arXiv subjects
Publications and source records attributed to A. K. Rajagopal.
A self-consistent thermodynamic framework is presented for power-law canonical distributions based on the generalized central limit theorem by extending the discussion given by Khinchin for deriving Gibbsian canonical ensemble theory. The thermodynamic Legendre transform structure is invoked in establishing its connection to nonextensive statistical mechanics.
It is shown that the distribution derived from the principle of maximum Tsallis entropy is a superposable Levy-type distribution. Concomitantly, the leading order correction to the limit distribution is also deduced. This demonstration fills an important gap in the derivation of the Levy-stable distribution from the nonextensive statistical framework.
Based on the form invariance of the structures given by Khinchin's axiomatic foundations of information theory and the pseudoadditivity of the Tsallis entropy indexed by q, the concept of conditional entropy is generalized to the case of nonadditive (nonextensive) composite systems. The proposed nonadditive conditional entropy is classically nonnegative but can be negative in the quantum context, indicating its utility for characterizing quantum entanglement. A criterion deduced from it for separability of density matrices for validity of local realism is examined in detail by employing a bipartite spin-1/2 system. It is found that the strongest criterion is obtained in the limit q going to infinity.
The group theory framework developed by Fukutome for a systematic analysis of the various broken symmetry types of Hartree-Fock solutions exhibiting spin structures is here extended to the general many body context using spinor-Green function formalism for describing magnetic systems. Consequences of this theory are discussed for examining the magnetism of itinerant electrons in nanometric systems of current interest as well as bulk systems where a vector spin-density form is required, by specializing our work to spin-density-functional formalism. We also formulate the linear response theory for such a system and compare and contrast them with the recent results obtained for localized electron systems. The various phenomenological treatments of itinerant magnetic systems are here unified in this group-theoretical description.
The problem of quantum state inference and the concept of quantum entanglement are studied using a non-additive measure in the form of Tsallis entropy indexed by the positive parameter q. The maximum entropy principle associated with this entropy along with its thermodynamic interpretation are discussed in detail for the Einstein-Podolosky-Rosen pair of two spin-1/2 particles. Given the data on the Bell-Clauser-Horne-Shimony-Holt observable, the analytic expression is given for the inferred quantum entangled state. It is shown that for q greater than unity, indicating the sub-additive feature of the Tsalls entropy, the entangled region is small and enlarges as one goes into super-additive regime where q is less than unity. It is also shown that quantum entanglement can be quantified by the generalized Kullback-Leibler entropy.
In this paper, the Green function theory of quantum many-particle systems recently presented is reworked within the framework of nonextensive statistical mechanics with a new normalized $q$-expectation values. This reformulation introduces a renormalization of temperature of the earlier theory and a self-consistency condition. The linear response theory is also presented, along with its two-particle Green function version. Finally, a Boltzmann transport-like equation is also developed here.
The form invariance of the statement of the maximum entropy principle and the metric structure in quantum density matrix theory, when generalized to nonextensive situations, is shown here to determine the structure of the nonextensive entropies. This limits the range of the nonextensivity parameter to so as to preserve the concavity of the entropies. The Tsallis entropy is thereby found to be appropriately renormalized.
The regions of independent quantum states, maximally classically correlated states, and purely quantum entangled (supercorrelated) states described in a recent formulation of quantum information theory by Cerf and Adami are explored here numerically in the parameter space of the well-known exactly soluable Jaynes-Cummings model for equilibrium and nonequilibrium time-dependent ensembles.
We show that the Jaynes principle is indeed a proper inference scheme when applied to compound systems and will correctly produce the entangled maximum entropy states compatible with appropriate data. This is accomplished by including the dispersion of the entanglement along with its mean value as constraints and an application of the uncertainty principle. We also construct a "thermodynamic-like" description of the entanglement arising out of the maximum entropy principle.
The Lie-Trotter formula $e^{\hat{A}+\hat{B}} = \lim_{N\to \infty} (e^{\hat{A}/N} e^{\hat{B}/N})^N$ is of great utility in a variety of quantum problems ranging from the theory of path integrals and Monte Carlo methods in theoretical chemistry, to many-body and thermostatistical calculations. We generalize it for the q-exponential function $e_q (x) = [1+ (1-q) x]^{(1/(1-q))}$ (with $e_1(x)=e^x$), and prove $e_q(\hat{A}+\hat{B}+(1-q) [\hat{A}\hat{B}+\hat{B}\hat{A}] /2) = \lim_{N\to \infty} {[e_{1-(1-q)N}(\hat{A}/N)] [e_{1-(1-q)N}(\hat{B}/N)]}^N$. This extended formula is expected to be similarly useful in the nonextensive situations