  We discuss the relation between the Wasserstein distance of order 1 between
probability distributions on a metric space, arising in the study of
Monge-Kantorovich transport problem, and the spectral distance of
noncommutative geometry. Starting from a remark of Rieffel on compact
manifolds, we first show that on any - i.e. non-necessary compact - complete
Riemannian spin manifolds, the two distances coincide. Then, on convex
manifolds in the sense of Nash embedding, we provide some natural upper and
lower bounds to the distance between any two probability distributions.
Specializing to the Euclidean space $R^n$, we explicitly compute the distance
for a particular class of distributions generalizing Gaussian wave packet.
Finally we explore the analogy between the spectral and the Wasserstein
distances in the noncommutative case, focusing on the standard model and the
Moyal plane. In particular we point out that in the two-sheet space of the
standard model, an optimal-transport interpretation of the metric requires a
cost function that does not vanish on the diagonal. The latest is similar to
the cost function occurring in the relativistic heat equation.
