I could not find the Boltzmann Gibbs equation for absolute entropy applied to DNA like this anywhere. It challenges the naturalistic assumption that semantic information can evolve from a chance combination of mass and energy.. by definition DNA is a low entropy state.. which must be paid for..
The Second Law condition for the random assembly of a string of semantic information of length p from an alphabet of m possible codes is..
n = m^p random trials.
(eg to throw a double [6] with 2 dice n = 6^2 = 36 throws or
for a string of 10 bases of DNA n = 4^10 = 1048576 random mutations)
It is the average occurrence of a specific sequence in an infinite number of random trials that determines the minimum number of trials (entropy cost) to meet the second law requirement, entropy must increase.
The probability of getting at least one occurrence of a specific string of length p codes from an alphabet of m possibilities in n = m^p random trials is..
Pr(at least one) = 1 - Pr(not getting any) = 1 - [(n-1) /n]^n
So for at least one (two heads) from n = 4 throws of a coin..
Pr(at least one 2xhead) = 1 - [(4 - 1) / 4]^4 = 0.6836
For at least one [double 6] from 2 dice in n = 6^6 = 36 throws..
Pr(at least one) = 1 - (35/36)^36 = 0.6372
For at least one 10 base DNA string from 1048576 random mutations..
Pr(at least one) = 1 - (1048575)/1048576]^1048576 = 0.6321
Note the probability of at least one occurrence as n gets large asymptotes toward a certain LIMITING value.. So what is it?
the limit of [1 - [ (n-1) / n ] ^ n] = 1 - 1/e approx = 0.6321
for n -> infinity
Its my number so.. The 'Bellamy limit' = 1 - 1/e is the lowest probability demanded by the Second Law for a randomly assembled string of semantic information length p from m codes in n = m^p tries to make it PROBABLE ENOUGH NOT to violate that law (for large n say > 50).
(Jan 2015: I now believe it applies to all logical states ie.. microstates)
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