Applications of the generalized Langevin equation: Towards a realistic description of the baths

The generalized Langevin equation (GLE) method, as developed previously [L. Stella et al., Phys. Rev. B 89, 134303 (2014)], is used to calculate the dissipative dynamics of systems described at the atomic level. The GLE scheme goes beyond the commonly used bilinear coupling between the central system and the bath, and permits us to have a realistic description of both the dissipative central system and its surrounding bath. We show how to obtain the vibrational properties of a realistic bath and how to convey such properties into an extended Langevin dynamics by the use of the mapping of the bath vibrational properties onto a set of auxiliary variables. Our calculations for a model of a Lennard-Jones solid show that our GLE scheme provides a stable dynamics, with the dissipative/relaxation processes properly described. The total kinetic energy of the central system always thermalizes toward the expected bath temperature, with appropriate fluctuation around the mean value. More importantly, we obtain a velocity distribution for the individual atoms in the central system which follows the expected canonical distribution at the corresponding temperature. This confirms that both our GLE scheme and our mapping procedure onto an extended Langevin dynamics provide the correct thermostat. We also examined the velocity autocorrelation functions and compare our results with more conventional Langevin dynamics.

Nonequilibrium processes from generalized Langevin equations: Realistic nanoscale systems connected to two thermal baths

We extend the generalized Langevin equation (GLE) method [L. Stella, C. D. Lorenz, and L. Kantorovich, Phys. Rev. B 89, 134303 (2014)] to model a central classical region connected to two realistic thermal baths at two different temperatures. In such nonequilibrium conditions a heat flow is established, via the central system, in between the two baths. The GLE-2B (GLE two baths) scheme permits us to have a realistic description of both the dissipative central system and its surrounding baths. Following the original GLE approach, the extended Langevin dynamics scheme is modified to take into account two sets of auxiliary degrees of freedom corresponding to the mapping of the vibrational properties of each bath. These auxiliary variables are then used to solve the non-Markovian dissipative dynamics of the central region. The resulting algorithm is used to study a model of a short Al nanowire connected to two baths. The results of the simulations using the GLE-2B approach are compared to the results of other simulations that were carried out using standard thermostatting approaches (based on Markovian Langevin and Nosé-Hoover thermostats). We concentrate on the steady-state regime and study the establishment of a local temperature profile within the system. The conditions for obtaining a flat profile or a temperature gradient are examined in detail, in agreement with earlier studies. The results show that the GLE-2B approach is able to treat, within a single scheme, two widely different thermal transport regimes, i.e., ballistic systems, with no temperature gradient, and diffusive systems with a temperature gradient.

Mapping the Education System in Northern Ireland

In this chapter we provide a broad overview of the education system discussing both the system of schooling and the network of both statutory and voluntary bodies that are responsible for the administration and management of the system. We begin with an examination of the administrative structures. This network of statutory and non-statutory bodies has increased in number over recent years, illustrating the diversity of vested interests in the system. This is followed by an examination of the system of schooling and the chapter concludes by exploring the significance of changes that have taken place in terms of its implications for a more pluralist society where each is accorded parity of esteem and equality of opportunity.

Analytically continued generalized continuum distorted waves

The continuum distorted-wave eikonal-initial-state (CDW-EIS) theory of Crothers and McCann (Crothers DSF and McCann JF, 1983 J. Phys. B: At. Mol. Opt. Phys. 16 3229 ) used to describe ionization in ion-atom collisions is generalized (G) to GCDW-EIS, to incorporate the azimuthal ange dependence into the final-state wavefunction. This is accomplished by the analytic continuation of hydrogenic-like wavefunctions from below to above threshold, using parabolic coordinates and quantum numbers including magnetic quantum numbers, thus providing a more complete set of states. At impact energies lower than 25 ke V u^{-1}, the total CDW-EIS ionization cross section falls off, with decreasing energy, too quickly in comparison with experimental data by Crothers and McCann. The idea behind and motivation for the GCDW-EIS model is to improve the theory with respect to experiment, by including contributions from non-zero magnetic quantum numbers. We also therefore incidentally provide a new derivation of the theory of continuum distorted waves for zero magnetic quantum numbers while simultaneously generalizing it.

Mapping cosmopolis: moral topographies of the medieval city

Cosmopolis is a concept that has a long history in many cultures around the globe. It is a mirroring of the 'social' and 'natural' worlds, such that in one is seen the order and the structures of the other -- a mutual 'mapping'. In this paper I examine how the presence of cosmopolis -- a Christianised cosmopolis of the European Middle Ages -- was made evident in the representation and formation of cities at that time. I reveal a dualism between the social and spatial ordering of both city and cosmos which defined and reinforced social and spatial boundaries in urban landscapes, evident for example in the 11th and 12th centuries. Recently, Toulmin (1992) has taken the idea of cosmopolis to argue that it has been a persistent presence in Western - Enlightenment science, philosophy, and religion -- a 'hidden agenda of modernity'. I contend that, as an idea, cosmopolis has a much earlier circulation in European thinking, not least in the Middle Ages. Locating cosmopolis in the medieval and the modern periods then begs a question of what is it that really makes the two distinct and separate? All too often human geographers have emphasised discontinuities between the 'medieval' and 'modern' age, locating the 'rise of modernity' some time in the Enlightenment period. However, what 'mapping' cosmopolis reveals are continuities, binding time and space together, which when looked at begin to help query the modernity concept itself.