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Author
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Topic: A 4th Law of Thermodynamics
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Rex Kerr
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Member # 632
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posted 27. October 2003 16:06
I don't think I'm really qualified to (and don't have the free time to become qualified). Was there something specific you had in mind?
Here's another interesting (to me) thought. In an expanding universe with locally attractive forces, the number of conceptually possible states increases faster than the number of physically attainable states. This may give the appearance of decreasing entropy relative to random-energy-soup. I'm sure physicists have worked through the implications of this before. Does anyone know what they concluded?
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kyle7
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posted 28. October 2003 06:51
Pim said originally,
quote:
I would say that the combination of dissipative systems, open systems and far from equilibrium systems offer many ways of information/entropy in a system to increase/decrease.
I responded by saying,
quote:
This may be true theoretically (for the systems act as simple auxiliary devices), but the probability is small that significant information is generated, just like the sand illustration above.
Pim responded by saying,
quote:
This seems to be begging the question. In fact the strawman of the need of an auxiliary device clearly has to be rejected given Kyle's own comments here (theoretically true) and by virtue of the fact that nothing in the SLOT makes such a requirement.
When I said that this is "theoretically true" I meant that nature does provide very crude auxiliary devices such as dissipative systems. The output of complexity of these systems is limited based on the complexity of the boundary conditions of these systems and the physics that constrains the processes. If I found a large mosaic in the desert showing a detailed picture, one could argue that nature has the potential to create such artifacts. But, most intelligent people would agree that natural explanations are ridiculous. Work has to be applied in specific ways to enable such artifacts. The auxiliary device enables the constraining of work in a specified fashion. Energy is constrained and controlled enabling the mosaic to be crafted. Dissipative systems cannot explain a detailed mosaic nor biological systems. Applicable to this discussion is the displacement problem of dissipative systems. Although dissipative systems could possibly explain an artifact, the probability is small and the problem is displaced to the boundary conditions. The boundary conditions need to be constrained in very specified ways.
Pim also says that there is nothing in the SLOT that makes such a requirement of an auxiliary device. The Clausius statement of the Second Law of Thermodynamics is the following:
quote:
It is impossible for any system to operate in such a way that the sole result would be an energy transfer by heat from a cooler to a hotter body.
This statement of the second law applies to open systems since energy is transferred across the boundary conditions. The notion of the auxiliary device is found in the words, "sole result". Although it is possible to transfer heat from the cold side to the hot side, this will not happen without an auxiliary device enabling some other effect within the system allowing the transfer of heat. For example, you could have a refrigerator that allows the transfer of heat. Energy is required along with the auxiliary device.
Gedanken,
Here is a thought game for you. Put building materials for a house in a pile. Put enough gasoline on the pile that contains enough energy to possibly build a house. Next, burn the fuel and see if the house will be built. Certainly there is an additional requirement beyond just energy to enable the house to be constructed. Energy has to be constrained in precise ways by an auxiliary device to enable the house to be built. Typically, humans act as the auxiliary device in construction as well as all the tools and equipment required in the building process. Engineers build and model a multitude of devices. Have you ever heard of bioengineering?
Erik says,
quote:
You did not clarify your claim, but if it is correctly construed in my previous question and if by "equilibrium" you mean "thermal equilibrium", then you are simply wrong. Moving away from thermal equilibrium is generally enough to decrease the entropy.
You are correct, but you need an auxiliary device (along with energy) to allow the movement away from equilibrium. Without the auxiliary device, you will not move away from equilibrium.
Again, spontaneous processes are not of interest. A rock rolling down a hill is expected. The origin of life and the development of life are not spontaneous. These processes are like the rock rolling up the hill. The spontaneous formation of crystals is expected and does not help explain the nonspontaneous formation of life.
Yes, you have a point that I need to evaluate my use of the term specification, but this is no major problem. I am sure the term can be used as Dembski defined it. I need to develop this concept with more clarity.
RBH claims that I am jumping back and forth between "Dembski-land" and thermodynamics. No I am not. Really the concepts are quite simple if you think about it. Would you expect a telephone to dance around the room and yodel? Why not? It is limited by the functionality of the telephone and lacks the mechanisms such as legs that would allow this. The simple mechanisms of nature are also extremely limited and don't have significant complexity to allow complex processes to occur. The complexity of the thermodynamic processes is a function of the complexity of the auxiliary device.
I said,
quote:
If life was programmed into nature, we would expect to see a plethora of different life-forms originating throughout time.
