An explicit model of exponential decay

E. Torrontegui, Juan Gonzalo Muga, Joan Martorell +1 more

We study the elusive transition from exponential to postexponential (algebraic) decay of the probability density of a quantum particle emitted by an exponentially decaying source in one dimension. The main finding is that the probability density at the transition time, and thus its observability, increases with the distance of the detector from the source up to a critical distance beyond which exponential decay is no longer observed. Solvable models provide explicit expressions for the dependence of the transition on resonance and observational parameters, facilitating the choice of optimal conditions.

A. Steyerl, S. S. Malik

Alexander Rudol'fovich Its, Anatoli G. Izergin, V. E. Korepin +1 more

This is a historical note. In 1993 we calculated space, time and temperature dependent correlation function in isotropic version of one dimensional XY spin chain. The correlation function decays exponentially with time and space separation. The rate of exponential decay was evaluated explicitly. Since that time similar results were obtained in other models: Bose gas with delta interaction, Ising model and strongly correlated electrons.

Moshe Schwartz

The decay of a general time dependent structure factors is considered. The dynamics is that of stochastic field equations of the Langevin type, where the systematic generalized force is a functional derivative of some classical field Hamiltonian with respect to the field. Equations of this type are generic and describe many physical systems. It is usually believed that simple non linear systems exhibit exponential decay in time, while non linear complex systems ,such as ferromagnets at their critical point,decay slowly as a power law or stretched exponential. A necessary condition for slow decay is that the eigenvalues of the appropriate Fokker- Planck operator accumulate at zero for each momentum q .This is a property of the system. I argue here that when the necessary condition is obeyed slow or fast (exponential) decay are both possible for simple linear systems as well as for complex non liner ones. The actual form of decay is determined by what structure factor we prefer to observe. An explicit family of linear models in which the "natural" structure factors decay exponentially is constructed. It is shown how a more complex structure factor decays in those models with a slow decay form. It is further shown how in general non linear systems it is always possible to find structure factors that decay exponentially. The question of actual observation of such structure factors in experiment and numerical simulations is discussed.

P. Thornton Greenland