A close examination of the validity of the Markovian approximation in the context of relaxat~on t?eory is presented. In particular, we examine the question of positivity of various approxImations to the reduced dynamics of an open system in interaction with a heat r.es~rvoir. It is show~ that the Markovian equations of motion obtained in the weak coupling lImIt .(Redfield equatIons) are a consistent approximation to the actual reduced dynamics only If supplemented by a slippage in the initial conditions. This slippage captures the effects of the non-Markovian evolution that takes place in a short transient time, of the order of the relaxation time of the isolated bath.

A systematic theory for the dynamics of concentrated hard-sphere suspensions of interacting Brownian particles with both hydrodynamic and direct interactions is presented. An alternative equation for the number density of particles is derived. The volume-fraction dependence of the shortand long-time self-diffusion coefficients is thus explored from a new point of view. Both the short-range and the long-range hydrodynamic interactions are shown to play an important role in both coefficients, while the direct interactions are reduced drastically by the hydrodynamic interactions.

Part 111. Boson Field in Interaction with a Given Source Distribution Expectation Values of the Energy a d the Number of BosonsThe simplest specific problem that may be posed in field quantum theory is"to determine what happens in a field under the influence of a given source distribution. In the subsequent discussion of this problem we shall employ the Heisenberg picture, unless otherwise stated. As before, the operator which corresponds to the "field quantity" will be denoted by ," and considered as a function Z(s, t ) of position z and time t. In case no external forces are present this function was supposed to satisfy the differential equationin which V, = a/at, V, = grad, and p is the "rest mass", see Section 3. In case the field is under the influence of a given distribution of charges with the density j = j(z, t), the function E(s, 2) should satisfy the differential equationThe term j on the right side is of course to be understood as the identity operator multiplied by the function j (~, t).The solution of this differential equation must satisfy initial conditions.In the present section we shall assume that the operators ," and V,Z are prescribed at a finite time, specifically, at t = 0. Thus we require Z to satisfy conditionin which Zn(z) and no(z) are two-improper-operators which obey the commutation ruleIn Section 16 we shall impose initial conditions which characterize the behavior of the operator E(z, t) as t + -a.The source distribution j will be assumed independent of the time,As, 0 = .w in the present and in the two subsequent sections. In Section 16 we shall show that, the results obtained can be used to deal with a general, time dependent, source distribution.

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