  The overdamped Brownian dynamics of a harmonic oscillator is a paradigmatic
system in non-equilibrium statistical mechanics, which reliably models relevant
stochastic systems such as colloidal particles submitted to optical
confinement. In this work, optimal thermal protocols are tailored to minimise
the connection time between equilibrium states of overdamped $d$-dimensional
oscillators. Application of control theory reveals that these optimal protocols
are of bang-bang type, that is, the temperature of the bath has to take
alternatively the minimum and maximum values allowed. Minimum connection times
increase with the considered dimension $d$. Remarkably, this is the case even
for symmetric oscillators, for example, with spherical symmetry -- in which the
degeneracy of the elastic constant along the $d$ possible directions seems to
imply a minimum connection time equal to that for the one-dimensional case.
This surprising unavoidable price to pay when increasing dimension is
thoroughly investigated and understood on a physical basis. Moreover,
information theory tools such as the thermodynamic length and its divergence
are analysed over the brachistochrone.
