arXiv subjects

Jan Naudts

Publications and source records attributed to Jan Naudts.

78 records · Page 5Linked to original sources

arXiv2000

C*-Multipliers, crossed product algebras, and canonical commutation relations

The notion of a multiplier of a group X is generalized to that of a C*-multiplier by allowing it to have values in an arbitrary C*-algebra A. On the other hand, the notion of the action of X in A is generalized to that of a projective action of X as linear transformations of the space of continuous functions with compact support in X and with values in A. It is shown that there exists a one-to-one correspondence between C*-multipliers and projective actions. C*-multipliers have been used to define twisted group algebras. On the other hand, the projective action tau can be used to construct the crossed product algebra A x_tau X. Both constructions are unified in the present approach. The results are applicable in mathematical physics. The multiplier algebra of the crossed product algebra A x_tau X contains Weyl operators {W(x),x in X}. They satisfy canonical commutation relations w.r.t. the C*-multiplier. Quantum spacetime is discussed as an example.

math-ph↗

arXiv1999

Rigorous results in non-extensive thermodynamics

This paper studies quantum systems with a finite number of degrees of freedom in the context of non-extensive thermodynamics. A trial density matrix, obtained by heuristic methods, is proved to be the equilibrium density matrix. If the entropic parameter q is larger than 1 then existence of the trial equilibrium density matrix requires that q is less than some critical value q_c which depends on the rate by which the eigenvalues of the hamiltonian diverge. Existence of a unique equilibrium density matrix is proved if in addition q<2 holds. For q between 0 and 1, such that 2<q+q_c, the free energy has at least one minimum in the set of trial density matrices. If a unique equilibrium density matrix exists then it is necessarily one of the trial density matrices. Note that this is a finite rank operator, which means that in equilibrium high energy levels have zero probability of occupancy.

math-ph↗

arXiv1999

Nonlinear von Neumann-type equations

We review some recent developments in the theory of nonlinear von Neumann equations. We distinguish between the von Neumann equation (which can be nonlinear) and the Liouville equation (which should be linear). Explicit examples illustrate the technique of binary Darboux integration of nonlinear density matrix equations and special attention is payed to the problem of how to find physically nontrivial `self-scattering' solutions.

quant-ph↗

arXiv1999

Dual description of nonextensive ensembles

In the case of a system with an unbounded hamiltonian the entropic index q of non-extensive thermodynamics has an upperbound q_c>1 beyond which the formalism becomes meaningless. The expression 1/(q_c-1) is the dimension of the state space (i.e. the manifold of density matrices) in the context of non-commutative geometry. For q=q_c an ultraviolet cutoff E< E_ub is needed to guarantee the existence of the equilibrium density matrix. Duality between q>1 and q<1-statistics via a q --> 1/q transformation is established. It leads to an overall picture in which the meaning of both q>1-statistics and q<1-statistics is clarified.

cond-mat.stat-mech↗

arXiv1999

Stretched Exponential Relaxation in the Biased Random Voter Model

We study the relaxation properties of the voter model with i.i.d. random bias. We prove under mild condions that the disorder-averaged relaxation of this biased random voter model is faster than a stretched exponential with exponent $d/(d+α)$, where $0<α\le 2$ depends on the transition rates of the non-biased voter model. Under an additional assumption, we show that the above upper bound is optimal. The main ingredient of our proof is a result of Donsker and Varadhan (1979).

math-ph↗

arXiv1998

Microscopic Foundation of Nonextensive Statistics

Combination of the Liouville equation with the q-averaged energy $U_q = _q$ leads to a microscopic framework for nonextensive q-thermodynamics. The resulting von Neumann equation is nonlinear: $i\dotρ=[H,ρ^q]$. In spite of its nonlinearity the dynamics is consistent with linear quantum mechanics of pure states. The free energy $F_q=U_q-TS_q$ is a stability function for the dynamics. This implies that q-equilibrium states are dynamically stable. The (microscopic) evolution of $ρ$ is reversible for any q, but for $q\neq 1$ the corresponding macroscopic dynamics is irreversible.

quant-ph↗