Digest: J. Math. Phys. 53

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AIP: Journal of Mathematical Physics

AIP online journals now offer MathJax to display mathematics

AIP Publishing has become a MathJax Partner, providing major funding to the MathJax initiative, and now offers MathJax to render mathematics in its online journals. MathJax is an open-source JavaScript display engine that produces high-quality math in all modern browsers, without plugins or other special set-up requirements.

To see MathJax in action, visit your favorite  Journal of Mathematical Physics article and select the Read Online option. Once in the HTML view, go to the navigation bar and turn on MathJax. From there you can copy and paste any equation into your favorite MathML-enabled editor.

jmp-82-103707The gap equation for spin-polarized fermions 
Abraham Freiji, Christian Hainzl, and Robert Seiringer

The authors study the BCS gap equation for a Fermi gas with unequal population of spin-up and spin-down states. This is the continuation of recent studies where the BCS gap equation in the balanced case for systems with general pair interaction potential V was investigated. The authors find that for cosh (δμ/T)⩽2, with T the temperature and δμ the chemical potential difference, the question of existence of non-trivial solutions can be reduced to spectral properties of a linear operator, similar to the unpolarized case studied previously. The authors derive upper and lower bounds for the critical temperature, and study their behavior in the small coupling limit.

J. Math. Phys. 53, 012101 (2012)

 

jmp-82-103707Proof of rounding by quenched disorder of first order transitions in low-dimensional quantum systems 
Michael Aizenman, Rafael L. Greenblatt, and Joel L. Lebowitz

The authors attempt to prove that for quantum lattice systems in d⩽2 dimensions the addition of quenched disorder rounds any first order phase transition in the corresponding conjugate order parameter, both at positive temperatures and at T=0. For systems with continuous symmetry the authors extend the proof up to d⩽4 dimensions. The authors achieve the extension of the proof to quantum systems by carrying out the analysis at the level of thermodynamic quantities rather than equilibrium states.

J. Math. Phys. 53, 023301 (2012)

 

Vortex dynamics in R4
Banavara N. Shashikanth

The author studies the vortex dynamics of Euler's equations for a constant density fluid flow in R4 with special focus on singular Dirac delta distributions of the vorticity supported on two-dimensional surfaces (membranes). The self-induced velocity field of a membrane has a logarithmic divergence. A regularization done via the LIA then shows that the regularized velocity field is proportional to the mean curvature vector field of the membrane rotated by 90° in the plane of normals. The author also presents a Hamiltonian structure for the regularized self-induced motion  of the membrane. The dynamics of the four-form ω ∧ ω is examined and it is shown that Ertel's vorticity theorem in R3 , for the constant density case, is a special case of this dynamics.

J. Math. Phys. 53, 013103 (2012)

Composite parameterization and Haar measure for all unitary and special unitary groups
Christoph Spengler, Marcus Huber, and Beatrix C. Hiesmayr

The authors adopt the concept of the composite parameterization of the unitary group U(d) to obtain a novel parameterization of the special unitary group SU(d), and furthermore, consider the Haar measure in terms of the introduced parameters. The authors show that the well-defined structure of the parameterization leads to a concise formula for the normalized Haar measure on U(d) and SU(d).

J. Math. Phys. 53, 013501 (2012)

jmp-82-103707The stochastisation hypothesis and the spacing of planetary systems
Jacky Cresson

The stochastisation hypothesis aims to provide a framework to deal with physical systems in random environment.It is applied here in two different cases: in the study of the dynamics of a protoplanetary nebula and in the chaotic long-term behaviour of a generic planetary system using previous works of Albeverio et al.

J. Math. Phys. 52, 113502 (2011)