The dynamics of liquids, regarded as strongly-interacting classical
particle systems, remains a field where theoretical descriptions are
So far, there is no microscopic theory starting from first principles
and using controlled approximations.
At the thermodynamic level, static equilibrium properties are well
understood in simple liquids only far from glassy regimes.
Here we derive, from first principles, the dynamics of liquids and
glasses using the limit of large spatial dimension, which provides a
well-defined mean-field approximation with a clear small parameter.
In parallel, we recover their thermodynamics through an analogy between
dynamics and statics.
This gives a unifying and consistent view of the phase diagram of these
systems. We show that this mean-field solution to the structural glass
problem is an example of the Random First-Order Transition scenario, as
conjectured thirty years ago, based on the solution of mean-field spin
These results allow to show that an approximate scale invariance of the
system, relevant to finite-dimensional experiments and simulations,
becomes exact in this limit.