N ( ˆ t ) ≃ [ f $^{2}$ξ$_{eq}$ (ˆ g )] − d ≃ f − 2 $^{d}$| Δˆ g | dν with f of the order of one. Knowing the τ$_{Q}$ -dependence of Δˆ g allows one to derive the τ$_{Q}$ -dependence of N : We stress that for each τ$_{Q}$ this expression should be mea- sured at the special instant ˆ t ( τ$_{Q}$ ) after the phase tran- sition. The system’s behaviour at t > ˆ t is not fully ad- dressed by KZ. In some publications it is assumed that any further evolution [15, 16] can be neglected, whereas in others it is reckoned that after ˆ t the system resumes its out of equilibrium evolution with a mechanism that depends on the problem at hand (domain growth, vortex- anti-vortex diffusion and annihilation, etc.) [9] and the density of defects may therefore continue to decrease al- though no detailed study was performed. Numerous nu- merical [15–19] and experimental [20–24] papers tested the quantitative consequences of the Kibble-Zurek mech- anism with variable results. While the numerical studies claimed that they successfully verified the predictions, the conclusions are less clear in the experimental works that studied vortex formation in superfluid $^{4}$He and $^{3}$He with null results in the former [20] and agreement with the KZ prediction in the latter [21]. See [22–24] for dis- cussions of some recent experimental results disagreeing with the KZ prediction. In the following we revisit the KZ scaling analysis. We focus on the dynamics of classical systems coupled to an environment, a setting in which the dynamics are stochastic and the energy, directly linked to the num- ber of topological defects, is not conserved. We use the temperature of the thermal bath as the control parame- ter driving the second order phase transition and a linear cooling rate T ( t ) = T$_{c}$ (1 − t/τ$_{Q}$ ). These choices are made to keep the discussion simple; extensions to more compli- cated protocols are straightforward and will be partially addressed later. Moreover, we restrict to systems with a unique equilibrium correlation length and a single dy- namic counterpart, the typical growing length. We solely deal with problems with power-law scaling laws, that link e.g. the correlation length to the distance from criticality, and length-scales to time-scales [25]. These restrictions exclude from the analysis complex systems with several competing lengths and problems with quenched disorder. Henceforth we focus on the growth law of the size of the correlated regions, R ( t ). This will allow us to discuss systems characterized by topological defects as well as those that are not from the same point of view. We shall explain below how the density of topological defects can be obtained from R ( t ). Let us start our analysis with some simple remarks. First, although the initial adiabatic regime and the de- parture from it are expected, the existence of the im- puse regime is questionable. During this regime, which would take place between − ˆ t and ˆ t , the system is sup- posed not to evolve [7–10]. However, it is well-known This imposes that f ( x ) be a constant for x ≪ − 1 and proportional to x 1 /z$_{d}$ for x ≫ 1. We expect the scaling function f ( x ) to be universal since it describes evolution on diverging time and length scales close to the critical point. A sketch of ξ$_{eq}$ and R is shown in Fig. 1. Our scal- ing assumption applies to coarsening with and without This asymptotic form encompasses equilibrium above the critical point, x ≡ t/τ$_{eq}$ ( T ( t )) ≪ − 1, critical coarsening, x ∝ O (1), and the cross-over to standard coarsening x ≫ 1. The limits of f ( x ) are obtained by requiring to find the expected adiabatic behavior and standard coarsening on the two extremes, We now endeavor to connect the dots and propose a general scenario for infinitely slow annealing. Our main conjecture, that is motivated by the previous discussion and the fact that the system stays for a very long time in the vicinity of the critical point, involves the growth of the length-scale R ( t ): Our second remark is that once getting across the crit- ical point, when the running temperature T ( t ) is far enough from T$_{c}$ , the dynamics crosses over to standard coarsening. In order to get a better insight into this pro- cess, let us recall that an infinitely rapid quench to a tem- perature T < T$_{c}$ leads to a growth law R ≃ λ ( T )Δ t 1 /z$_{d}$ [1]. Now z$_{d}$ is the dynamic exponent that, quite gener- ally, is different from z$_{eq}$ and depends on the dynamic rules. The prefactor vanishes at T$_{c}$ and is characterized by a singular power law λ ( T ) ≃ | T − T$_{c}$ | ν ( − 1+ z$_{eq}$/z$_{d}$ ) = ξ 1 − z$_{eq}$/z$_{d}$ eq [28]. If the annealing rate is finite, one natu- rally expects the growth law at long times and far from the critical point to be R ≃ λ ( T ( t ))Δ t 1 $^{/z$_{d}$}$. The reason is that the dynamical process renormalizing the value of λ ( T ( t )) should be finite and, hence, evolve on a much faster timescale than the coarsening one which instead is of the order of the age of the system and diverges with t . that after a rapid quench into the critical region any system undergoes critical coarsening described by the growth of a typical linear length-scale for correlated re- gions, R (Δ t ) ∼ Δ t 1 /z$_{eq}$ where Δ t is the time spent in the critical region and z$_{eq}$ is the exponent that links the equilibrium relaxation time and correlation length close to criticality [26, 27]. Above criticality, the major differ- ence between slow and rapid quenches is in the extension of the adiabatic regime. The slower the quench or the an- nealing, the closer the system gets to the critical point in equilibrium. However, also for very slow annealing, the system eventually departs from the adiabatic evolution and has to undergo critical coarsening. 2 N ( ˆ t ) ∝ τ − dν/ (1+ νz$_{eq}$ ) Q at ˆ t ∝ τ νz$_{eq}$/ (1+ νz$_{eq}$ ) Q . (1) R ( t ) ≃ ξ$_{eq}$ ( T ( t )) f [ t τ$_{eq}$ ( T ( t )) ] . (2) R ( t ) ≃ { ξ$_{eq}$ ( T ( t )) t ≪ − τ$_{eq}$ ( T ( t )) , [ ξ$_{eq}$ ( T ( t ))] 1 − z eq z d t 1 z d t ≫ τ$_{eq}$ ( T ( t )) . (3)