content_image/421830.jpg content_image/421837.jpg content_image/421849.jpg content_image/421829.jpg content_image/421836.jpg content_image/421847.jpg content_image/421848.jpg content_image/421845.jpg content_image/421846.jpg
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Using α = ( S$_{d}$Gβmρ$_{0}$ ) 1 / $^{2}$R and introducing the dimen- sionless temperature (51), we obtain 

This equation gives the relation between the inverse tem- perature η and α for the n -th branch. In Figs. 22, 23, 24 and 25, we plot the inverse temperature η as a func- tion of α for the first two branches n = 1 , 2 in different dimensions of space d = 1 , 2 , 3. The discussion is similar to the one given in Sec. III C. We have also represented the branch corresponding to the homogeneous solution. Its equation is given by η = α $^{2}$/d . The branch n = 1 of inhomogeneous solutions connects the branch of homo- geneous solutions at α 2 c = dη ∗ c (see Appendix F). 

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We must also relate α to the energy E . The total energy is given by 

Using the Maxwell-Boltzmann distribution (10), the ki- netic energy is simply 

Using the screened Poisson equation (64) and integrating by parts, the potential energy can be written 

<img src='content_image/421847.jpg'>

The total energy 

<img src='content_image/421848.jpg'>

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## D. The energy 

<img src='content_image/421846.jpg'>

