Non-extensive statistical entropy,quantum groups and quantum entanglement

In this work, we explore a new connection between quantum groups and Tsallis entropy through the energy spectrum of a Hamiltonian with S_Uq(2)$S{U}_q\left(2\right)$ symmetry. Identifying the deformation parameter of the entropy with the parameter of deformation of the associated quantum group, we deduce Tsallis entropy for states related to such a system with S_Uq(2)$S{U}_q\left(2\right)$ symmetry and conducted an investigation of quantum entanglement.     

On a nonstandard two-parametric quantum algebra and its connections withU p,q (gl(2)) andU p,q(gl(1/1))

A quantum algebraU _{p, q} (,H,X _±) associated with a nonstandardR-matrix with two deformation parameters (p, q) is studied and, in particular, its universal -matrix is derived using Reshetikhin's method. Explicit construction of the (p, q)-dependent nonstandardR-matrix is obtained through a coloured generalized boson realization of the universal -matrix of the standardU _{p, q}(gl(2)) corresponding to a nongeneric case. General finite dimensional coloured representation of the universal -matrix ofU _{p, q}(gl(2)) is also derived. This representation, in nongeneric cases, becomes a source for various (p, q)-dependent nonstandardR-matrices. Superization ofU _{p, q}(,H,X _±) leads to the super-Hopf algebraU _{p, q}(gl(1/1)). A contraction procedure then yields a (p, q)-deformed super-Heisenberg algebraU _{p, q}(sh(1)) and its universal -matrix.     

The effect of nonextensivity on the time development of quantum systems

In this study, the classical evolution operator has been generalized within Tsallis thermostatistics. By using the generalized evolution operator, the q-dependent transition probability of a quantal system, namely Fermi Golden rule, has been introduced as w (n → m) = ‖[U _q(t, t ₀)]_mn‖². In order to make the situation concrete, a simple example from quantum mechanics has been solved in the frame of this formalism and the effect of q-index has been clearly illustrated in the figures.     

Inhomogeneous Tsallis distributions in the HMF model

We study the maximization of the Tsallis functional at fixed mass and energy in the Hamiltonian Mean Field (HMF) model. We give a thermodynamical and a dynamical interpretation of this variational principle. This leads to q-distributions known as stellar polytropes in astrophysics. We study phase transitions between spatially homogeneous and spatially inhomogeneous equilibrium states. We show that there exists a particular index q _c = 3 playing the role of a canonical tricritical point separating first and second order phase transitions in the canonical ensemble and marking the occurence of a negative specific heat region in the microcanonical ensemble. We apply our results to the situation considered by Antoni and Ruffo [Phys. Rev. E 52, 2361 (1995)] and show that the anomaly displayed on their caloric curve can be explained naturally by assuming that, in this region, the QSSs are polytropes with critical index q _c = 3. We qualitatively justify the occurrence of polytropic (Tsallis) distributions with compact support in terms of incomplete relaxation and inefficient mixing (non-ergodicity). Our paper provides an exhaustive study of polytropic distributions in the HMF model and the first plausible explanation of the surprising result observed numerically by Antoni and Ruffo (1995). In the course of our analysis, we also report an interesting situation where the caloric curve presents both microcanonical first and second order phase transitions.     

Modified exponential function: the connection between nonextensivity and q-deformation

Tsallis thermostatistics has deep-rooted connection with quantum group formalism. Assuming that the modification of the standard exponential function considered in Tsallis thermostatistics has the same functional form as the one appearing in the q-calculus formalism and using the appropriate internal energy constraint, we derive the temperature dependent connection between the nonextensivity parameter and deformation parameter.     

Specific heat in the nonextensive statistics: effective temperature and Lagrange parameter β

We analyse the specific heat and the fluctuation–dissipation theorem by considering the effective temperature, T_eff≡(Trρ_q^q)/β, in the Tsallis statistics. In particular, the results show that the specific heat is nonnegative for q∉[0,1). We also investigate how to obtain a family of entropies employing the condition C_q=−β²(∂U_q/∂β)⩾0 for q>0, S_q=S_q(Trρ_q^q) and the normalized constraints.     

Self organized criticality in a modified Olami-Feder-Christensen model

A modified version of the Olami-Feder-Christensen model has been introduced to consider avalanche size differences. Our model well demonstrates the power-law behavior and finite size scaling of avalanche size distribution in any range of the adding parameter p _{a d d} of the model. The probability density functions of the avalanche size differences at consecutive time steps (defined as returns) appear to be well approached, in the thermodynamic limit, by q-Gaussian shape with appropriate q values which can be obtained a priori from the avalanche size exponent τ. For small system sizes, however, return distributions are found to be consistent with the crossover formulas proposed recently in Tsallis and Tirnakli [J. Phys. Conf. Ser. 201, 012001 (2010)]. Our results strengthen recent findings of Caruso et al. [Phys. Rev. E 75, 055101(R) (2007)] on the real earthquake data which support the hypothesis that knowing the magnitude of previous earthquakes does not make the magnitude of the next earthquake predictable.     

