  A small quantum scattering system (the microsystem) is studied in interaction
with a large system (the macrosystem) described by unknown stochastic
variables. The interaction between the two systems is diagonal for the
microsystem in a certain orthonormal basis, and the interaction gives an
imprint on the macrosystem. Moreover, the interaction is assumed to involve
only small transfers of energy and momentum between the two systems (as
compared to typical energies/momenta within the microsystem). The analysis is
carried out within scattering theory. Calculated in the conventional way, the
transition amplitude for the whole system factorizes. The interaction taking
place within the macrosystem is assumed to depend on the stochastic variables
in such a way that, on the average, no particular basis vector state of the
microsystem is favoured. The density matrix is studied in a formalism which
includes generation of the ingoing state and absorption of the final state.
Then the dependence of the final state on the conventional scattering amplitude
for the microsystem is highly non-linear.
  In the thermodynamic limit of the macrosystem, the density matrix of the
ensemble (of microsystem plus macrosystem) develops into a final state which
involves a set of macroscopically distinguishable states, each with the
microsystem in one of the basis vector states and the macrosystem in an
entangled state.
  For an element of the ensemble, i.e., for a single measurement, the result is
instead a random walk, where the microsystem ends up in one of the basis vector
states (reduction of the wave packet).
