Read e-book online Aspects of polaron theory: equilibrium and nonequilibrium PDF

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 procedure. It serves as an ordeal method or zero-level approximation in lots of refined equipment of polaron research. This ebook analyzes, particularly, the potential for aid of the total polaron Hamiltonian to the linear one, and introduces a distinct approach to calculating thermodynamical features according to the calculation of the averages of T-products. This T-product formalism looks a simpler means of doing related calculations related to Feynman's direction quintessential process.

This ebook follows a step by step procedure, from relatively easy actual rules to a transparent figuring out of refined mathematical instruments of research in smooth polaron physics. The reader is ready to evaluate the actual perspective with equipment proposed within the publication, and even as grab the underlying arithmetic.

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

Contents: Linear Polaron version; Equilibrium Thermodynamic kingdom of Polaron process; Kinetic Equations in Polaron conception.

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5. Average Values of T-products 43 Setting t = −is and choosing real s to be the ordering parameter, we define the T-product as rα (−is)rα (−iσ) if s > σ, rα (−iσ)rα (−is) if σ > s. 72) if α = α , and, in the opposite case, T {[rα (−is) − rα (−iσ)]2 } = eq +∞ 2(1 − e−ω|s−σ| ) i¯h ∫ 2π −∞ 1 mΩ − η + Ω (Ω) 2 −β¯ hω 1−e 2 ω+i0 ω−i0 dω. 73) Consider the single-frequency case: E(ω) = K02 {δ(ω − ν0 ) + δ(ω + ν0 )}. 57). Rigorously speaking, we mean that rα (t)rα (τ ) eq = x(t)x(τ ) eq . 58) that x=q+ M Q.

51) gives us ∂H(λ) ∂λ = −3 λ,eq = −3 Ωλ (Ω) i¯h +∞ 1 ∫ 2π −∞ 1 − e−β¯hω mΩ2 − η 2 + λ2 Ω (Ω) λ (Ω) i¯h +∞ Ω ∫ 2π −∞ 1 − e−β¯hΩ mΩ2 − η 2 + λ2 Ω (Ω) ω+i0 ω−i0 ω+i0 ω−i0 dω dω . 54) Using a similar approach for the calculation of the correlation function pα (t)pα (τ ) eq , we find in the limit η → 0, V → ∞ that ∂H(λ) ∂λ =− λ,eq λ ∞ (Ω) Ω 3i¯h +∞ 1 ∫ 2π −∞ 1 − e−β¯hΩ Ω mΩ + λ2 ∞ (Ω) ω+i0 ω−i0 dω . 55) The right-hand side of this equation does not depend on ε when 0 < ε < < 2π/¯hβ. 50) Fint = − +∞ λ ∞ (Ω) Ω 3i¯h 1 1 ∫ dλ ∫ −β¯ hΩ Ω 2π 0 mΩ + λ2 ∞ (Ω) −∞ 1 − e ω+i0 ω−i0 dω.

133) (f ) Let us consider β¯ h β¯ h ∫ ds T e 0 Λf (s)Qf (s) −βH(Σ) (f ) H(Σ) where Qf (s) = exp = sH(Σ) ¯h Tr e ∫ ds T {e 0 (f ) Tr e−βH(Σ) qf exp − sH(Σ) ¯h . 6. 131) we can treat Λf (s) as the usual C-functions. 82), we get β¯ h ∫ ds T e0 Λf (s)Qf (s) (f ) H(Σ) ¯ hβ ¯ hβ 1 ∫ ds ∫ dσ 2 0 0 exp T Λf (s)Qf (s) (f ) Λf Qf (σ) . 130) that T Λf (s)Qf (s) Λf Qf (σ) H(Σ) f (f ) = Λf (s)Λ−f (σ) T {Qf (s)Q−f (σ)} H(Σ) (f ) Λf (s)Λ−f (σ) = (f ) ¯h (1 − e−βωf ¯h )−1 (e−ωf |s−σ| + e−ωf β¯h+ωf |s−σ| ). 2ωf Thus β¯ h ∫ ds T e0 Λf (s)Qf (s) (f ) H(Σ) ¯ hβ ¯ hβ 0 0 = exp ∫ ds ∫ dσ (f ) ¯hΛf (s)Λ−f (σ) (e−ωf |s−σ| + e−ωf β¯h+ωf |s−σ| ) .

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


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