The requirement is often made in non-equilibrium statistical mechanics that a transport equation should be derived as that which governs the subdynamics relative to a (small) part of a (large) conservative dynamical system close to equilibrium. We show that such a requirement on the Markovian relaxation of a 89 imposes that this process be described by a Bloch equation of a very specific form, which we call standard. We show that this reduced dynamics is quasi-free if, and only if, the relaxation time is maximally anisotropic.
The Bloch equation for a 89 reads:
k=l with ~-/(0) being the usual Paul/matrices. We say that this equation is standard if there exists a unitary transformation U on r such that o/(t) = Ur/(t)U -1 satisfies (d/dt)cq (t) = -~ol (t) -cooz(t), (d/dt)o 2 (t) = + wol (t) -~,o~ (t), (2) (d/d0 [03(0 -el] = -g [ o 3 ( t ) -el], with -1 < e < 1, co real, and 0 ~<# ~< 2•. e is the equilibrium value (03) of the component 03 of the spin, whereas (oa) = (02) = 0; co is the transverse frequency; there are only two relaxation times TII =//-1, and T• = X-1, which moreover satisfy the very remarkable relation TII ) T• Favre and Martin [6] seem to have been the first authors to argue that the Bloch equation would take this standard form whenever the 1-spin system is 'weakly coupled' to a bath at 'high' temperature. More recently Gorini et al. [7, 15] gave a thermodynamical, model-independent argument showing that if T/-1 = X/is the inverse relaxation time relative to the ]th component of *Research supported in part by NSF grant MCS 76 07286.