Re: Is evolution an example of decreasing entropy?


Robert thus chastized:

Thanks you,
that is all.

> wade wrote:
> > Robert J. Kolker wrote:
> > > wade wrote:
> > > >
> > > > Informatic entropy is seriously misnamed and not associated with
> > > > the 2nd law of thermodynamics.
> > > >
> > >
> > > Statistical entropy and thermodynamic entropy are one and the same.
> > >
> > > See:
> > >
> > > http://www.nyu.edu/classes/tuckerman/stat.mech/lectures/lecture_6/node3.html
> >
> > They are decidedly not one in the same.
>
> You are both wrong. Thermodynamic and statistical entropy are not "one
> and the same", but they are deeply and intimately related.
>
> Statistical entropy is a mathematical concept - to any probability
> distribution, one may assign an "entropy", given by the Gibbs-Shannon
> formula. When the probability distribution under consideration is the
> distribution of a physical system over its possible microstates, the
> corresponding statistical entropy is precisely the entropy as given by
> statistical _mechanics_ - which in the limit of a large system becomes
> equal to the thermodynamic entropy, as shown in the Tuckerman lecture
> that Kolker cites.
>
> To repeat:
>
> The Entropy of Statistics is a mathematical concept.
>
> The Entropy of Statistical Mechanics is a physical concept - an
> application of statistical
> entropy to a certain class of physical systems that are described in
> statistical language.
>
> 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.
>
> The Entropy of Statistics does not obey the Second Law of
> Thermodynamics or any other physical law, since it is not a physical
> concept.
>
> 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.)
>
> The Entropy of Thermodynamics obeys the Second Law exactly, since the
> fluctuations vanish in the limit of a macroscopic system.
>
> In another post in this thread, Wade said:
>
> "Informatic entropy is seriously misnamed and not associated with
> the 2nd law of thermodynamics."
>
> While the entropy as used in information theory - another application
> of statistical entropy need not obey the 2nd Law - and in the class of
> problems studied by Shannon, it does not - it does NOT follow that it
> was "misnamed." Shannon and Von Neumann knew exactly what they were
> doing when they chose their nomenclature. The connection between
> thermodynamics and information theory goes back to Szilard in the
> 1920's. It was most extensively explored by E. T. Jaynes and his
> successors, who reformulated the whole of statistical mechanics using
> the language of information theory:
>
> "The essential point... is that we accept the von Neumann-Shannon
> expression for entropy, very literally, as a measure of the amount
> of uncertainty represented by a probability distribution; thus entropy
> becomes the primitive concept with which we work, more fundamental
> even than energy. If in addition we reinterpret the prediction problem
> of statistical mechanics in the subjective sense, we can derive the
> usual relations in a very elementary way without any consideration of
> ensembles or appeal to the usual arguments concerning ergodicity or
> equal a priori probabilities."
> [E. T. Jaynes, Information Theory and Statistical Mechanics, _Phys.
> Rev._ _106_, 620 (1957),]
>
> Wade again:
> > Thermodynamic entropy is a real thing that has real units associated with it
> > and these units include a temperature dependence as was mentioned in the part of my
> > post you deceiptfully deleted.
>
> Baloney. The fact that thermodynamic entropy, as conventionally
> defined, has units is nothing more than a historical artifact, arising
> from the arbitrariness of our temperature units. If we choose to
> measure temperature in energy units (as is done in many textbooks on
> Statistical Mechanics written by physicists, most notably the
> celebrated treatise by Landau and Lifshitz), entropy becomes a
> dimensionless quantity. In fact, the most rigorous treatments of
> thermodynamics (such as presented in the books by Callen and by
> Landsberg) regard entropy as a fundamental quantity, and *define*
> temperature in terms of it.
>
> Wade is right to draw a distinction between thermodynamic entropy (a
> physical concept) and statistical entropy (a mathematical concept,
> which becomes equal to the former when the mathematics is applied to a
> particular class of physical systems) but he is wrong in attributing
> any sort of significance to this business of units. It's a red herring.
> Kolker is right to insist upon the equivlance (for macroscopic systems)
> between the entropy of statistical mechanics and the entropy of
> thermodynamics, but incorrectly conflates the entropy of statistical
> mechanics with the general, mathematical concept of statistical
> entropy.
>
> --------
> Robert

.

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