Abstract It is well-known that the invocation of ‘equilibrium processes’ in thermodynamics is oxymoronic. However, their prevalence and utility, particularly in elementary accounts, presents a problem. We consider a way in which their role can be played by sets of sequences of processes demarcated by curves carrying the property of accessibility. We also examine the vexed question of whether equilibrium processes are necessarily reversible and the revision of this property in relation to sets of sequences of such processes.