In statistical physics, we have a system interacting with a (random) medium. This is what shifts and broadens the system states and this is well understood. Clearly the random interaction is responsible for the shift and broadening(because if it were nonrandom, it would simply cause a renormalization of the system with infinitely sharp states)Randomness is interpreted as follows: I have a large number of systems(such as an impurity) and each one sees a DIFFERENT local field. The KEY for "memory loss" or broadening is the RANDOM interaction with the medium. Now I am trying to draw an analogy with decoherence where we have a system interacting with a medium, in this case the "environment". To get decoherence in statistical physics we still need the system-environment interaction to be random. I interpret this to mean that the environment consists of a large number of microscopic entities interacting with the system via microscopic forces. In contrast to statistical physics though we no longer have an ensemble of systems(e.g. Schroedinger cats), but only one. However, again the interaction with the environment is crucial-switch it off and one has no decoherence- So what I am trying to understand is in what way can we understand the system -environment interaction in decoherence to be random. I.e. if we look at Schroedinger's cat case, where we do NOT have a large number of cats, each one seing a different local environment(e.g. a broken or unbroken poison bottle), in what sense is that cat-environment interaction "random"