  Observations reveal that the peaks of the X-ray map and the
Sunyaev-Zel'dovich (SZ) effect map of some galaxy clusters are offset from each
other. In this paper, we perform a set of hydrodynamical simulations of mergers
of two galaxy clusters to investigate the spatial offset between the maxima of
the X-ray and the SZ surface brightness of the merging clusters. We find that
significantly large SZ-X-ray offsets (>100kpc) can be produced during the major
mergers of galaxy clusters. The significantly large offsets are mainly caused
by a `jump effect' occurred between the primary and secondary pericentric
passages of the two merging clusters, during which the X-ray peak may jump to
the densest gas region located near the center of the small cluster, but the SZ
peak remains near the center of the large one. Our simulations show that
merging systems with higher masses and larger initial relative velocities may
result in larger offset sizes and longer offset time durations; and only nearly
head-on mergers are likely to produce significantly large offsets. We further
investigate the statistical distribution of the SZ-X-ray offset sizes and find
that (1) the number distribution of the offset sizes is bimodal with one peak
located at low offsets ~0 and the other at large offsets ~350-450kpc/h, but the
objects with intermediate offsets are scarce; and (2) the probabilities of the
clusters in the mass range higher than 2x10^{14}Msun/h that have offsets larger
than 20, 50, 200, 300, and 500kpc/h are 34.0%, 11.1%, 8.0%, 6.5%, and 2.0%
respectively at z=0.7. The probability is sensitive to the underlying pairwise
velocity distribution and the merger rate of clusters. Future observations on
the offsets for a large number of clusters may put strong constraints on the
cosmic velocity fields on the cluster scale and the cluster merger rate.
(Abridged)
