Physics - Molecular Theory of the Living Cell - page 11

Boltzmann (1844-1906), Eq. 2.11 , after replacing W with D, where W is the
number of possible microstates compatible with the macrostate of a system
( http://en.wikipedia.org/wiki/Boltzmann's_entropy_formula ):

¼

S k logW (2.11)

Schr ˝ dinger then concluded that negative entropy is a measure of order :
Hence the awkward expression “negative entropy” can be replaced by a better one: entropy,
taken with the negative sign, is itself a measure of order. (2.12)
Evidently Eq. 2.10 defining the concept of “negative entropy” violates the Third
Law according to which entropy cannot be negative; see Eq. 2.8 . In other words, the
concept of “negative entropy” violates the Third Law of thermodynamics. Since
Eq. 2.10 violates the Third Law, its consequent, Statement 2.12, must also violate
the same law, leading to the following conclusion:
Negative entropy cannot be a measure of order. (2.13)
So we are now confronted with the following question: How can the same
equation, Eq. 2.10 , give rise to two opposing conclusions, Statements 2.12 and
2.13? One possible answer to this puzzle may be provided if it can be assumed (1)
that there are two aspects to Eq. 2.10 - the formal (or syntactic) and physical (or
semantic) aspects - and (2) that Eq. 2.10 is true formally (since it can be derived
from Eq. 2.11 logically, by multiplying both sides of the equation with 1 and

equating W with D) but untrue physically (since it violates the Third Law of
thermodynamics). In other words, Eq. 2.10 that Schr˝dinger made famous among
the generations of physicists, chemists, and biologists since 1944 is paradoxical .
For this reason, we may refer to Eq. 2.10 and its equivalents as Schr ˝ dinger's
paradox which can be defined more broadly as follows:
Schr˝dinger's paradox refers to the mathematical equations, concepts, or general
statements that are formally true but physically meaningless. (2.14)
Although “negative entropy (NE)” (an absolute value) cannot exist, due to the
Third Law, the concept of “negative entropy changes (NEC)” (a differential value)
is both formally sound and physically true and meaningful since NEC can be
equated with order or organization without violating any laws of physics.
In other words, although entropy cannot be negative, entropy change can be
negative, zero, or positive:

D S ¼ S A S B > 0 (2.15)

D S ¼ S A S B ¼ 0 (2.16)

D S ¼ S A S B < 0 (2.17)