New approach in theory of Clebsch-Gordan coefficients foru(n) andU q(u(n))

A new method for calculation of Clebsch-Gordan coefficients (CGCs) of the Lie algebrau(n) and its quantum analogU _q(u(n)) is developed. The method is based on the projection operator method in combination with the Wigner-Racah calculus for the subalgebrau(n−1) (U _q(u(n−1))). The key formulas of the method are couplings of the tensor and projection operators and also a tensor form of the projection operator ofu(n) andU _q(u(n)). It is shown that theU _q(u(n)) CGCs can be presented in terms of theU _q(u)(n−1)) q−9j-symbols. Presented at the 9th International Colloquium: “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000. Supported by Russian Foundation for Fundamental Research, grant 99-01-01163. Supported in part by the U.S. National Science Foundation under Grant PHY-9970769 and Cooperative Agreement EPS-9720652 that includes matching from the Louisiana Board of Regents Support Fund.     

An extension of the Borel-Weil construction to the quantum groupU q (n)

The Borel-Weil (BW) construction for unitary irreps of a compact Lie group is extended to a construction of all unitary irreps of the quantum groupU _q(n). Thisq-BW construction uses a recursion procedure forU _q(n) in which the fiber of the bundle carries an irrep ofU _q(n–1)×U _q(1) with sections that are holomorphic functions in the homogeneous spaceU _q(n)/U _q(n–1)×U _q(1). Explicit results are obtained for theU _q(n) irreps and for the related isomorphism of quantum group algebras.Supported in part by the National Science Foundation, No. PHY-9008007     

The maximization of Tsallis entropy with complete deformed functions and the problem of constraints

By only requiring the q deformed logarithms (q exponentials) to possess arguments chosen from the entire set of positive real numbers (all real numbers), we show that the q-logarithm (q exponential) can be written in such a way that its argument varies between 0 and 1 (among negative real numbers) for 1?q<2, while the interval 0<q?1 corresponds to any real argument greater than 1 (positive real numbers). These two distinct intervals of the nonextensivity index q, also the expressions of the deformed functions associated with them, are related to one another through the relation (2−q), which is so far used to obtain the ordinary stationary distributions from the corresponding escort distributions, and vice versa in an almost ad hoc manner. This shows that the escort distributions are only a means of extending the interval of validity of the deformed functions to the one of ordinary, undeformed ones. Moreover, we show that, since the Tsallis entropy is written in terms of the q-logarithm and its argument, being the inverse of microstate probabilities, takes values equal to or greater than 1, the resulting stationary solution is uniquely described by the one obtained from the ordinary constraint. Finally, we observe that even the escort stationary distributions can be obtained through the use of the ordinary averaging procedure if the argument of the q-exponential lies in (−∞,0]. However, this case corresponds to, although related, a different entropy expression than the Tsallis entropy.     

DNA molecule as an elastic Heisenberg chain

A DNA molecule is simulated by an anisotropic elastic fiber which defines the configuration of the molecule central line and is supplemented with a chain of quantum two-level systems imitating hydrogen bonds between two polynucleotide chains in the DNA double helix. The system Hamiltonian consists of Kirchhoff’s classical elastic energy and the energy of a quantum anisotropic chain of “spins” 1/2. The two-level systems and macroscopic vector variables which determine the conformation of the central line are coupled by a classical vector field q, which is introduced to take into account the existence of two polynucleotide strands. Averaging over fast (microscopic) variables yields an effective potential U(q). In the approximation of weak coupling between the systems, the spectrum of elementary excitations and effective potential U(q) have been calculated in explicit form. The relation between elementary excitations in the “magnetic” subsystem and so-called breathing modes [C. Mandel, N. R. Kallenbach, and S. W. Englander, J. Mol. Biol. 135, 391 (1980); G. Manning, Biopolymers 22, 689 (1983)] corresponding to low-frequency excitations in DNA molecules is discussed. Zh. éksp. Teor. Fiz. 111, 1833–1844 (May 1997)     

Nonlinear diffusion under a time dependent external force: q-maximum entropy solutions

Nonlinear diffusion equations provide useful models for a number of interesting phenomena, such as diffusion processes in porous media. We study here a family of nonlinear Fokker-Planck equations endowed both with a power-law nonlinear diffusion term and a drift term with a time dependent force linear in the spatial variable. We show that these partial differential equations exhibit exact time dependent particular solutions of the Tsallis maximum entropy (q-MaxEnt) form. These results constitute generalizations of previous ones recently discussed in the literature [C. Tsallis, D.J. Bukman, Phys. Rev. E 54, R2197 (1996)], concerning q-MaxEnt solutions to nonlinear Fokker-Planck equations with linear, time independent drift forces. We also show that the present formalism can be used to generate approximate q-MaxEnt solutions for nonlinear Fokker-Planck equations with time independent drift forces characterized by a general spatial dependence. Received 25 April 2001 and Received in final form 6 June 2001     

Nonstandard U q(so3) and U q(so4): tensor products of representations, oscillator realizations and roots of unity

Nonstandard q-deformed algebras U _q(so₃) and U _q(so₄), which can be embedded into U _q(sl₃) and U _q(sl₄) and are coideals in them, are considered. It is shown how to multiply finite dimensional representations of U _q(so₃) when q is positive. Homomorphisms from U _q(so₃) and U _q(so₄) to the q-oscillator algebras are given. By making use of these homomorphisms, irreducible representations of U _q(so₃) and U _q(so₄) for q equal to a root of unity are obtained.     