We don't observe life originating from inanimate matter over and over again as would be expected if life was programmed into nature. This is the point I was trying to make.
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gedanken
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posted 28. October 2003 12:35
Kyle7, so much seems mixed up, I recommend the following:
Entropy and the Second Law of Thermodynamics
And as to:
quote:
These processes are like the rock rolling up the hill.
Ever heard of volcanos? Plate tectonics?. (All about rocks and hills forming themselves.)
Or did you think that all the rocks started out at the tops of hills, and only rolled down?
[ 28. October 2003, 12:43: Message edited by: gedanken ]
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RBH
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posted 28. October 2003 13:47
Referring to my interest in his working out the implications of his remark that "Thermodynamic entropy is the limiting case as information representation becomes maximally efficient," Rex wrote quote:
I don't think I'm really qualified to (and don't have the free time to become qualified). Was there something specific you had in mind?
Here's another interesting (to me) thought. In an expanding universe with locally attractive forces, the number of conceptually possible states increases faster than the number of physically attainable states. This may give the appearance of decreasing entropy relative to random-energy-soup. I'm sure physicists have worked through the implications of this before. Does anyone know what they concluded?
I'm not wholly sure what I have in mind - it's an unverbalizable intuition at the moment. I'm now trying to interest a quantum information theorist in the physics department at the college in this set of questions. I'll keep you posted as and if I'm successful.
RBH
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Rex Kerr
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posted 28. October 2003 18:39
Kyle is apparently arguing for vitalism, since if we start out with one bacterium and end up with two, all with only the "crude auxiliary devices" present as natural mechanisms, there couldn't be any appropriate auxilliary devices to reduce the entropy and yet we have an apparent local decrease in entropy.
Thus, there must be some non-natural, or supernatural device necessary for reproduction and growth. Hence, vitalism.
If Kyle disagrees, perhaps he can answer the following question: what are the natural auxilliary devices that allow living cells to divide/reproduce?
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Salvador T. Cordova
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posted 29. October 2003 21:19
In regards to Maxwell's Demon and Adami's work, there are some considerations.
First of all, I commend Adami for his hard work for the National Academy of Sciences. I esteem the computer simulation he created in an attemt to create an environment and organism where Maxwell's demon dwells. I do not discount Adami's work immediately on the grounds it was a computer simulation, not a physical experiment.
If Adami's work is true, the increase in information occurring in a biological system through natural selection does not rule out
the possibility of intelligent design of the organism. In fact, it can be argued that this is an indication of intelligent design of the organism.
In the vein of Adami's thought experiment, imagine an intelligently designed self-replicating von-Neumann machine with say a billion free bytes of information storage. It makes data measurements on the environment and hands it down to it's descendants (such is implied as a possibility by our esteemed ISICD fellow Frank Tipler).
That is approximately how Adami modeled this "make believe" digital life form with a finite pre-defined storage space. In fact, what impressed me was that this "digital creature" had to be specified and complex to begin with to do it's wonderful task of increasing information in it's natrual environment.
The capacity for information acquisition and increase was designed into the system from the beginning.
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Erik
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posted 30. October 2003 16:28
quote:
Pim van Meurs: You are now confusing to separate concepts, the Shannon entropy of the message versus the termodynamical entropy of the tombstone.
you state that the general relationship between Shannon entropy and thermodynamical entropy is indefensible and yet the derivation of said relationship is straightforward WHEN applied to the same entities namely the entropy of the information/complexity in question.
Confusing separate concepts is precisely what I'm not doing. I pointed out from the start that the relation between Shannon entropy and thermodynamical entropy is only valid for one particular stochastic variable (namely the QM energy eigenstate of the system). The alleged relation is indefensible when the stochastic variable we study is not the exact QM energy eigenstate. The article by Christoph Adami on evolution in Avida is about the Shannon entropy of a stochastic variable having nothing to do with any QM energy eigenstate and therefore there is no link between that entropy and thermodynamical entropy.
BTW, what do you mean by "the entropy of the information/complexity"? Shannon entropy is defined for stochastic variables, nothing else. And what does the term "information" refer to here? It seems obvious that you do not mean a decrease in Shannon entropy in this case (lest the statement could be expanded to "the Shannon entropy of the decrease in Shannon entropy").
quote:
Pim van Meurs: Erik suggests there is a relationship between H(N), the shannon entropy of the text versus the thermodynamical entropy of the book.
