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

3.4 Semigroups and the Emergence of Order 77
3.4 Semigroups and the Emergence of Order
In physical systems, ‘ordered’ states are characterized by low entropy. Order
may appear in the form of simple structures (such as regular lattices)
or complex ones (organisms). For example, the rectifier discussed in the previous
section as replacing Maxwell’s demon must display ordered dynamical
behavior. The emergence of order from disorder in Nature, also called selforganization
of matter, may appear to contradict the Second Law with its
general trend towards disorder and chaos. This has often been misunderstood
as a ‘discrepancy between Clausius and Darwin’. However, the fundamental
phenomenological equation (3.1) allows entropy to decrease locally. A negative
first term would allow physical entropy to flow into the environment. If
this environment is not in complete thermal equilibrium, and characterized
by at least two different temperatures, T 1 and T 2 ,alocallossofentropy,
dS ext =dQ 1 /T 1 +dQ 2 /T 2 < 0, would not even require any net loss of heat,
dQ 1 +dQ 2 < 0. (Here differentials are always meant to refer to positive time
increments dt.) This local decrease of entropy is thus not in conflict with its
global increase according to the Second Law – see also Sect. 5.3.
In statistical terms, the number of states in a dynamically representative
ensemble (see Sect. 3.1.2) may decrease locally in accordance with determinism
and intuitive causality, provided the ensemble characterizing the state of the
environment increases accordingly – precisely as during the ‘reset’ of a memory
device, indicated in Fig. 3.5. In this Laplacean description, the outcome of
evolution would be determined by the microscopic initial state of the whole
Universe.
An important special case is a steady state of non-equilibrium, characterized
by dS =dS int +dS ext = 0 in spite of non-vanishing entropy production,
dS int > 0 (Bertalanffi 1953). It may support ordered states as dissipative
structures. The standard example, known as Bénard’s instability, describes
convective heat transfer through a thin horizontal layer of a liquid in the
form of spatially ordered convection cells, which optimize the process of thermal
equilibration between two reservoirs at different temperatures. In a finite
universe, this stationary situation can only represent a transient local
phenomenon. The emergence of structure is often connected with symmetry
breaking (in particular of translational symmetry), related to a phase transition.
In a deterministic description, an initial microscopic fluctuation would
thereby become unstable and be amplified to a macroscopic scale. In quantum
theory, it may also require an indeterministic collapse of the wave function
(see Sect. 4.1.2).
For similar reasons, Boltzmann suggested that biological processes here
on earth are facilitated by the temperature difference between the sun (with
its 6000 K surface temperature) and the dark Universe (at 2.7 K, as we know
today). At the distance of the earth, the solar radiation has an energy density
much lower than that of a black body with the same spectrum (temperature).
Since photon number is not conserved (in general not even a robust