Stochastic spreading and relaxation ()

Mathematical pendulum in the upper position is unstable. But it is claimed that if you ''watch'' the pendulum, then it will become stable due to the so called Quantum Zeno Effect (QZE). In fact there is nothing quantum in the QZE. To watch the pendulum is like to introduce noise; and consequently the dynamics in the vicinity of the upper position becomes a stochastic squeeze process, which is described by the equations

\dot{x} \ = \ \ \ w x \ - \ \omega(t) y
\dot{y} \ = \ -w y \ + \ \omega(t) x

where w induces stretching, while \omega(t) induces random rotations. The animation shows the evolution of circular cloud. We have managed to solve analytically how the pendulum spreads away from the unstable position. In particular we were able to calculate the radial diffusion coefficient D_r . Its dependence on the noise intensity D is shown in the figure (analytical red line agree with the numerical points). The traditional quantum analysis (green line) is a valid approximation only for very strong noise, i.e. in the limit were the spreading is strongly suppressed.

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