That's not why I suggest.
quote:
Pim van Meurs: "The Second Law of Thermodynamics may be rephrased to state that correlations are highly unlikely to arise spontaneously, and that the natural course of evolution of a system is one in which correlations diminish."
Thus processes which increase correlations such as selection seem to be sufficient to explain the decrease in entropy in a system.
1. The second law of thermodynamics cannot generally be rephrased like that.
2. Natural selection is not sufficient to explain the decrease in thermodynamical entropy of a system (which is fine because there's really no entropy decrease that is in need of such an explanation, anyway). However, it may well be sufficient to explain that some non-QM stochastic variables become correlated. I once again remind you of the importance of remembering which stochastic variables we calculate the Shannon entropy for. The Shannon entropy of stochastic variables defined in terms of, say, DNA sequences have no obvious relation with the thermodynamical entropy, and 2LOT is only about the latter entropy.
quote:
Rex Kerr: 1. Something that cannot vary has no Shannon entropy. So yes, H(X) is and should be zero in this case. Likewise, there is one microstate, and thermodynamic entropy is zero.
2. Assuming equal energies for both spin states, S = k_B * sigma = k_B log g(N,U) = k_B log 2 and H(X) = - 0.5 log 0.5 + 0.5 log 0.5 = log 2
3. Same thing.
4. This is exactly covered by Schneider's derivation that was linked to above, unless I misunderstand what you're asking for.
5. Okay, this breaks down. However, H(N) isn't really the information present. In fact, there is much more inherent in the physical system, but we don't care about it.
6. I'm not sure what to do with H(S), aside from noting that there is much more information in a system, in general, than a scalar representing its entropy. That's interesting, actually. If I get time, I'll think about it more.
1. The last sentence is incorrect. There's not necessarily just one microstate. The thermodynamical entropy of an isolated hydrogen atom known to be in the ground state is not zero (there is still two possible spin states), yet the Shannon entropy of the energy is zero.
2. I didn't ask for the values. I asked for the relation between the entropies.
3. Yes. Now consider this. What is the thermodynamical entropy of an isolated hydrogen atom that is known to be in its ground state and is known to have spin up? What is the Shannon entropy of the outcome of spin measurement in the left/right direction?
4. You probably misunderstood, because it is not covered by Schneider's "derivation". The parity of a wave function Y(r) is even if Y(r) = +Y(-r) and odd if Y(r) = -Y(-r).
5. No, H(N) is the entropy of the stochastic variable N. It is unclear to me both what you mean by "information" and why find it relevant to bring it up. Regardless of what's that about, your use of the word "information" indicates that you have several concepts of "information" that you equivocate between (just as in your arguments in your first two posts).
6. -
quote:
[b]Rex Kerr[b]: Case (5) is the interesting one. Here we have an encoding that, as thermodynamic entropy increases, could apparently decrease Shannon entropy. For example, as "e" starts to decay, it might start to look like "c" instead, a lower-probability and therefore higher-information character. But all we've really learned is that we're not encoding information efficiently; we use quintillions of ink molecules to represent "e" instead of a few spin states or position of a molecule.
Thermodynamic entropy is the limiting case as information representation becomes maximally efficient.
We've also learned a much more important lesson, namely that the Shannon entropy of a stochastic variable associated with a system is not always proportional to the thermodynamical entropy of the system.
I think I understand what you mean by that last comment (namely that using all of a system's microstates to represent different messages/data is a limiting case in data storage), but I'm not sure I'd express it like that.
Finally, RBH may be interested in the article The Physical Limits of Communication.
Erik
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Rex Kerr
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posted 30. October 2003 21:13
Whoops. When Erik said "parity", I thought "spin". So that isn't exactly the Schneider case. And in at least one spot I said "information" when I meant "entropy".
That said, I think I now see the fundamental issue that Erik was getting at.
If K is the set of possible microstates of a system all with equal probability, and there are N microstates total, then the thermodynamic entropy is given by
S = - k_b sum [1..N] (1/N * log 1/N) = k_b log N
And if X is a stochastic variable that specifies the state, then the Shannon entropy of that variable is
H(X) = - sum [1..N] ( 1/N * log 1/N / log 2 ) = log N / log 2
so that
S = k_b * log 2 * H(X)
However, with Shannon entropy, we often use some other variable that groups many states together (for instance, we group together particles of spin up and spin down into one cateogry, or we ignore angular momentum, or whatever). Then we have a stochastic variable Y where Y is a subset of K, and {Y_i} forms a partition of K (that is, for each state k in K, there exists a unique Y_i such that k is in Y_i). Let's say we've grouped n_i states together and labeled it with one variable value Y_i. Then
H(Y) = - sum [1..M] p(Y_i) log p(Y_i) = - sum [1..M] n_i/N log(n_i/N)
which in general is smaller than H(X) since
-(n_i / N) * log ( n_i / N ) = -(n_i/N)*log(n_i) +(n_i/N)log(N)
The (n_i/N)log(N) term is what we would have gotten had we not grouped variables, but we have a correction factor of -(n_i/N)*log(n_i) for grouping.
