Annealing is a physical process that motivates our definition of a Gibbs measure, which is a certain probability measure on Euclidean space. In this paper we examine a sequence of Gibbs measures characterized by the distance function. In Chapter 2 we conclude that the sequence of measures converge to a Hausdorff probability measure equally distributed along self-similar fractals with Hutchinson's Open Set Condition. In Chapter 3 we study spaces of concentric circles (which we call targets) in the plane, and examine how the sequence of probability measures distributes over the targets. By varying the number of targets and the size of the circles, we find probability measures that divide their mass between different point masses and spaces. Finally, in Chapter 4 we conclude that the measure will distribute evenly over the highest-dimensional strata of any semi-algebraic set.