content_image/404023.jpg content_image/404043.jpg content_image/404042.jpg content_image/404047.jpg content_image/404046.jpg content_image/404048.jpg content_image/404044.jpg content_image/404045.jpg content_image/404041.jpg content_image/404040.jpg
<img src='content_image/404023.jpg'>

## III. THERMODYNAMIC POTENTIAL DIAGRAMS 

The thermodynamic potential of the system is deter- mined by the connected part of the mean value of the evolution operator [11 , 12] 

Let us consider a more general quantity first 

and put then λ = 1. 

By using the perturbation theory we have obtained the first orders of diagrams for 〈 U$_{λ}$ ( β ) 〉 c $_{0}$, depicted in Fig. 3. 

In order to obtain the better understanding of these diagrammatic contributions we examine the expression 

<img src='content_image/404043.jpg'>

where double repeated indices suppose summation and integration. Consequently (16) is equal to 

<img src='content_image/404042.jpg'>

Here we have carried out the integration by time. 

From diagrammatic point of view equation (16) implies the procedure of locking of the external lines of propaga- tors G$_{λ}$ diagrams depicted on the Fig. 1 with the tun- neling matrix element t ( − → x ′ − − → x ) and obtaining in such a way the diagrams without external lines similar with ones for 〈 U$_{λ}$ ( β ) 〉 c 0 depicted on Fig. 3. These two series of diagrams differ by coefficients in front of them. 

<img src='content_image/404047.jpg'>

<img src='content_image/404046.jpg'>

In expression (17) the coefficients 1 n before each dia- gram are absent, where n is the order of perturbation theory. These coefficients are present in Fig. 3. In order to restore these 1 n coefficients in (17) and obtain the co- incidence with 〈 U$_{λ}$ ( β ) 〉 c 0 series it is enough to integrate by λ the expression (17) and obtain: 

<img src='content_image/404048.jpg'>

The expression (18) displayed in a diagrammatic repre- sentation coincides exactly with mean value of the evo- lution operator: 

In Fourier representation we have 

<img src='content_image/404044.jpg'>

From (15) and (20) we obtain 

Using the definition (14), the equation (21) can be writ- ten in the form: 

<img src='content_image/404045.jpg'>

<img src='content_image/404041.jpg'>

<img src='content_image/404040.jpg'>

