Escape times for rigid Brownian rotators in a bistable potential from the time evolution of the Green function and the characteristic time of the probability evolution

The greatest relaxation time for an assembly of three- dimensional rigid rotators in an axially symmetric bistable potential is obtained exactly in terms of continued fractions as a sum of the zero frequency decay functions (averages of the Legendre polynomials) of the system. This is accomplished by studying the entire time evolution of the Green function (transition probability) by expanding the time dependent distribution as a Fourier series and proceeding to the zero frequency limit of the Laplace transform of that distribution. The procedure is entirely analogous to the calculation of the characteristic time of the probability evolution (the integral of the configuration space probability density function with respect to the position co-ordinate) for a particle undergoing translational diffusion in a potential; a concept originally used by Malakhov and Pankratov (Physica A 229 (1996) 109). This procedure allowed them to obtain exact solutions of the Kramers one-dimensional translational escape rate problem for piecewise parabolic potentials. The solution was accomplished by posing the problem in terms of the appropriate Sturm-Liouville equation which could be solved in terms of the parabolic cylinder functions. The method (as applied to rotational problems and posed in terms of recurrence relations for the decay functions, i.e., the Brinkman approach c.f. Blomberg, Physica A 86 (1977) 49, as opposed to the Sturm-Liouville one) demonstrates clearly that the greatest relaxation time unlike the integral relaxation time which is governed by a single decay function (albeit coupled to all the others in non-linear fashion via the underlying recurrence relation) is governed by a sum of decay functions. The method is easily generalized to multidimensional state spaces by matrix continued fraction methods allowing one to treat non-axially symmetric potentials, where the distribution function is governed by two state variables. (C) 2001 Elsevier Science B.V. All rights reserved.

A variational approach to the relaxation of single domain magnetic particles based on Brown's model

Brown's model for the relaxation of the magnetization of a single domain ferromagnetic particle is considered. This model results in the Fokker-Planck equation of the process. The solution of this equation in the cases of most interest is non- trivial. The probability density of orientations of the magnetization in the Fokker-Planck equation can be expanded in terms of an infinite set of eigenfunctions and their corresponding eigenvalues where these obey a Sturm-Liouville type equation. A variational principle is applied to the solution of this equation in the case of an axially symmetric potential. The first (non-zero) eigenvalue, corresponding to the largest time constant, is considered. From this we obtain two new results. Firstly, an approximate minimising trial function is obtained which allows calculation of a rigorous upper bound. Secondly, a new upper bound formula is derived based on the Euler-Lagrange condition. This leads to very accurate calculation of the eigenvalue but also, interestingly, from this, use of the simplest trial function yields an equivalent result to the correlation time of Coffey et at. and the integral relaxation time of Garanin. (C) 2004 Elsevier B.V. All rights reserved.

Religious Geopolitics of Palestinian Christianity: Palestinian Christian Zionists, Palestinian Liberation Theologists, and American Missions to Palestine

The introduction of Protestantism into the Middle East by American missionaries in the nineteenth century met with limited success while the responses and internalizations of local converts proved incredibly diverse. The two resultant theological descendants are Palestinian Christian Zionists and Palestinian Liberation Theologists. The article provides a short history of these two movements and highlights influential voices through interviews and media analysis. This article argues that hybrid religious identifications with nation and place has transcended, in some cases, political struggle for territory.

Religious Nationalism and Christian Zionist Pilgrimages to Holy Landscapes

Mapping the End Times:American Evangelical Geopolitics and Apocalyptic Visions

Over the last quarter-century, evangelicalism has become an important social and political force in modern America. Here, new voices in the field are brought together with leading scholars such as William E. Connolly, Michael Barkun, Simon Dalby, and Paul Boyer to produce a timely examination of the spatial dimensions of the movement, offering useful and compelling insights on the intersection between politics and religion. This comprehensive study discusses evangelicalism in its different forms, from the moderates to the would-be theocrats who, in anticipation of the Rapture, seek to impose their interpretations of the Bible upon American foreign policy. The result is a unique appraisal of the movement and its geopolitical visions, and the wider impact of these on America and the world at large.