In this paper, we develop a deterministic annealing approach for the design of entropy-constrained vector quantizers. Here, the quantization problem is reformulated within a probabilistic setting and the principle of maximum entropy is invoked. The resulting method is connected to statistical physics in two important respects. First, there is annealing in the process controlled by a `temperature' parameter, useful for avoiding local optima of the cost. Second, the number of distinct codevectors grows by a sequence of phase transitions as the temperature is varied. When the temperature is lowered to zero, the method searches for the optimal entropy-constrained vector quantizer. Conditions for bifurcation in the process provide insights into the `right' codebook size for a given problem.