It is shown that the weak-coupling limit is not valid for the third-order term in the density expansion of quantum or classical transport coefficients. The weak-coupling limit is not valid in the sense that it does not preserve the qualitative features of real gases and, hence, is useless within the context of the transport divergence problem. In fact, it is shown that, although the weak-coupling limit of the recollision part of the quantum four-particle collision operator diverges, its contribution to transport coefficients actually converges (whereas this contribution is known to diverge logarithmically when no weak-coupling limit is taken). A rather curious behavior is uncovered for the quantum weak-coupling limit of the recollision part of the four-particle collision operator. It is found that the matrix elements of this operator grow as ${t}^{\frac{1}{2}}$ and oscillate rapidly. This growth, which is faster than logarithmic in $t$ , is associated with a spatial overlapping of broadening wave packets. Because of the accompanying oscillation, however, it does not present any divergence problem to transport theory.It is shown that the weak-coupling limit is not valid for the third-order term in the density expansion of quantum or classical transport coefficients. The weak-coupling limit is not valid in the sense that it does not preserve the qualitative features of real gases and, hence, is useless within the context of the transport divergence problem. In fact, it is shown that, although the weak-coupling limit of the recollision part of the quantum four-particle collision operator diverges, its contribution to transport coefficients actually converges (whereas this contribution is known to diverge logarithmically when no weak-coupling limit is taken). A rather curious behavior is uncovered for the quantum weak-coupling limit of the recollision part of the four-particle collision operator. It is found that the matrix elements of this operator grow as ${t}^{\frac{1}{2}}$ and oscillate rapidly. This growth, which is faster than logarithmic in $t$, is associated with a spatial overlapping of broadening wave packets. Because of the accompanying oscillation, however, it does not present any divergence problem to transport theory.

Physical ReviewMay 5, 19686 citations

Strong Coupling Limit and Some Exact Results

Lecture notes in physics. M, MonographsJan 1, 19950 citations

On the relation between the singular and the weak coupling limits

Luigi Accardi, A. Frigerio, Yun Gang Lu

Acta Applicandae MathematicaeMar 1, 19928 citations

Renormalization-group approach to superconductivity: from weak to strong electron–phonon coupling

Shan-Wen Tsai, A. H. Castro Neto, R. Shankar +1 more

We present the numerical solution of the renormalization group (RG) equations derived in Tsai, Castro Neto, Shankar, et al. [Phys. Rev. B to appear], for the problem of superconductivity in the presence of both electron–electron and electron–phonon coupling at zero temperature. We study the instability of a Fermi liquid to a superconductor and the RG flow of the couplings in the presence of retardation effects and the crossover from weak to strong coupling. We show that our numerical results provide an ansatz for the analytic solution to the problem in the asymptotic limits of weak and strong coupling.We present the numerical solution of the renormalization group (RG) equations derived in Tsai, Castro Neto, Shankar, et al. [Phys. Rev. B to appear], for the problem of superconductivity in the presence of both electron–electron and electron–phonon coupling at zero temperature. We study the instability of a Fermi liquid to a superconductor and the RG flow of the couplings in the presence of retardation effects and the crossover from weak to strong coupling. We show that our numerical results provide an ansatz for the analytic solution to the problem in the asymptotic limits of weak and strong coupling.

arXiv:cond-mat/0505426The Philosophical Magazine A Journal of Theoretical Experimental and Applied PhysicsMay 15, 200614 citations

An invitation to the weak coupling and the low density limit

Luigi Accardi, Robert Alicki, A. Frigerio +1 more

Jan 1, 199123 citations

{t}^{\frac{1}{2}}

and oscillate rapidly. This growth, which is faster than logarithmic in

t

, is associated with a spatial overlapping of broadening wave packets. Because of the accompanying oscillation, however, it does not present any divergence problem to transport theory.