The integral has a simple asymptotic behavior. Indeed, changing η → ξ via η = 2 τ ( ξ/ ln τ ) 1 /β we recast the exponential factor in the integrand into In the τ → ∞ limit, the exponential factor (33) is asymp- totically equal to 1 when ξ < b and zero when ξ > b . Therefore the RHS is asymptotically The LHS of (29) is asymptotically 1/2 as before. Equat- ing it to (34) we arrive at the announced result (15). A Kibble-Zurek (KZ) argument has been proposed [2, 3] to determine the final density of defects (domain walls in our case). The KZ argument leads to qualita- tively correct results, yet it involves an uncontrolled ap- proximation and cannot predict more subtle quantitative features, e.g. amplitudes in the scaling laws. In most ex- amples, it is impossible to estimate the quantitative error as we do not possess (asymptotically) exact results. The ferromagnetic Ising chain provides a rare exception and hence it allows us to probe the quality of an approxima- tion provided by the KZ argument. The KZ argument suggests to estimate ρ ( τ ) in the fol- lowing way: First, we need to find the equilibrium solution. In other words, we must solve the stationary version of Eqs. (2). These equations admit an exponential equilibrium solu- tion: By inserting G$_{k}$ = η k into (2) we find that the 1. During the initial regime the density of domain walls ρ ( t ) is assumed to be given by its equi- librium value at the corresponding temperature, ρ ( t ) = ρ$_{eq}$ [ T ( t )]. This assumption is natural since the cooling rate is very slow. 2. During the final regime the density of domain walls ρ ( t ) is assumed to be constant, ρ ( t ) = ρ$_{∗}$ where ρ$_{∗}$ = ρ$_{eq}$ ( T$_{∗}$ ) with T$_{∗}$ = T ( t$_{∗}$ ). This freezing also seems plausible as the density ρ$_{∗}$ is very small and during the final regime the density should not de- crease much. 3. The Ising model endowed with a Glauber dynam- ics is characterized by the typical time r$_{eq}$ ( T ) to relax to equilibrium. We now postulate that the separation time is determined from the criterion r$_{eq}$ ( T$_{∗}$ ) = τ − t$_{∗}$ . V. APPROXIMATE ANALYSIS exp [ − 2 B bβ ( ξ ln τ ) 1+1 /β τ 1 − b/ξ ] (33) ρ ( τ ) ∫ 2 τ ( b/ ln τ ) 1 /β 0 dη √$_{2}$$_{πη}$ = 2 ρ ( τ ) √ τ π ( b ln τ ) 1 /β (34) right-hand side vanishes when 2 = γ ( η + η − $^{1}$). From this relation, η = γ − 1 [ 1 − √ 1 − γ 2 ] , and therefore 5 r$_{eq}$ = 1 2[1 − γ ] (36) ρ$_{eq}$ = 1 − η 2 = 1 2 [ 1 − 1 − √ 1 − γ 2 γ ] (35) 1 2[1 − γ$_{∗}$ ] = τ − t$_{∗}$ , γ$_{∗}$ = γ ( t$_{∗}$ ) (37) ρ$_{∗}$ = 1 2 [ 1 − 1 − √ 1 − γ 2 ∗ γ$_{∗}$ ] ≃ 2 − 3 / $^{4}$τ − 1 / 4 (38) C$_{KZ}$ = 2 − 1 / $^{2}$A1 / (2+2 α ) (39) C C$_{KZ}$ = √$_{π}$ 2 [ Γ ( 3 + 2 α 2 + 2 α )] − $^{1}$[ 1 1 + α ] 1 2(1+ α ) (40) ρ$_{∗}$ ≃ 1 √$_{4}$$_{τ}$ ( ln τ b ) 1 2 β (41) For the family of cooling procedures (14) we find This ratio approaches to 1 as α → 0 and to √ π/ 4 ≑ 0 . 8862269 when α → ∞ . The largest deviation from unity is 0 . 7913 . . . , so the maximal quantitative mistake of the KZ argument is about 20%. Interestingly, the KZ argument correctly predicts even the dependence on A : The ratio of the exact and ap- proximate amplitudes is the function of α alone For the family of cooling schemes (11) we recover the correct scaling law (12), but with a slightly erroneous amplitude Comparing (10) and (38) we see that the exact amplitude C = π 1 / $^{2}$/ Γ(1 / 4) ≑ 0 . 48887 is smaller than the predic- tion from the KZ argument C$_{KZ}$ = 2 − 3 / 4 ≑ 0 . 59460. Let us begin with the special cooling procedure (1). Then γ$_{∗}$ = t$_{∗}$/τ and therefore (37) gives 1 − γ$_{∗}$ = 1 $_{/}$√$_{2}$$_{τ}$$_{.}$ Hence as a reasonable definition. Combining (36) and the cri- terion r$_{eq}$ ( T$_{∗}$ ) = τ − t$_{∗}$ we arrive at The relaxation time is to a degree the matter of agree- ment. Equation (2) suggests that A. Algebraic Cooling B. Exponential Cooling