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Many-Body Boson Systems: Half a Century Later by André F. Verbeure

By André F. Verbeure

Many-body Boson platforms: part a Century Later bargains a contemporary method of facing the issues of equilibrium states of Bose structures. beginning with the difference precept of statistical mechanics and the energy-entropy stability precept as equilibrium standards, effects for normal boson platforms and types are explicitly derived utilizing uncomplicated sensible analytic calculus.

Bridging the space among idea’s of normal theoretical physics and the phenomenological learn within the box of Bose structures, this booklet presents an perception into the attention-grabbing quantum global of bosons. Key subject matters comprise the incidence of BEC and its intimate structural relation with the phenomena of spontaneous symmetry breaking and off-diagonal lengthy variety order; the condensate equation; the problem in regards to the number of boundary stipulations; solvable as opposed to non-solvable boson versions; the set of quasi-free boson states; the function of dissipative perturbations; and the astonishing yet common relation among basic quantum fluctuations and boson systems.

Only a few wisdom of quantum mechanics and undergraduate algebra and research is believed. This textbook brings scholars and researchers easily from basic strategies to shiny applications.

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Later on we discuss in some detail the effects due to alternative choices. Getting now to the point, consider a system of bosons of mass m enclosed in the cubic boxes V ⊂ Rd , where d is the dimension of the system. The notation V denotes the subsets of the space Rd as well as its volume V = Ld ; L is the side length of the boxes. We consider the dual volume V ∗ = k ∈ R d ; kα = 2π nα ; nα = 0, ±1, . . , α = 1, . . d . 3) k∈V ∗ h¯ 2 k2 , h¯ = 1, μ is the chemical potential, and use the notation 2m a∗k = a∗ ( fk ) = V eikx dx √ a∗ (x) V 1 fk (x) = √ eikx (the individual particle wave function) V with the boson commutation relations [ak , a∗k ] = δk,k ; [ak , ak ] = 0 , Note that the wave functions fk with k = 0 are periodic functions describing particular periodic localizations of the boson particles in the space variables x.

Before proceeding to the next chapter, we derive from the EEB criterion a number of generally valid properties of equilibrium states for any quantum system as well as, of course, any boson system. As a first result following from this criterion, the stationarity or the time invariance of all equilibrium states is derived. Although this property may come over as an obvious physical statement, its derivation is not sufficiently visible in the physics literature. 8. Let ωβ be a state, a solution of the EEB criterion Eq.

32 3 Equilibrium States As a matter of introducing this subject, we consider again the case of density matrix states for finite systems with a Hamiltonian H(μ ) as we did in the previous subsection in order to argue the variational principle Eq. 1). 5. The necessary and sufficient condition in order that a state ω is the Gibbs state ωβ , is that it satisfies the following correlation inequalities, called from now on the energy-entropy balance correlation inequalities: For any observable A, β ω (A∗ [H(μ ), A]) ≥ ω (A∗ A) ln ω (A∗ A) ω (AA∗ ) where the function f (u, v) = u ln uv is well defined for all real u, v > 0 and where f (u, v) = 0 if u = v = 0.

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