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Next: 9. How to measure Up: 4. Notes Previous: 7. Nonunitarity

8. Quantum characteristic exponent

Averaging our nonlinear PDP over individual histories one gets a linear Liouville equation for the density matrix of the total system. Tracing over the classical system is, in our case, easily performed and we then get:
\begin{displaymath}
{\dot \rho}=\kappa (\sum_{i=1}^N P({\bf n}[i],\alpha)\rho
P(...
...pha)-\frac{1}{2}\{\sum_{i=1}^N P({\bf n}[i],\alpha)^2
,\rho\})
\end{displaymath} (23)

where $\{\/ ,\/ \}$ stands for anti-commutator, and $\kappa$ is a coupling constant. For $\rho$ written as $\rho=\frac{1}{2}(I+\sigma({\bf m})$, ${\bf m}^2=m_1^2+m_2^2+m_3^2\leq 1$ after some calculations we get a very simple time evolution:
\begin{displaymath}
{\bf m}(t)=\exp \left(-\frac{N}{3}\kappa\alpha^2 t\right) {\bf m}(0).
\end{displaymath} (24)

The quantum characteristic exponent, as defined in Ref [24], is thus $\frac{2N}{3}\kappa$ - not a very useful quantity in our case. The Hausdorff dimension of the limit set, for the tetrahedral case, has been numerically estimated in Ref. [25] and shown to decrease from 1.44 to 0.49 while $\alpha$ increases from 0.75 to 0.95. We hope that by publishing the generating algorithm we will create interest in confirming these as well as obtaining new results in this field.
next up previous
Next: 9. How to measure Up: 4. Notes Previous: 7. Nonunitarity

2002-04-11