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

3 The Thermodynamical Arrow of Time 41
such an assumption would simply be taken for granted: a consequence of
operations to be performed in time. In a cosmological context it requires a
cosmic initial condition. This has occasionally been called the Kaltgeburt (cold
birth) of the Universe, although a low temperature (kT much smaller than
energies of mechanical degrees of freedom) need not be its essential aspect
– see Sect. 5.3. However, this initial assumption appears quite unreasonable
precisely for statistical reasons, since (1) there are just as many processes of
the type
and (2) far more of the kind
probable state −→
t
improbable state ,
probable state −→
t
probable state .
The latter describe equilibrium. Hence, for statistical reasons we should expect
the world to be in the situation of a heat death, while the required improbable
initial condition needs an explanation that does not presume causality.
The first of these two arguments is the ‘reversibility objection’ (Umkehreinwand),
formulated by Boltzmann’s friend and teacher Johann Joseph
Loschmidt. It is based on the fact that each trajectory has precisely one
time-reversed counterpart. 1 If, for example, z(t) ≡{q i (t),p i (t)} i=1,...,3N describes
a trajectory in 6N-dimensional phase space (Γ -space) according to
the Hamiltonian equations, then the time-reversed trajectory, z T (−t) ≡
{q i (−t), −p i (−t)}, is also a solution of the equations of motion. If the entropy
S of a state z can be defined as a function of this state, S = F (z), with
F (z) =F (z T ), then Loschmidt’s objection means that for every solution with
dS/dt >0 there is precisely one corresponding solution with dS/dt