Transition state theory for wave packet dynamics. II. Thermal decay of Bose-Einstein condensates with long-range interaction

We apply transition state theory to coupled Gaussian wave packets and calculate thermal decay rates of Bose-Einstein condensates with additional long-range interaction. The ground state of such a condensate is metastable if the contact interaction is attractive and a sufficient thermal excitation may lead to its collapse. The use of transition state theory is made possible by describing the condensate within a variational framework and locally mapping the variational parameters to classical phase space as has been demonstrated in the preceding paper [A. Junginger, J. Main, and G. Wunner, submitted to J. Phys. A]. We apply this procedure to Gaussian wave packets and present results for condensates with monopolar 1/r-interaction comparing decay rates obtained by using different numbers of coupled Gaussian trial wave functions as well as different normal form orders.Comment: 14 pages, 4 figures, submitted to J. Phys.

Variational calculations on multilayer stacks of dipolar Bose-Einstein condensates

We investigate a multilayer stack of dipolar Bose-Einstein condensates in terms of a simple Gaussian variational ansatz and demonstrate that this arrangement is characterized by the existence several stationary states. Using a Hamiltonian picture we show that in an excited stack there is a coupled motion of the individual condensates by which they exchange energy. We find that for high excitations the interaction between the single condensates can induce the collapse of one of them. We furthermore demonstrate that one collapse in the stack can force other collapses, too. We discuss the possibility of experimentally observing the coupled motion and the relevance of the variational results found to full numerical investigations.Comment: 9 pages, 9 figure

Transition state theory for wave packet dynamics. I. Thermal decay in metastable Schr\"odinger systems

We demonstrate the application of transition state theory to wave packet dynamics in metastable Schr\"odinger systems which are approached by means of a variational ansatz for the wave function and whose dynamics is described within the framework of a time-dependent variational principle. The application of classical transition state theory, which requires knowledge of a classical Hamilton function, is made possible by mapping the variational parameters to classical phase space coordinates and constructing an appropriate Hamiltonian in action variables. This mapping, which is performed by a normal form expansion of the equations of motion and an additional adaptation to the energy functional, as well as the requirements to the variational ansatz are discussed in detail. The applicability of the procedure is demonstrated for a cubic model potential for which we calculate thermal decay rates of a frozen Gaussian wave function. The decay rate obtained with a narrow trial wave function agrees perfectly with the results using the classical normal form of the corresponding point particle. The results with a broader trial wave function go even beyond the classical approach, i.e., they agree with those using the quantum normal form. The method presented here will be applied to Bose-Einstein condensates in the following paper [A. Junginger, M. Dorwarth, J. Main, and G. Wunner, submitted to J. Phys. A].Comment: 21 pages, 3 figures, submitted to J. Phys.

Transition states and thermal collapse of dipolar Bose-Einstein condensates

We investigate thermally excited, dipolar Bose-Einstein condensates. Quasi-particle excitations of the atomic cloud cause density fluctuations which can induce the collapse of the condensate if the inter-particle interaction is attractive. Within a variational approach, we identify the collectively excited stationary states of the gas which form transition states on the way to the BEC's collapse. We analyze transition states with different $m$-fold rotational symmetry and identify the one which mediates the collapse. The latter's symmetry depends on the trap aspect ratio of the external trapping potential which determines the shape of the BEC. Moreover, we present the collapse dynamics of the BEC and calculate the corresponding decay rate using transition state theory. We observe that the thermally induced collapse mechanism is important near the critical scattering length, where the lifetime of the condensate can be significantly reduced. Our results are valid for an arbitrary strength of the dipole-dipole interaction. Specific applications are discussed for the elements $^{52}$Cr, $^{164}$Dy and $^{168}$Er with which dipolar BECs have been experimentally realized.Comment: 10 pages, 6 figure

Producer income instability and farmers' risk response: The case of major Kenyan export crops

The instability of export earnings in LDCs and its presumably harmful economic effects have been broadly discussed in the economic literature and among policy makers in international meetings. In analyzing these effects, the destabilization of producer incomes and farmers' risk response play a prominent role. Producer incomes may be destabilized by either domestic factors on the supply side (yield instability due to weather, crop diseases, etc.), or by fluctuating producer prices reflecting the instability of international primary commodity markets. If unstable producer incomes induce risk aversion among farmers, the sectoral factor input will be reduced and will be suboptimal from a welfare point of view, thus possibly hampering economic growth. The purpose of this paper is to quantify the effects of producer income instability on farmers' planting and long-run supply decisions in the coffee, tea, and sisal production of the Kenyan large farm sector. Coffee, tea, and sisal are the leading Kenyan export crops, the domestic consumption of which is negligible. About half of the Kenyan coffee and tea, and all the sisal are grown in the large farm sector, and nearly always on plantations. Coffee, tea, and sisal are permanent crops the planting of which requires long-run decisions. It is the long-run we shall focus on in this paper; hence the influence of income instability on short-term production planning will be neglected. The analysis will be based on a time series approach covering the period 1951-1975. In the following section we shall develop the methodological framework of how to measure the risk response of farmers. Next the estimation equations will be specified, and the estimation techniques will be demonstrated. Subsequently, the regression results are presented and interpreted. Some tentative conclusions are drawn in the final section.