So the result is basically that if your stochastic variable Y doesn't actually specify the full state of the system, but leaves M possible states even given the value of that stochastic variable, then
S ~= k_b * log(2) * (H(Y) + log M)
(Of course, normally you don't get exactly M unspecified states for each value of the s.v., so the equation becomes a much more complex sum.)
Unfortunately, I don't have time right now to explore the consequences of the second law on systems where microstates are combined. Someone must have done this already, though I can't find it. Does anyone know?
[ 30. October 2003, 21:15: Message edited by: Rex Kerr ]
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RBH
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posted 31. October 2003 00:12
Erik wrote quote:
Finally, RBH may be interested in the article The Physical Limits of Communication.
Yup. Thanks! Rex will be interested in it too, given his remark about thermodynamic entropy as a limiting case when information representation is maximally efficient.
RBH
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kyle7
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posted 02. November 2003 03:10
Gedanken,
I am familiar with Prof. Lambert's web site on thermodynamics and take issue with him on a number of points. In his discussion about evolution he fails to mention the notion of equilibrium, which has implications on the supposed natural origin of life. Perhaps you could specify a point that you think disproves my posts. I would be glad to respond.
The earth at times acts as a auxiliary device by constraining energy and producing mountains either by plate tectonics (where plates colide)or by volcanoes (where pressure within the earth forces rock and lava out forming mountains). The thermodynamic expected direction is for rocks to roll down hill. Another way to say this is the statement, "you can't push water uphill". Those who work with thermodynamics are familiar with these statements.
Rex,
The cell acts as the auxiliary device, but it is limited by the functionality built into the cell. The SLoT prevents the formation of the first cells and cuts off the possiblity of the initial conditions containing the specification of life, without the intervention of an intelligent being.
The Second Law of Thermodynamics also prevents the development of life. The same probability problem that is at the heart of the Second Law of Thermodynamics, is found in Darwin's Demon. The number of possible mutations is large compared to the required specified mutations. The time required for the evolution of earthworms and humans is greater than the available time. Life-forms can evolve in a limited sense, but only to what is programmed into the DNA of the lifeform.
[ 02. November 2003, 03:15: Message edited by: kyle7 ]
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Rex Kerr
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posted 02. November 2003 05:16
Let's suppose that the first cell had to be designed. Why can't it have been designed in order to incorporate information in the environment such that all forms today are a result of evolution from that primative cell that contains an appropriate auxilliary device?
(The argument against certainly won't have anything to do with thermodynamics, as 99.9999% of the thermodynamic work is done in replicating faithfully. The variation on top of that is insignificant.)
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gedanken
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posted 02. November 2003 10:10
Kyle7,
"The earth at times acts as a auxiliary device ..." But that is just the point, the Earth, the Sun, they act as the "auxiliary" device. More closely, they receive energy into local environments, and then the local environment contains one or many "auxiliary" devices (or their equivalent).
Since Lambert reaches a conclusion differing from yours, perhaps you would care to list a single step of reasoning in detail that you disagree with, or a single predicate of his. I do not see your point as well reasoned, (it seems to jump straignt from premise to the conclusion) so with the reasoning steps missing I can't point to a single "missing" or incorrect step. It's too much work for me to bother until you identify a specific step in Lambert's reasoning to which you disagree. Then we can speak more specifically.
As to equilibrium, you will have to bring up references. But be very careful--as the typical argument given against development of life with regard to thermodynamic "laws" are arguments that seem to show that life itself cannot exist. So show that what you understand does not prevent life itself, yet has the effects you propose.
