The Physical Basis of The Direction of Time (The Frontiers ...

78 3 The Thermodynamical Arrow of Time
quantity), a canonical distribution exp(−H/kT) in the occupation number
representation determines not only the spectral distribution as a function of
temperature, but also the intensity (photon density). A gas with conserved
particle number would instead allow one independently to choose the mean
density – either by fixing the particle number by closing the vessel, or by
fixing the chemical potential (in a grand canonical ensemble) by connecting
the vessel to a particle reservoir. In contrast, a photon from the sun can be
transformed very efficiently into many soft photons, which together possess
much higher physical entropy.
Although order appears to be an objective property, an absolute concept
of order that is not simply defined by means of phenomenological entropy is
as elusive as an objective concept of information or relevance (see Denbigh
1981, p. 147, or Ford 1989). For reasons already mentioned in Sect. 3.3.1, the
definition of order in terms of computability would depend on the choice of
‘relevant coordinates’. For example, the obvious order observed in a crystal
lattice is not invariant under general canonical transformations. How, then,
may the order of an organism be conceptually distinguished from the ‘chaotic’
correlations arising from molecular collisions in a gas?
Many self-organizing systems include chemical reactions. They are phenomenologically
described by irreversible rate equations, which define the
dynamics of concentrations X,Y,... These concentrations are ‘macroscopic’
variables, called α in Sects. 3.2 and 3.3.1. In statistical terms, rate equations
can be derived from a generalized Stoßzahlansatz that includes rearrangement
collisions between different kinds of molecules, which are usually assumed to
be already in thermal equilibrium with one another. These rate equations are
therefore special master equations (as derived in Sect. 3.2) for these ‘relevant’
degrees of freedom X,Y,...
Rate equations determine trajectories in the configuration space of concentrations.
11 For closed systems, these trajectories may eventually approach that
point in their configuration space which describes equilibrium. Reversible determinism
must come to an end at such attractors (see Fig. 3.7a), although this
may require infinite time. A mechanical example of an attractor in the presence
of friction is the phase space point characterized by v := dx/dt = 0 and
V (x) =V min . The corresponding equation of motion, mdv/dt = −av −∇V ,
neglects any stochastic response from the energy-absorbing microscopic degrees
of freedom, which is in principle required by the fluctuation–dissipation
theorem. Similar to the LAD equation of Sect. 2.3, this equation is, therefore,
deterministic, even though it is asymmetric under time reversal.
Points in the space of macroscopic variables X, Y or x, v (‘macroscopic
states’ α, in general) describe the physical states incompletely. They represent
large subspaces of the complete Γ -space (that may realistically even have
to include the environment). Volume elements of the same size in macroscopic
11 As the rate equations are of first order in time, this macroscopic configuration
space is often called a phase space.