9 . Irreversibility in stochastic dynamics ∗

Over recent decades, some approaches to non-equilibrium statistical mechanics, that differ decidedly in their foundational and philosophical outlook, have nevertheless converged in developing a common unified mathematical framework. I will call this framework ‘stochastic dynamics’, since the main characteristic feature of the approach is that it characterizes the evolution of the state of a mechanical system as evolving under stochastic maps, rather than under a deterministic and time-reversal invariant Hamiltonian dynamics.1 The motivations for adopting this stochastic type of dynamics come from at least three different viewpoints. 1. “Coarse graining”(cf. van Kampen 1962, Penrose 1970): In this view one assumes that on the microscopic level the system can be characterized as a (Hamiltonian) dynamical system with deterministic time-reversal invariant dynamics. However, on the macroscopic level, one is only interested in the evolution of macroscopic states, i.e. in a partition (or coarse graining) of the microscopic phase space into discrete cells. The usual idea is that the form and size of these cells are chosen in accordance with the limits of our observational capabilities. On the macroscopic level, the evolution now need no longer be portrayed as deterministic. When only the macrostate of a system at an instant is given, it is in general not fixed what its later macrostate will be, even if the underlying microscopic evolution is deterministic. Instead, one can provide transition probabilities, that specify how probable the transition from any given initial macrostate to later macrostates is. Although it is impossible, without further assumptions, to say anything general about the evolution of the macroscopically characterized states, it is possible to describe the evolution of an ensemble or a probability distribution over these states, in terms of a stochastic process. 2. “Interventionism”, or “open systems” (cf. Blatt 1959, Davies 1976, Lindblad 1976, Lindblad 1983, Ridderbos 2002): On this view, one assumes that the system to be described is not isolated but in (weak) interaction with the environment. It is assumed that the total system, consisting of the system of interest and the environment can be described as a (Hamiltonian) dynamical system with a time-reversal invariant and deterministic dynamics. If we represent the state of the system by x ∈ Γ and that of the environment by y ∈ Γ, their joint evolution is