We study the one-dimensional version of Axelrod’s model of cultural transmission from the point of view of optimization dynamics. We show the existence of a Lyapunov potential for the dynamics. The global minimum of the potential, or optimum state, is the monocultural uniform state, which is reached for an initial diversity of the population below a critical value. Above this value, the dynamics settles in a multicultural or polarized state. These multicultural attractors are not local minima of the potential, so that any small perturbation initiates the search for the optimum state. Cultural drift is modelled by such perturbations acting at a finite rate. If the noise rate is small, the system reaches the optimum monocultural state. However, if the noise rate is above a critical value, that depends on the system size, noise sustains a polarized dynamical state.