  The connection between dissipation and symmetry breaking is a long-standing
enigma in statistical physics. It is intimately connected to the quest of a
non-equilibrium functional whose minimization gives the non-equilibrium steady
state (NESS). Writing down such a functional, we show that, in the presence of
additive noise, any NESS is characterized by the minimum entropy production
compatible with the maximum dissipation along cycles in the trajectory space.
This result sheds light on the excess entropy production principle and the
onset of chiral symmetry breaking out-of-equilibrium, indicating that the
housekeeping dissipation is connected with the tendency of performing cycles in
a preferential direction. Finally, when multiplicative noise is present, we
find that the non-equilibrium functional has two dissipative symmetry-breaking
contributions, one stemming from cycles and the other from a
thermophoresis-like effect. Our framework paves the way to understand selection
phenomena as symmetry-breaking processes driven by non-equilibrium dissipation.