And another point. You have now disagreed with Lambert, as opposed to invoking Dr. Dembski's "4th law". So you need to make up your mind if it is thermodynamics, or some other argument that you are using. Lambert does not consider a further limitation based on "4th law", or the like. So it is possible that Dr. Dembski might be right (though I firmly believe not). In that case there is nothing wrong with Lambert's reasoning. In other words there is nothing in thermodynamics that prevents development of life, only something in Dr. Dembski's "4th law". In that case, you must either admit that Lambert is correct in his reasaoning as far as it goes--that thermodynamics per se makes no contradiction--or show the problem in thermodynamics itself for which you disagree with Lambert. But if it is only Dr. Dembski's "4th law", then no argument of Lambert has been violated, and you don't in fact disagree with Lambert's reasoning (only claiming its incompleteness, and not any error). In that case, nothing in termodynamics prevents development of life, and a "4th law" like Dr. Dembski's is needed to show such a result--and you should admit that and move on to claiming that your point is based solely on Dr. Dembski's "4th law". (Or present even further evidence).
[ 02. November 2003, 10:19: Message edited by: gedanken ]
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Erik
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posted 08. November 2003 09:19
In statistical mechanics, we think of macrosystems as being "kicked around" between different microstates. Even when the macroscopic properties are stable, the system will randomly wander between different microstates. In statistical mechanics, the entropy is a measure of how many microstates a system randomly wanders between. After the arrival of quantum mechancs, we now take the microstates of a system to be the energy eigenstates of the system.
Shannon entropy of X
H(X) = -sum P(X=x) log(P(X=x))
The Shannon information about X associated with the event E
R(X;E) = H(X) - H(X | E),
where H(X | E) = -sum P(X=x | E) log(P(X = x | E))
Thermodynamical entropy of a system
We can imagine that we give each possible microstate an individual label (say a number) and we let MICROSTATE be a stochastic variable indicating the current microstate of a system. With a suitable choice of units, the thermodynamical entropy can be defined as
S(system) = H(MICROSTATE)
Maximum entropy distribution over MICROSTATES
For an isolated system, the maximum entropy distribution over MICROSTATES is a uniform probability distribution. For closed a system, the maximum entropy distribution over MICROSTATES is given by
p(MICROSTATE = s) = exp(-E(s) / kT) / Z,
where E(s) is the energy of microstate s, k is Boltzmann's constant, T is the temperature, and Z is just a normalization constant. For an open system capable of exchanging only one kind of particles, the maximum entropy distribution over MICROSTATES is given by
p(MICROSTATE = s) = exp((m N(s) - E(s)) / kT) / Z,
where N(s) is the number of particles in the system and m is the chemical potential.
Specified information of an event w.r.t. a hypothesis and specification
Given a hypothesis H and a specification K, the "specified information" of the event E (for which K is a valid specification) is given by
si(E;K,H) = -log(P(K | H)).
General conditions for the Shannon entropy of a stochastic variable X to be equal to the thermodynamical entropy of a system
The equality H(X) = S(system) holds if
* X = MICROSTATE.
General conditions for the "specified information" to equal the thermodynamical entropy
The equality si(E;K,H) = S(system) holds if all of the following conditions are satisfied
* The probability distribution over the microstates is uniform (in practice this will typically happen when the system is described by the maximum entropy distribution for an isolated system).
* The probability distribution conferred on MICROSTATE by H is uniform.
* The event E and the specification K are equal and correspond to a single microstate (this will never happen for the systems Dembski is interested in). That is, E = K = {MICROSTATE = s}, for some microstate s.
General conditions for the "specified information" to equal the Shannon entropy
The equality si(E;K,H) = H(X) holds if all of the below conditions are satisfied:
* The probability distribution over X used to calculate H(X) is uniform.
* The probability distribution conferred on X by the hypothesis H is uniform.
* The events E and K are equal and correspond to a single value of X. That is, E = K = {X = x} for some x.
General conditions for the "specified information" to equal the Shannon information in an event
The equality si(E;K,H) = R(X;E) holds if all of these conditions are satisfied:
* For the probability used to evaluate R(X;E) we must have that P(X=x) is uniform and the conditional probability P(X=x | E) is zero for all x inconsistent with E and uniform over those x that are consistent with E.
* The observed event E and its specification K are equal.
* The probability distribution conferred on X by the hypothesis H is equal the probability distribution used to evaluate R(X;E). In practice one will probably choose the latter probability distribution to be exactly the former, so this need not be a problem. (I note this condition despite its relatively unproblematic nature, because it is not satisfied in Schneider's analysis of the Ev program. Schneider evaluated R(X;E) using a probability distribution describing the relative frequency of DNA sequences in the population at a given time. This is perfectly fine considering his goals, but anyone who wanted to compare R(X;E) to "specified information" should use a probability distribution describing the origin of sequences, not their current distribution.)
