  We discuss the notion of nonequilibrium chemical potential in gases of
non-interacting active particles filling two compartments separated by a
potential energy barrier. Different types of active particles are considered:
run-and-tumble particles, active Brownian particles, and active Brownian
particles with a stochastic reorientation along an external field. After
recalling some analytical results for run-and-rumble particles in one
dimension, we focus on the two-dimensional case and obtain a perturbative
expression of the density profile in the limit of a fast reorientation
dynamics, for the three models of active particles mentioned above. Computing
the chemical potentials of the non-equilibrium systems in contact from the
knowledge of the stationary probability distribution of the whole system
---which agrees with a recently proposed general definition of the chemical
potential in non-equilibrium systems in contact--- we generically find that the
chemical potential lacks an equation of state, in the sense that it depends on
the detailed shape of the potential energy barrier separating the compartments
and not only on bulk properties, at odds with equilibrium. This situation is
reminiscent of the properties of the mechanical pressure in active systems. We
also argue that the Maxwell relation is no longer valid and cannot be used to
infer the nonequilibrium chemical potential from the knowledge of the
mechanical pressure.
