Fractional powers of operators of Tsallis ensemble and their parameter differentiation

Rajagopal, A. K.

doi:10.1590/s0103-97331999000100006

Cited by 3 publications

(3 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(v) Path integral and Bloch equation [53], as well as related properties [54]; (vi) Quantum statistics and those associated with the Gentile and the Haldane exclusion statistics [55][56][57]; (vii) Simulated annealing and related optimization, Monte Carlo and Molecular dynamics techniques [58][59][60][61][62][63][64][65][66][67][68].…”

Section: B Canonical Ensemblementioning

confidence: 99%

“…Finally, let us mention various important theoretical tools which enable the thermostatistical discussion of complex nonextensive systems, and which are now available (within the unnormalized and/or normalized versions for the q-expectation values) for arbitrary q. We refer to (i) Linear response theory [35]; (ii) Perturbation expansion [51]; (iii) Variational method (based on the Bogoliubov inequality) [51]; (iv) Many-body Green functions [52]; (v) Path integral and Bloch equation [53], as well as related properties [54]; (vi) Quantum statistics and those associated with the Gentile and the Haldane exclusion statistics [55][56][57]; (vii) Simulated annealing and related optimization, Monte Carlo and Molecular dynamics techniques [58][59][60][61][62][63][64][65][66][67][68].…”

Section: B Canonical Ensemblementioning

confidence: 99%

See 1 more Smart Citation

Nonextensive statistics: theoretical, experimental and computational evidences and connections

Tsallis¹

1999

Braz. J. Phys.

551

200

View full text Add to dashboard Cite

The domain of validity of standard thermodynamics and Boltzmann-Gibbs statistical mechanics is discussed and then formally enlarged in order to hopefully cove r a v ariety o f anomalous systems. The generalization concerns nonextensive systems, where nonextensivity is understood in the thermodynamical sense. This generalization was rst proposed in 1988 inspired by the probabilistic description of multifractal geometries, and has been intensively studied during this decade. In the present e ort, after introducing some historical background, we brie y describe the formalism, and then exhibit the present status in what concerns theoretical, experimental and computational evidences and connections, as well as some perspectives for the future. In addition to these, here and there we point out various possibly relevant questions, whose answer would certainly clarify our current understanding of the foundations of statistical mechanics and its thermodynamical implications.

show abstract

Section: B Canonical Ensemblementioning

confidence: 99%

Section: B Canonical Ensemblementioning

confidence: 99%

Nonextensive statistics: theoretical, experimental and computational evidences and connections

Tsallis¹

1999

Braz. J. Phys.

551

200

View full text Add to dashboard Cite

show abstract

“…Most of the results that have been achieved provide the derivative as an integral expression 1,2 , or by means of an integral representation 3 of the function in the complex plane. Such elegant procedure is very useful to prove mathematical properties but, in many cases, to calculate the remaining integrals is not an easy task.…”

mentioning

confidence: 99%

Functions of linear operators: Parameter differentiation

Prato

Tsallis

2000

Journal of Mathematical Physics

View full text Add to dashboard Cite

We derive a useful expression for the matrix elements $[\frac{\partial f[A(t)]}{\partial t}]_{i j}$ of the derivative of a function $f[A(t)]$ of a diagonalizable linear operator $A(t)$ with respect to the parameter $t$. The function $f[A(t)]$ is supposed to be an operator acting on the same space as the operator $A(t)$. We use the basis which diagonalizes A(t), i.e., $A_{i j}=\lambda_i \delta_{i j}$, and obtain $[\frac{\partial f[A(t)]}{\partial t}]_{i j}=[\frac{\partial A}{\partial t}]_ {i j}\frac{f(\lambda_j) - f(\lambda_i)} {\lambda_j - \lambda_i}$. In addition to this, we show that further elaboration on the (not necessarily simple) integral expressions given by Wilcox 1967 (who basically considered $f[A(t)]$ of the exponential type) and generalized by Rajagopal 1998 (who extended Wilcox results by considering $f[A(t)]$ of the $q$-exponential type where $\exp_q(x) \equiv [1+(1-q)x]^{1/(1-q)}$ with $q \in {\cal {R}}$; hence, $\exp_1 (x)=\exp(x))$ yields this same expression. Some of the lemmas first established by the above authors are easily recovered.Comment: No figure

show abstract

II. Quantum Density Matrix Description of Nonextensive Systems

Rajagopal

2001

Nonextensive Statistical Mechanics and Its Applications

View full text Add to dashboard Cite

Fractional powers of operators of Tsallis ensemble and their parameter differentiation

Cited by 3 publications

References 0 publications

Nonextensive statistics: theoretical, experimental and computational evidences and connections

Nonextensive statistics: theoretical, experimental and computational evidences and connections

Functions of linear operators: Parameter differentiation

II. Quantum Density Matrix Description of Nonextensive Systems

Contact Info

Product

Resources

About