Re: Is evolution an example of decreasing entropy?


J. J. Lodder wrote:
> Robert <rparson00@xxxxxxxxx> wrote:


> > The Entropy of Thermodynamics is a physical concept, which emerges from
> > the entropy of statistical mechanics when the probability distribution
> > collapses to a peak in accordance with the law of large numbers.
>
> That is a mistake, as it stands.
> When the probability distribution collapses to a narrow peak
> the system approaches a pure state,
> and the entropy goes to zero.
> Near equilibrium the system tends to a thermal state.

I concede that my language was imprecise. The distribution that
"collapses to a peak" is not the fine-grained distribution over
individual quantum states, but the coarse-grained distribution of
macroscopically measurable quantities, such as energy and pressure. The
probability that a system has energy E and pressure p collapses to its
peak value in the thermodynamic limit. Mathematically, this is a very
subtle problem, but in practice it works out like this: the density of
states of a large system depends *exponentially* upon the number of
degrees of freedom, whereas fluctuations in the macroscopically
measurable extensive variables are proportional to the *square root* of
this number, and the magnitude of the fluctuations relative to the mean
go as one over the square root. So the individual quantum states
"quickly" merge into an effective continuum as the system size
increases, whereas fluctuations persist somewhat "longer".

> > The Entropy of Statistical Mechanics obeys the Second Law
> > _approximately_ (it is subject to fluctuations, whose magnitude is
> > proportional to the square root of the number of particles in the
> > system.)
>
> Not approximately, statistically.
> The statistical equivalent of the second law is that motion of the
> system in the direction of increasing entropy
> is more probably that motion in the opposite direction.

Yes, that is more accurate way to describe it. Any individual system
may deviate from the average trend by an arbitrarily large amount, but
as the system size increases, the fraction that deviate tends to zero.

------
Robert

.

Relevant Pages

  • Re: behavior as mapping
    ... estimating a probability distribution, the distribution ... sequence with equal probability - since you have microsecond temporal ... reduction of the entropy Pto the entropy P ... If there were 4 genes we would need 2 bits of binding site info. ...
    (comp.ai.philosophy)
  • Computational secure entropy extraction
    ... distilling entropy from an unknown distribution. ... there existed some universal entropy distiller that could be used on all input ... D is -secure if given that k is drawn from any distribution ... Let's define that a "hit" is the case ...
    (sci.crypt)
  • Re: behavior as mapping
    ... estimating a probability distribution, the distribution ... sequence with equal probability - since you have microsecond temporal ... reduction of the entropy Pto the entropy P ... If there were 4 genes we would need 2 bits of binding site info. ...
    (comp.ai.philosophy)
  • Re: new /dev/random
    ... >A c1,c2 entropy generator takes any input k and produces a string of ... >fixing any probability distribution it likes over those strings, ... >long as the entropy exceeds k. ... our mixer to do a good job with, we can then ask whether applying SHA1 ...
    (sci.crypt)
  • Re: new /dev/random
    ... > optimistic dreams for what one would like from an entropy distiller. ... SHA1 is a secure hash, though, which would also be very exciting. ... > to any such distribution yields pseudorandom outputs. ...
    (sci.crypt)
Loading