Radial velocities of open stellar clusters: A new solid constraint favouring Tsallis maximum entropy theory

In the present study we apply a Tsallis maximum entropy distribution law to the study of the stellar residual radial velocity in a sample of 13 stellar open clusters. From a comparison between results obtained from the analysis based on Tsallis law and on the one based on the Maxwellian law we show that the generalized Tsallis distribution fits more closely the observed distribution of the stellar residual radial velocities for these stellar clusters. We have found clear evidences that the q-parameter in the Tsallis generalized distribution depends on stellar cluster ages for clusters older than . There is also some indication that q increases with cluster galactocentric distance. The results obtained in this work represent an additional solid constraint in the stellar astrophysics favoring the Tsallis maximum entropy theory.     

Circular-like maps: sensitivity to the initial conditions, multifractality and nonextensivity

Dissipative one-dimensional maps may exhibit special points (e.g., chaos threshold) at which the Lyapunov exponent vanishes. Consistently, the sensitivity to the initial conditions has a power-law time dependence, instead of the usual exponential one. The associated exponent can be identified with 1/(1-q), where q characterizes the nonextensivity of a generalized entropic form currently used to extend standard, Boltzmann-Gibbs statistical mechanics in order to cover a variety of anomalous situations. It has been recently proposed (Lyra and Tsallis, Phys. Rev. Lett. 80, 53 (1998)) for such maps the scaling law , where and are the extreme values appearing in the multifractal function. We generalize herein the usual circular map by considering inflexions of arbitrary power z, and verify that the scaling law holds for a large range of z. Since, for this family of maps, the Hausdorff dimension d_f equals unity for all z in contrast with q which does depend on z, it becomes clear that d_f plays no major role in the sensitivity to the initial conditions. Received 5 February 1999     

Centre and representations of small quantum superalgebras at roots of unity

When the deformation parameter is a root of unity, the centre of a quantum group can be described by a set of generators and non trivial relations. In the case ofU _q (sl(N)), these relations simply derive from the expressions of the deformed Casimir operators. In the case ofU _q (osp(1|2)), the relation is simple if we use an operator which anticommutes with the fermionic generators and whose square is the quadratic Casimir. This operator also simplifies the classification of finite dimensional irreducible representations. In the case ofU _q (sl(1|2)), the relations derive from the (infinite set of) standard Casimir operators.Presented at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.     

On the evolution of the ground state in the system U x La 1 - x S: Polarized neutron diffraction and X-ray magnetic circular dichroism study

In the U_xLa_1-xS system there is an abrupt loss of the long-range ferromagnetic ordering found in pure US at a critical concentration x _c ∼ 0.57, which is far above the percolation limit. As the magnetic ground state in such a system can be strongly affected by small variations of the 5f localization, we have investigated a set of samples with different x by polarized neutron diffraction and X-ray magnetic circular dichroism (XMCD). The neutron results are consistent with early measurements performed on pure US. Even at the lowest U content (x = 0.15, below x _c ) the shape of the induced form factor (f ( Q )) is comparable with that found for x = 1 and is well reproduced by either a U⁴⁺ or a U³⁺ state. The ratio between the orbital and the effective spin moments in the XMCD measurements confirms this result, but the evolution of the shape at the M₅ edge suggests an abrupt change in the distribution of the electrons (holes) in the 5 f density of states around x _c . Received 31 January 2001     

The imprints of superstatistics in multiparticle production processes

We provide an update of the overview of imprints of Tsallis nonextensive statistics seen in a multiparticle production processes. They reveal an ubiquitous presence of power law distributions of different variables characterized by the nonextensivity parameter q > 1. In nuclear collisions one additionally observes a q-dependence of the multiplicity fluctuations reflecting the finiteness of the hadronizing source. We present sum rules connecting parameters q obtained from an analysis of different observables, which allows us to combine different kinds of fluctuations seen in the data and analyze an ensemble in which the energy (E), temperature (T) and multiplicity (N) can all fluctuate. This results in a generalization of the so called Lindhard’s thermodynamic uncertainty relation. Finally, based on the example of nucleus-nucleus collisions (treated as a quasi-superposition of nucleon-nucleon collisions) we demonstrate that, for the standard Tsallis entropy with degree of nonextensivity q < 1, the corresponding standard Tsallis distribution is described by q′ = 2 − q > 1.     

Realization ofU q (so(N)) within the differential algebra onR q N

We realize the Hopf algebraU _q–1 (so(N)) as an algebra of differential operators on the quantum Euclidean spaceR _q ^N . The generators are suitableq-deformed analogs of the angular momentum components on ordinaryR ^N . The algebra Fun(R _q ^N ) of functions onR _q ^N splits into a direct sum of irreducible vector representations ofU _q–1 (so(N)); the latter are explicitly constructed as highest weight representations.