arXiv:1001.0741v1 [cond-mat.stat-mech] 5 Jan 2010 The quenching procedure that is used in numerous studies of coarsening is usually based on the following protocol: (i) Start at a high initial temperature T$_{i}$ > T$_{c}$ , where spins are disordered; (ii) Instantaneously cool the system to a final temperature that is lower than the crit- ical, T$_{f}$ < T$_{c}$ . The most popular choice of the initial temperature is T$_{i}$ = ∞ . Cooling to the critical tem- perature, T$_{f}$ = T$_{c}$ , is also often studied, especially for one-dimensional systems where T$_{c}$ = 0 or in quantum quenches. After such instantaneous quench, ordered re- gions begin to grow and it takes a long time (an infinite time in the thermodynamic limit) to reach the equilib- rium. Understanding of the dynamics of phase-ordering kinetics, particularly the scaling regime that develops long after the quench, is the main goal [1]. Although the very fast cooling is natural in a num- ber of settings, there are numerous applications (ranging from the formation of glasses to the expansion of the Universe) where slow cooling is appropriate. A popu- lar gradual cooling scheme posits that the temperature vanishes linearly in time, so as time varies in the interval (0 , τ ), the temperature falls from T$_{i}$ = T$_{0}$ to T$_{f}$ = 0. To characterize the cooling scheme (1) or other gradual cooling procedure, it is nat- ural to focus on the properties of the state at the end of the cooling procedure, particularly to find the difference between this final state [for the cooling scheme (1), it cor- responds to t = τ ] and a ground state. If the cooling rate 1 /τ is large, the problem is not interesting as during time interval (0 , τ ) the system has hardly evolved. Interesting behaviors arise if cooling is very slow, τ ≫ 1, so that the system has evolved over long time interval (0 , τ ) and its state is close to a ground state. Then the problem is to describe the deviation from the ground state. The limit of slow cooling has been investigated in nu- merous papers. Earlier work has been mostly driven by applications to cosmology and proposals to study analogs of the cosmological phase transitions in the laboratory The following section II we write-down the governing equations for the two-spin correlation function, outline the problems in solving these equations, and present our chief results. We then provide a complete derivation for a special cooling scheme (section III). In section IV we analyze an arbitrary cooling scheme and derive various asymptotic results announced in section II. Section V is devoted to the comparison of exact results with ap- proximate predictions following from the Kibble-Zurek argument. Finally in section VI we give a summary and briefly discuss how the cooling protocol can affect the probability for high-dimensional Ising ferromagnets to (eventually) fall into a ground state. Unfortunately exact results haven’t been obtained even for the Ising-Glauber spin chain [8]. Here we show that this model is tractable for essentially arbitrary cooling procedure. More precisely, we establish an integral equa- tion for the density of domain walls. This equation is closed and linear. Although it is generally unsolvable, in the interesting slow cooling limit it is possible to deduce analytically an exact asymptotic for the final density of domain walls. Overall, studies of slow quenching tend to rely on un- controlled approximations. This is not so surprising since little can be done analytically in the high-dimensional setting. Some of the uncontrolled approximations, e.g. the so-called Kibble-Zurek mechanism [2, 3], rely on sound physical ideas, yet we still want to better under- stand their possible limitations. Numerical (see e.g. [11– 13]) and laboratory (see e.g. [14–16]) experiments help in this regard, but large size simulations can be challenging (particularly for systems exhibiting quantum phase tran- sitions), while real system always have features beyond those that are captured by theoretical models. [2, 3]. Recently, similar ideas and methods have been applied to the dynamics of slow quantum annealing, see e.g. [4–7]. A surprisingly little work has been done in the more standard context of thermal phase transitions. Among a few exceptions is Ref. [8] that studies the Ising chain supplemented with Glauber’s spin-flip dynamics [9] and Ref. [10] which investigates high-dimensional sys- tems where T$_{c}$ > 0, so that the critical temperature is passed during the quenching procedure. Abstract. A ferromagnetic Ising chain which is endowed with a single-spin-flip Glauber dynamics is investigated. For an arbitrary annealing protocol, we derive an exact integral equation for the domain wall density. This integral equation admits an asymptotic solution in the limit of extremely slow cooling. For instance, we extract an asymptotic of the density of domain walls at the end of the cooling procedure when the temperature vanishes. Slow annealing is usually studied using a Kibble-Zurek argument; in our setting, this argument leads to approximate predictions which are in good agreement with exact asymptotics. Keywords: phase transitions, slow annealing, Glauber dynamics I. INTRODUCTION T ( t ) = T$_{0}$ · ( 1 − t τ ) (1) Slow Cooling of an Ising Ferromagnet P. L. Krapivsky 1 $^{1}$Department of Physics, Boston University, Boston, MA 02215, USA