  The entropy distribution of the intracluster medium and the shape of its
confining potential well completely determine the X-ray properties of a relaxed
cluster of galaxies, motivating us to explore the origin of intracluster
entropy and to describe how it develops in terms of some simple models. We
present an analytical model for smooth accretion, including both preheating and
radiative cooling, that links a cluster's entropy distribution to its mass
accretion history and shows that smooth accretion overproduces the entropy
observed in massive clusters by a factor ~2-3, depending on the mass accretion
rate. Any inhomogeneity in the accreting gas reduces entropy production at the
accretion shock; thus, smoothing of the gas accreting onto a cluster raises its
entropy level. Because smooth accretion produces more entropy than hierarchical
accretion, we suggest that some of the observed differences between clusters
and groups may arise because preheating smooths the smaller-scale lumps of gas
accreting onto groups more effectively than it smooths the larger-scale lumps
accreting onto clusters. This effect may explain why entropy levels at the
outskirts of groups are ~2-3 times larger than expected from self-similar
scaling arguments. The details of how the density distribution of accreting gas
affects the entropy distribution of a cluster are complex, and we suggest how
to explore the relevant physics with numerical simulations.
