We consider a system perturbed by an external field and subject to dissipative processes. From the von Neumann equation for such a system in the weak coupling limit we derive an inhomogeneous master equation, i.e., a master equation with dissipative terms and streaming terms, using Zwanzig’s projection operator technique in Liouville space. From this equation the response function, as well as expressions for the generalized conductivity and susceptibility, is obtained. It is shown that for large times only the diagonal part of the density operator is required. The various expressions are found to be in complete harmony with previous results (Part I) obtained via the van Hove limit of the Kubo–Green linear response formulas. In order to account for the properties at quantum frequencies, the evolution of the nondiagonal part in the weak coupling limit is also established. The complete time dependent behavior of the dynamic variables in the van Hove limit is expressed by B (t) =exp[−(Λd−iℒ0) t] B, where Λd is the master operator and L0 the Liouville operator in the interaction picture. The cause of irreversibility is discussed. Finally, the inhomogeneous master equation is employed to obtain as first moment equation a Boltzmann equation with streaming terms, applicable to quantum systems.

1.
K. M.
van Vliet
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J. Math. Phys.
19
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2.
R.
Kubo
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J. Phys. Soc. Jpn.
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3.
L.
van Hove
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Physica (Utrecht)
21
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517
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1955
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4.
Dγ can also act on any nondiagonal operator; it then destroys the nondiagonal part: DγK = 〈γ|K|γ〉 = 〈γ|Kd|γ〉.
5.
L. van Hove, in Les Gaz Neutres et Ionises, edited by C. de Witt and J. F. Detoeuf (Hermann, Paris, 1960), p. 159.
6.
Occasionally other boundary conditions are encountered, as for the Hermite polynomials or associated Laguerre polynomials for magnetic phenomena. Since these involve an infinite domain, they usually present no boundary term problems.
7.
U. Fano, Lectures on the Many‐Body Problem, edited by E. R. Caianiello (Academic, New York, 1964), Vol. 2.
8.
As in Paper I, a bracketed operator A(t) is a time dependent operator, either of the Heisenberg type AH(t), the reduced type AR(t), or of the interaction type AI(t). Unbracketed operators are time‐independent, and, if non‐superscripted, they are Schrödinger operators AS. Naturally AH(0) = AI(0) = AS = A.
9.
I owe this remark to M. Dresden, Vosbergen Conference, 1968.
10.
In a previous version of this manuscript we solved for the Green’s function of (3.4), defined by
∂K(t,t′)∂t+i(1−P)LK(t,t′)−1ℏF(t)(1−P)[A,K(t,t′)] = Jδ(t−t′)
. The solution is
K(t,t′) = u(t−t′)ei(1−P)(1−P)′×exp[i ∫ γ′dτF(τ)(1−P)CA(t′−τ)]
where CA is a superoperator such that
CA(t)B = (1/ℏ)eu′(1−P)[A,eu(1−P)′B].
While this solution separates more completely the diagonal and nondiagonal parts, it is of little value, for the linearized equations reduce again to (3.11) and (3.12).
11.
R. Zwanzig, in Lectures in Theoretical Physics, Vol. III, edited by W. E. Britten and J. Downs, Boulder, Colorado, 1960 (Interscience, New York, 1961), pp. 106–141.
12.
ρ is bounded; for arbitrary φ of ℋ we have
(ρφ,ρφ) =  ∑ γ|(Q,γ)|2(ρ2γ,γ)< ∑ γ|(φ,γ)|2(φγ,γ)⩽Trρ = 1.
13.
We prefer the notation JA over ȦR. However, to keep the equations in Kubo‐form appearance, we mainly use the notation ȦR, in Secs. 6 and 7.
14.
In Paper I we computed BdR(Δt), where we assumed from the outset that B was diagonal. If this assumption is dropped, then I, Eq. (6.11) will read 〈γ|U(0)†BU(0)|γ〉 = 〈γ|BI(Δt)|γ〉. In the higher order terms the diagonal approximation is allowed. With this modification, I, (6.37) will read BR(Δt) = BI(Δt)−AdBdΔt. subtracting from both sides B(0), dividing by Δt, and operating with 𝒫, the statement (4.28) follows if we set B→A. In principle, we can also obtain (4.28) from the van Hove limit of the Heisenberg equation of motion.
15.
We assume that B is a vector of ℋ, i.e., translationally invariant over the dimensions of the system.
16.
H. Grad, in Encyclopedia of Physics (Springer, New York, 1958) Vol. 12.
17.
L. Bohzmann Lectures on Gas Theory (University of California, Berkeley, 1964)
[translated from Vorlesungen uber Gastheorie (Barth, Leipzig, 1896, 1898)].
18.
See, for example, A. H. Kahn and H. P. R. Frederickse, in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic, New York, 1959), Vol. 9.
19.
D. Kennedy, C.‐T. Hsing, A. Sutherland, and K. M. van Vliet, Phys. Status Solidi (a), in press;
also, Chi‐Tien Hsing, Ph.D. Thesis, University of Florida (1977).
20.
M. Charbonneau and K. M.van Vliet, to be published.
21.
See, for example, C. Kittel, Quantum Theory of Solids (Wiley, New York, 1963).
22.
K. M.
van Vliet
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Phys. Status Solidi (b)
78
,
667
(
1976
).
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23.
F. Riesz and B. Sz. Nagy, Lecpns d’analyse Fonctionelle (Gauthiers‐Villars, Paris, 1965), 4th edition.
24.
R. H. Fowler, Statistical Mechanics (Cambridge University, Cambridge, England, 1936)
[(reprinted by Dover, New York, 1966)]; Eqs. (2114) and (2115).
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