arXiv subjects
Publications and source records attributed to Tatsuaki Wada.
We consider a system described by a Fokker-Planck equation with a new type of momentum-dependent drift coefficient which asymptotically decreases as $-1/p$ for a large momentum $p$. It is shown that the steady-state of this system is a $κ$-generalized Gaussian distribution, which is a non-Gaussian distribution with a power-law tail.
We derive the multiplicative duality "q<->1/q" and other typical mathematical structures as the special cases of the (mu,nu,q)-relation behind Tsallis statistics by means of the (mu,nu)-multinomial coefficient. Recently the additive duality "q<->2-q" in Tsallis statistics is derived in the form of the one-to-one correspondence between the q-multinomial coefficient and Tsallis entropy. A slight generalization of this correspondence for the multiplicative duality requires the (mu,nu)-multinomial coefficient as a generalization of the q-multinomial coefficient. This combinatorial formalism provides us with the one-to-one correspondence between the (mu,nu)-multinomial coefficient and Tsallis entropy Sq, which determines a concrete relation among three parameters mu, nu and q, i.e., nu(1-mu)+1=q which is called "(mu,nu,q)-relation" in this paper. As special cases of the (mu,nu,q)-relation, the additive duality and the multiplicative duality are recovered when nu=1 and nu=q, respectively. As other special cases, when nu=2-q, a set of three parameters (mu,nu,q) is identified with the q-triplet (q_{sen},q_{rel},q_{stat}) recently conjectured by Tsallis. Moreover, when nu=1/q, the relation 1/(1-q_{sen})=1/alpha_{min}-1/alpha_{max} in the multifractal singularity spectrum f(alpha) is recovered by means of the (mu,nu,q)-relation.
We derive a scaling property from a fundamental nonlinear differential equation whose solution is the so-called q-exponential function. A scaling property has been believed to be given by a power function only, but actually more general expression for the scaling property is found to be a solution of the above fundamental nonlinear differential equation. In fact, any power function is obtained by restricting the domain of the q-exponential function appropriately. As similarly as the correspondence between the exponential function and Shannon entropy, an appropriate generalization of Shannon entropy is expected for the scaling property. Although the q-exponential function is often appeared in the optimal distributions of some one-parameter generalized entropies such as Renyi entropy, only Tsallis entropy is uniquely derived from the algebra of the q-exponential function, whose uniqueness is shown in the two ways in this paper.
Based on the one-parameter generalization of Shannon-Khinchin (SK) axioms presented by one of the authors, and utilizing a tree-graphical representation, we have further developed the SK Axioms in accordance with the two-parameter entropy introduced by Sharma-Taneja, Mittal, Borges-Roditi, and Kaniadakis-Lissia-Scarfone. The corresponding unique theorem is proved. It is shown that the obtained two-parameter Shannon additivity is a natural consequence from the Leibniz rule of the two-parameter Chakrabarti-Jagannathan difference operator.
The thermodynamic stability conditions (TSC) of nonadditive and composable entropies are discussed. Generally the concavity of a nonadditive entropy with respect to internal energy is not necessarily equivalent to the corresponding TSC. It is shown that both the TSC of Tsallis' entropy and that of the $κ$-generalized Boltzmann entropy are equivalent to the positivity of the standard heat capacity.
A brief review of the interaction-round-a-face (IRF) density-matrix renormalization-group (DMRG) method. We have demonstrated the numerical superiority of IRF-DMRG method applied to SU(2) invariant quantum spin chains over the conventional DMRG. The ground state energy densities and the gap energies of both $S = 1$ and $S = 2$ spin chains can be calculated using the IRF-DMRG without extensive computations. We have also studied the effect of tuning boundary interaction $J_{\rm end}^{~}$ at both ends of the chain from the IRF view point. It is clearly observed that the magnon distribution is uniform when the best $J_{\rm end}^{~}$ is chosen.
An interaction-round-a-face density-matrix renormalization-group (IRF-DMRG) method is developed for higher integer spin chain models which are rotational invariant. The expressions of the IRF weights associated with the nearest-neighbor spin-S interaction ${\bf S}_{i} \cdot {\bf S}_{i+1}$ are derived explicitly. Using the IRF-DMRG with these IRF weights, the Haldane gaps $Δ$ and the ground state energy densities $e_{0}$ for both S=1 and S=2 isotropic antiferromagnetic Heisenberg quantum spin chains are calculated by keeping up to only m=90 states.