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tokens: considering equation ( 4 ) and measured P values of ∼1–10 h for AMXPs (Salvo Sanna 2020 ). Figure 2. Numerically computed neutron star spin frequency ( ν ) evolution curves due to transient accretion (see Sections 3.1 and 3.2 ). The dotted grey horizontal line shows a constant long-term average accretion rate () of 5 × 10 15 g s −1 . The solid grey curve shows an evolving with a fast decay (see Section 3.1 ) in the RLDP on the right of the dotted vertical line. Various ν evolution curves are for g s −1 , and based on if EM spin-down and evolution are included not, as mentioned on the plot. This figure shows that ν can increase in the RLDP (see Section 3.2 ). 3.2 Spin evolution of transient sources First, we study how ν evolves if the neutron star accretes transiently throughout the LMXB phase (see Fig. 2 ). For an evolution duration of two billion years and for the examples given in this figure, ν remains much lower than ν eq,eff (not shown in the figure) for g s −1 . Note that an –10 16 g s −1 is common for many neutron star LMXBs (Lamb Yu 2005 ), although could be significantly higher in the initial stage of the LMXB phase (e.g. see Tauris 2018 ; Chen, Liu Wang 2020 ). The aim here is to find out how ν evolves in the RLDP. Fig. 2 shows that ν can increase in the RLDP, even if decreases drastically, and regardless of the inclusion of an additional spin-down (e.g. due to the EM torque (see Bhattacharyya Chakrabarty 2017 ) in the quiescent phase; see Section 1 ). This is not unexpected, as ν neither approaches nor tracks ν eq,per during a transient accretion (see Section 2 ). However, we note that ν either approaches or tracks ν eq,eff for such an accretion. Hence, ν either increases or decreases due to transient accretion, both in the RLDP and before the RLDP, depending on whether increases or decreases (see Section 2 and equation 5 ). Therefore, ν can decrease in the RLDP, if decreases. But, since is expected to depend on the binary orbital period P (with an estimated relation of , Lasota 2001 ), and the change of P is moderate (e.g. Bhattacharya van den Heuvel 1991 ), ν should not considerably decrease in the RLDP. 3.3 Spin evolution of persistent sources How does ν evolve for a persistent source? The neutron star first spins up relatively quickly towards the spin equilibrium frequency ( ν eq,per ), and then tracks it (see Introduction and Section 2 , Fig. 3 ). In the RLDP, if decreases rapidly, ν decreases for a persistent source, but cannot track ν eq,per anymore, which was shown by Tauris ( 2012 ), and is confirmed in Fig. 3 . But in reality, a persistent source becomes a transient for , and consequently, ν increases, as it does not track the lower spin equilibrium frequency ν eq,per anymore, and now approaches a higher effective spin equilibrium value ν eq,eff (see Fig. 3 , and also Section 2 ). Therefore, since even an initially persistent source should become a transient at least in the RLDP due to the decrease of , ν should increase in