THE CONNECTED DIAGRAM EXPANSION OF THE GRAND PARTITION FUNCTION AND THE STATISTICAL MECHANICS OF THE ELECTRON GAS (thesis)

Technical Report · Sun May 01 04:00:00 EDT 1960

OSTI ID:4076636

Sakakura, A Y

>The connected diagram expansion of the grand partition function, generalized to include many-component systems with external fields, was obtained by means of second quantization and Wick's theorem. This was accomplished by expanding the iterative solution of the Bloch equation into a sum of normal products and by directly evaluating the traces of these normal products; the contraction symbol for fermions and bosons is the sum of two contraction symbols, one arising from the expansion in normal products and the other from the evaluation of the traces. An alternative derivation for the fermions based on the hole formalism was also given. Rules for constructing the contribution to log Z from a particular diagram were given in both the configuration and momentum spaces. For the case of pair interactions alone, the diagrams were simplified by the separation into the irreducible diagrams and the polarization parts in analogy with the ground-state theories. The integral equation satisfied by the polarization parts was derived, and the rules for computing the eigenvalues of this integral equation were given. It was shown that the lowest order eigenvalues yielded the Montroll-Ward ring contributions. With the inclusion of the first- and second-order irreducible diagrams, the ground-state energy of the electron gas was derived to second order. The role of the chemical potential in the elimination of contributions not encountered in the direct ground-state energy derivations was discussed. The ring contribution to the grand partition function containing all possible combinations of the firstorder exchange interactions is obtained by including the eigenvalues corresponding to kernels with one internal line. From this the equation of state and the internal energy of the electron gas in the near classical limit were extended to include terms of orders LAMBDA gamma , gamma /sup 4/, and LAMBDA l gamma /sup 2/. A method of absorbing all passive particle interaction into a contraction symbol was given, and a simple ladder diagram was summed. (auth)

• Research Organization: Colorado. Univ., Boulder
• NSA Number: NSA-15-011781
• OSTI ID: 4076636
• Report Number(s): TID-11028
• Country of Publication: Country unknown/Code not available
• Language: English

Related Subjects

BLOCH THEORY
BOSONS
CONFIGURATION
DIAGRAMS
DIFFERENTIAL EQUATIONS
DISTRIBUTION
ELECTRIC FIELDS
ELECTRONS
ENERGY
FERMIONS
FIELD THEORY
GASES
INTERACTIONS
LATTICES
MAGNETIC FIELDS
MANY BODY PROBLEM
MECHANICS
MOMENTUM
PARTITION FUNCTION
PERTURBATION THEORY
PHYSICS
POLARIZATION
QUANTUM MECHANICS
STATISTICS
THERMODYNAMICS
WICK THEOREM