6 Note that the ratio of the asymptotically exact density given by (15) to the prediction (41) of the KZ argument is equal to √ π/ 4; it is a universal number as it does not depend on the parameters B, b, β of the cooling proce- dures (14). The same ratio characterizes algebraic cool- ing schemes (11) in the α → ∞ limit. This seems to be a general feature: If γ ( t ) approaches to unity faster than any power law, then the KZ argument quickly gives a prediction and the exact answer is just √ π/ 4 times smaller. geneous integral equation. It does not seem feasible to express a solution in quadratures, but interesting asymp- totically exact results have been extracted. We have also given expressions of other two-spin correlations functions via the density of domain walls. What would happen if we shall continue beyond the point t = τ , that is, we shall follow the zero-temperature evolution for t > τ . In one dimension, domain walls will continue to diffuse and annihilate upon collisions; eventually the system will reach a ground state. In the case of instantaneous cooling, higher-dimensional behav- iors are different: In two dimensions, the Ising ferromag- net may not reach a ground state (it can get trapped in a metastable stripe state); in three dimensions, a ground state is never reached [18]. It seems plausible that for slow cooling the system may not get trapped, or the probability of getting trapped decreases as τ → ∞ . This appears an interesting challenge and in two dimensions at least, one may even hope for theoretical progress [19]. This work was inspired by the desire to test the validity of the Kibble-Zurek argument for the spin chain supple- mented with non-conservative (Glauber) dynamics. The conclusion is that in this setting, the KZ argument is very good, namely it gives correct exponents and even the amplitudes are approximated reasonably well. The chief outcome of this work is actually unrelated to the KZ argument or any other approximation, namely we have shown that the Ising-Glauber chain remains solvable in the general case of an arbitrary (externally imposed) tem- perature drive T = T ( t ). The term ‘solvable’ here means that for the most interesting characteristic, the density of domain walls, we have found a closed linear inhomo- I am grateful to Leticia Cugliandolo for discussions that revitalized my interest to the cooling problem. This work has been supported by NSF grant CCF-0829541. Acknowledgments VI. DISCUSSION [1] A. J. Bray, Adv. Phys. 43 , 357 (1994). [2] T. W. B. Kibble, J. Phys. A 9 , 1387 (1976); Phys. Rep. 67 , 183 (1980). [3] W. H. Zurek, Nature 317 , 505 (9185); Phys. Rep. 276 , 177 (1996). [4] W. H. Zurek, U. Dorner, and P. Zoller, Phys. Rev. Lett. 95 , 105701 (2005). [5] A. Polkovnikov, Phys. Rev. 72 , 161201(R) (2005). [6] J. Dziarmaga, Phys. Rev. Lett. 95 , 245701 (2005). [7] R. W. Cherng and L. S. Levitov, Phys. Rev. A 73 , 043614 (2006). [8] S. Suzuki, J. Stat. Mech. P03032 (2009). [9] R. J. Glauber, J. Math. Phys. 4 , 294 (1963). [10] G. Biroli, L. F. Cugliandolo, and A. Sicilia, arXiv:1001.0693. [11] P. Laguna and W. H. Zurek, Phys. Rev. Lett. 78 , 2519 (1997). [12] A. Yates and W. H. Zurek, Phys. Rev. Lett. 80 , 5477 (1998). [13] M. Hindmarsh and A. Rajantie, Phys. Rev. Lett. 85 , 4660 (2000). [14] M. I. Bowick et al., Science 263 , 943 (1994). [15] M. E. Dodd et al., Phys. Rev. Lett. 81 , 3703 (1998). [16] A. Maniv, E. Polturak, and G. Koren, Phys. Rev. Lett. 91 , 197001 (2003). [17] P. L. Krapivsky, S. Redner, and E. Ben-Naim, A Kinetic View of Statistical Physics (Cambridge University Press, Cambridge, 2010). Chapter 8 describes the Glauber dy- namics and an exact analysis of the Ising-Glauber chain. [18] V. Spirin, P. L. Krapivsky, and S. Redner, Phys. Rev. E 63 , 036118 (2001); Phys. Rev. E 65 , 016119 (2002). [19] K. Barros, P. L. Krapivsky, and S. Redner, Phys. Rev. E 80 , 040101(R);