content_image/404314.jpg content_image/404313.jpg content_image/404310.jpg content_image/404309.jpg content_image/404312.jpg content_image/404302.jpg content_image/404308.jpg content_image/404315.jpg content_image/404316.jpg content_image/404304.jpg content_image/404311.jpg content_image/404303.jpg content_image/404306.jpg content_image/404307.jpg content_image/404305.jpg
γ ( t ) near the end of the cooling procedure is exponential rather than algebraic 

<img src='content_image/404314.jpg'>

This suggests to analyze a family of cooling laws with an asymptotic behavior of γ ( t ) of the form 

<img src='content_image/404313.jpg'>

with arbitrary positive parameters B, b, β . In this sit- uation, the asymptotic behavior of the final density of domain walls is 

This is a remarkably universal asymptotic: The leading τ − 1 / 2 algebraic factor does not depend on the parameters B, b, β and the logarithmic prefactor depends only on one parameter β . 

We now turn to derivations of the announced results. 

In this section we consider the special cooling proce- dure (6). First, we employ an exact analysis and then turn to the asymptotic behavior of the final density of domain walls. 

To handle an infinite set of ordinary differential equa- tions (8) we recast it into a single differential equation using a generating function technique. Multiplying (8) by z k and summing over all k ≥ 1 we find that the gen- erating function 

satisfies 

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Solving (17) subject to 

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where η = ( t 2 − t ′ $^{2}$) /τ . Recalling that the exponential factor exp [ η z + z 1 2 ] is the generating function for the modified Bessel functions 

## III. SPECIAL COOLING SCHEME 

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## A. Exact Analysis 

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we conclude that 

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Thus to determine G$_{1}$ we need to solve an integral equation 

<img src='content_image/404316.jpg'>

Higher correlation functions G$_{k}$ with k ≥ 2 are then ex- pressed via G$_{1}$ and (18). 

The above results are exact. Now we turn to the most interesting case of slow cooling, τ ≫ 1, and compute the final density of domain walls. 

Using (4), we re-write the integral equation (19) in terms of the density of domain walls: 

We are mostly interested in the final density ρ ( τ ). Hence we specialize (20) to t = τ . It is also convenient to change the integration variable, t ′ → η = ( τ 2 − t ′ $^{2}$) /τ . We arrive at the integral equation 

<img src='content_image/404304.jpg'>

where we used the shorthand notation 

The chief contribution to the integral on the right-hand side of (21) is gathered when η ∼ √$_{τ}$ [see Eq. (26) below]. Therefore we can replace ρ ( τ √ 1 − η/τ ) by ρ ( τ ) and the exponential factor E ( η, τ ) by 

as it follows by expanding the term inside the square brackets in Eq. (22). Thus the integral on the right-hand side of (21) becomes 

We can further use the asymptotic formula for the Bessel function, 

## B. Final Density of Domain Walls 

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