By N N Bogolubov, Nickolai N Bogolubov Jr

ISBN-10: 9812833986

ISBN-13: 9789812833983

The linear polaron version is a superb instance of an precisely soluble, but nontrivial polaron process. It serves as an ordeal method or zero-level approximation in lots of subtle tools of polaron research. This ebook analyzes, specifically, the opportunity of relief of the entire polaron Hamiltonian to the linear one, and introduces a distinct approach to calculating thermodynamical features in line with the calculation of the averages of T-products. This T-product formalism looks a less difficult manner of doing comparable calculations related to Feynman's direction critical method.

This booklet follows a step by step technique, from relatively basic actual rules to a transparent realizing of refined mathematical instruments of research in smooth polaron physics. The reader is ready to examine the actual standpoint with tools proposed within the ebook, and whilst seize the underlying arithmetic.

a few familiarity with quantum statistical mechanics is fascinating in analyzing this ebook.

**Contents: Linear Polaron version; Equilibrium Thermodynamic nation of Polaron approach; Kinetic Equations in Polaron thought.
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4. Free Energy Calculation 41 Fig. 3. Fig. 2. with center at the point iε and of radius L (see Fig. 2). On this contour, F (Ω) = O 1 L2 so we can see that ∫ F (Ω) dΩ = O C Therefore iε+∞ ∫ iε−∞ 1 L , → 0. F (Ω) dΩ = 0. The same considerations may be applied to prove that −iε+∞ ∫ −iε−∞ F (Ω) dΩ. In fact, F (Ω) — is a regular analytic function on the lower half-plane Im Ω −ε < 0. Therefore it is sufficient to choose the proper contour (see Fig. 3) and to repeat all the previous reasoning. 64) into account, we have Fint = 0.

2Mµ a† a = 1 e β¯ hµ −1 , aa† = 1 + 1 e β¯ hµ −1 . 6. 76) that T {[rα (−is) − rα (−iσ)]2 } =− + µ2 − ν02 ¯h mµ3 ν0 µ eq 2 ¯h ϑ (s − σ)2 + m m ν0 m 2 |s − σ| 1 1 (1 − e−µ|s−σ| ) − β¯hµ (eµ|s−σ| − 1) 1 − e−β¯hµ e −1 . 80) where β = 1/ϑ and ϑ = Kβ T. 6. Averaged Operator T-Product Calculus for Some Model Oscillatory Systems Let the Hamiltonian Γ be a quadratic positive-definite form composed of Bose operators bα , b†α . We denote the statistical sum as Z = Tr e−βΓ and consider linear forms composed of the Bose operators bα , b†α : A1 , A2 , ...

These relations can be generalized for a broader set of functionals F (R) if one approximates these functionals by corresponding sequences of the above-mentioned “special functionals” with subsequent passage to the limit N → ∞. 129) 58 Ch. 1. Linear Polaron Model and approximate the functional F (R) in question by the form F (RN ), belonging obviously to the class of “special functionals”. We should like to stress here that the technique of functional integration was developed first by R. Feynman.

### Aspects of polaron theory: equilibrium and nonequilibrium problems by N N Bogolubov, Nickolai N Bogolubov Jr

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