.For the modified anti-de Sitter model we discuss the representation and wave functions of the momentum operator for a one-dimensional box problem.Also,we consider modified Snyder and anti-Snyder models for the generalized uncertainty principle.Then,we assume the Hamiltonian with different potential and solve the Heisenberg algebra for the modified(anti)-de Sitter and(anti)-Snyder models in both position and in the momentum space.     

Exploring the Neighborhood of q-Exponentials

The q-exponential form eqx≡[1+(1−q)x]1/(1−q)(e1x=ex) is obtained by optimizing the nonadditive entropy Sq≡k1−∑ipiqq−1 (with S1=SBG≡−k∑ipilnpi, where BG stands for Boltzmann–Gibbs) under simple constraints, and emerges in wide classes of natural, artificial and social complex systems. However, in experiments, observations and numerical calculations, it rarely appears in its pure mathematical form. It appears instead exhibiting crossovers to, or mixed with, other similar forms. We first discuss departures from q-exponentials within crossover statistics, or by linearly combining them, or by linearly combining the corresponding q-entropies. Then, we discuss departures originated by double-index nonadditive entropies containing Sq as particular case.     

A nonextensive approach to the dynamics of financial observables

We present results about financial market observables, specifically returns and traded volumes. They are obtained within the current nonextensive statistical mechanical framework based on the entropy . More precisely, we present stochastic dynamical mechanisms which mimic probability density functions empirically observed. These mechanisms provide possible interpretations for the emergence of the entropic indices q in the time evolution of the corresponding observables. In addition to this, through multi-fractal analysis of return time series, we verify that the dual relation q_stat+q_sens=2 is numerically satisfied, q_stat and q_sens being associated to the probability density function and to the sensitivity to initial conditions respectively. This type of simple relation, whose understanding remains ellusive, has been empirically verified in various other systems.     

量子变换的李代数方法

利用李代数生成元构成幺正算符,对耦合玻色体系给出了一种简单的对角化方法.     

Auxiliary ghost fields in statistical dynamics

The path integral formulation of statistical dynamics involves a functional determinant whose role within the theory has remained somewhat unclear. This has occasionally led to incorrect generalizations of the formalism to the case of multiplicative random forces. We present a hidden symmetry of the theory and show how it can be used to clarify these issues. Important further applications are also pointed out.