content_image/430436.jpg content_image/430444.jpg content_image/430446.jpg content_image/430447.jpg content_image/430445.jpg content_image/430435.jpg content_image/430439.jpg content_image/430452.jpg
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Introducing the dimensionless variables defined previ- ously, recalling that r = ξR/α and µ = k$_{0}$R , and in- troducing the normalized energy (57), we obtain 

<img src='content_image/430444.jpg'>

Using the expressions of κ and λ following Eq. (68), we find that 

so that, finally, 

<img src='content_image/430446.jpg'>

This equation gives the relation between the energy Λ and α for the n -th branch. In Figs. 26, 27, 29 and 30, we plot the energy Λ as a function of α for the first two branches n = 1 and n = 2 in different dimensions of space d = 1 , 2 , 3. The discussion is similar to the one given in Sec. III D. We have also represented the branch corresponding to the homogeneous solution. Us- ing Eq. (87) and η = α $^{2}$/d , its equation is given by Λ = − d $^{2}$/ (2 α $^{2}$) + d/ (2 µ $^{2}$). 

## E. The entropy and the free energy 

Finally, we relate α to the entropy S and to the free energy F . The entropy is given by 

<img src='content_image/430447.jpg'>

<img src='content_image/430445.jpg'>

<img src='content_image/430435.jpg'>

<img src='content_image/430439.jpg'>

<img src='content_image/430452.jpg'>

