3,978 research outputs found
Separation of variables and the XXZ Gaudin magnet
In this work we generalise previous results connecting (rational) Gaudin
magnet models and classical separation of variables. It is shown that the
connection persists for the case of linear r-matrix algebra which corresponds
to the trigonometric 4x4 r-matrix (of the XXZ type). We comment also on the
corresponding problem for the elliptic (XYZ) r-matrix. A prescription for
obtaining integrable systems associated with multiple poles of an L-operator is
given.Comment: 11 pages, AMS-Te
Models of q-algebra representations: Tensor products of special unitary and oscillator algebras
This paper begins a study of one- and two-variable function space models of irreducible representations of q analogs of Lie enveloping algebras, motivated by recurrence relations satisfied by q-hypergeometric functions. The algebras considered are the quantum algebra Uq(su2) and a q analog of the oscillator algebra (not a quantum algebra). In each case a simple one-variable model of the positive discrete series of finite- and infinite-dimensional irreducible representations is used to compute the ClebschâGordan coefficients. It is shown that various q analogs of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group and the corresponding matrix elements of the ``group operators'' on these representation spaces are computed. It is shown that the matrix elements are polynomials satisfying orthogonality relations analogous to those holding for true irreducible group representations. It is also demonstrated that general q-hypergeometric functions can occur as basis functions in two-variable models, in contrast with the very restricted parameter values for the q-hypergeometric functions arising as matrix elements in the theory of quantum groups
Models of q-algebra representations: Matrix elements of the q-oscillator algebra
This article continues a study of function space models of irreducible representations of q analogs of Lie enveloping algebras, motivated by recurrence relations satisfied by q-hypergeometric functions. Here a q analog of the oscillator algebra (not a quantum algebra) is considered. It is shown that various q analogs of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group and the corresponding matrix elements of the ``group operators'' on these representation spaces are computed. This ``local'' approach applies to more general families of special functions, e.g., with complex arguments and parameters, than does the quantum group approach. It is shown that the matrix elements themselves transform irreducibly under the action of the algebra. q analogs of a formula are found for the product of two hypergeometric functions 1F1 and the product of a 1F1 and a Bessel function. They are interpreted here as expansions of the matrix elements of a ``group operator'' (via the exponential mapping) in a tensor product basis (for the tensor product of two irreducible oscillator algebra representations) in terms of the matrix elements in a reduced basis. As a by-product of this analysis an interesting new orthonormal basis was found for a q analog of the BargmannâSegal Hilbert space of entire functions
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Explanation of the Gibbs paradox within the framework of quantum thermodynamics
The issue of the Gibbs paradox is that when considering mixing of two gases
within classical thermodynamics, the entropy of mixing appears to be a
discontinuous function of the difference between the gases: it is finite for
whatever small difference, but vanishes for identical gases. The resolution
offered in the literature, with help of quantum mixing entropy, was later shown
to be unsatisfactory precisely where it sought to resolve the paradox.
Macroscopic thermodynamics, classical or quantum, is unsuitable for explaining
the paradox, since it does not deal explicitly with the difference between the
gases. The proper approach employs quantum thermodynamics, which deals with
finite quantum systems coupled to a large bath and a macroscopic work source.
Within quantum thermodynamics, entropy generally looses its dominant place and
the target of the paradox is naturally shifted to the decrease of the maximally
available work before and after mixing (mixing ergotropy). In contrast to
entropy this is an unambiguous quantity. For almost identical gases the mixing
ergotropy continuously goes to zero, thus resolving the paradox. In this
approach the concept of ``difference between the gases'' gets a clear
operational meaning related to the possibilities of controlling the involved
quantum states. Difficulties which prevent resolutions of the paradox in its
entropic formulation do not arise here. The mixing ergotropy has several
counter-intuitive features. It can increase when less precise operations are
allowed. In the quantum situation (in contrast to the classical one) the mixing
ergotropy can also increase when decreasing the degree of mixing between the
gases, or when decreasing their distinguishability. These points go against a
direct association of physical irreversibility with lack of information.Comment: Published version. New title. 17 pages Revte
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