--------------------------------------------------------------------
The above conditions are "if" conditions (as opposed to "if and only if" conditions). This means that there may be other conditions which also enable the equalities to hold. However, I strongly doubt that they will hold except under fortitious circumstances (e.g. for a particular event and a particular probability distribution). The conditions I have noted are likely the most general conditions.
People who insist on using some or all of the terms "thermodynamical entropy", "Shannon entropy", "Shannon information", and "specified information" interchangeably should be aware of the highly unrealistic conditions that must be satisfied for such identifications to be correct.
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Pim van Meurs
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posted 08. November 2003 13:32
I now understand Erik's confusion it all comes down to the obvious statement that
quote:
General conditions for the Shannon entropy of a stochastic variable X to be equal to the thermodynamical entropy of a system
The equality H(X) = S(system) holds if
* X = MICROSTATE.
Of course Shannon information of a system is NOT equivalent to its thermodynamic information but that was not the argument, the argument is that Shannon entropy can be expressed in thermodynamical entropy. Thus the thermodymical entropy of information. This of course does not mean that thermodynamical entropy and shannon entropy of a system are equivalent. The argument is that Shannon entropy and thermodynamical entropy are equivalent (multiplier) and thus any law for conservation of information can be translated in a similar law in thermodynmics, thus my statement that the LCI is nothing more than the SLOT of thermodynamics. Of course we have to compare apples with apples and as Erik has shown, shannon entropy is a subset of the thermodynamical entropy of a system.
Thus I am not saying that by measuring the thermodynamical changes one can predict the Boltzman entropy, that only applies when the only change is in the shannon entropy. But the SLOT applies equally to such sub systems.
[ 08. November 2003, 13:37: Message edited by: Pim van Meurs ]
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Erik
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posted 08. November 2003 15:59
Pim van Meurs, my confusion comes down to your very imprecise way of expressing yourself and the fact that you do not clarify yourself. Your latest reply quotes some of my comments on Shannon entropy and thermodynamical entropy. Then you proceed to comment on Shannon information and "thermodynamical information" (what is that?) as if that had something to do with the quoted statement.
Next, you claim that Shannon entropy can be expressed in thermodynamical entropy. However, you do not make clear if you make that claim only true for a particular stochastic variable or if you make that claim for all stochastic variables. (The claim is in fact only correct for the stochastic variable MICROSTATE and false for other stochastic variables.)
The second sentence is incomplete or nonsensical.
The third sentence is strange because Shannon entropy is defined for stochastic variables not systems.
The first part of the fourth sentence appears to contradict the third sentence, and you have yet again failed to make clear which stochastic variable the Shannon entropy concerns. The second part asserts that any law for the conservation of "information" can be translated into a similar law in thermodynamics. It is unclear what you mean by "information" here. The assertion is nonsensical if you mean Shannon information, because Shannon information is not a conserved quantity. You go on to write: "thus my statement that the LCI is nothing more than the SLOT of thermodynamics". I challenge you to show in detail how to translate Dembski's LCI into the second law of thermodynamics. Show this in your own words and show it
I...N......N.......DDD......EEEE...TTTTTT...A.......I...L
I...NN....N.......D.....D...E............T......A.A.....I...L
I...N...N.N.......D.....D...EEE.........T.....A...A....I...L
I...N...N.N.......D.....D...E............T....AAAAA...I...L
I...N......N.......DDD......EEEE.......T...A........A..I...LLLLL
Do not allow yourself to write (e.g.) "Shannon entropy" instead of "Shannon entropy of [insert stochastic variable here]". Such convenience is for situations where the coherence of your ideas is not in question. I question the coherence of your ideas, so any reply that takes such liberties with the terminology is worthless.
The last sentence in the paragraph claims that I have shown Shannon entropy to be a subset of thermodynamical entropy. I have not. If anything, I have shown that Shannon entropy is the more general concept of the two.
The second paragraph is incomprehensible (e.g. what are the are subsystems you refer to? what is a thermodynamical change? the Shannon entropy of which stochastic variable? etc., etc.).
In summary, your posts would become comprehensible if you
* replaced every occurance of the unqualified term "information" with the full term you had in mind (e.g. "Shannon information", "specified information", etc.)
* for every occurance of the term "Shannon entropy" made clear which stochastic variable it concerns
* for every occurance of the term "Shannon information" made clear which stochastic variable and which event it concerns
* for every occurance of the term "specified information" made clear which hypothesis and specification you use to evaluate it.
That is what my confusion comes down to.
Erik
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