The common belief is that adding disorder to a designed object will destroy the design (like a tornado passing through a city, to paraphrase Hoyle). Now if increasing entropy implies increasing disorder, creationists will often reason that “increasing entropy of an object will tend to destroy its design”. This essay will argue mathematically that this popular notion among creationists is wrong.

The correct conception of these matters is far more nuanced and almost the opposite of (but not quite) what many creationists and IDists believe. Here is the more correct view of entropy’s relation to design (be it man-made or otherwise):

1. increasing entropy can increase the capacity for disorder, but it doesn’t necessitate disorder

2. increasing an object’s capacity for disorder doesn’t imply that the object will immediately become more disordered

3. increasing entropy in a physical object is a necessary (but not sufficient) condition for increasing the complexity of the design

4. contrary to popular belief, a complex design is a high entropy design, not a low entropy design. The complex organization of a complex design is made possible (and simultaneously improbable) by the high entropy the object contains.

5. without entropy there is no design

If there is one key point it is: Entropy makes design possible but simultaneously improbable. And that is the nuance that many on both sides of the ID/Creation/Evolution controversy seem to miss.

The notion of entropy is foundational to physics, engineering, information theory and ID. These essays are written to provide a discussion on the topic of entropy and its relationship to other concepts such as uncertainty, probability, microstates, and disorder. Much of what is said will go against popular understanding, but the aim is to make these topics clearer. Some of the math will be in a substantially simplified form, so apologies in advance to the formalists out there.

Entropy may refer to:

1. Thermodynamic (Statistical Mechanics) entropy – measured in Joules/Kelvin, dimensionless units, degrees of freedom, or (if need be) bits

2. Shannon entropy – measured in bits or dimensionless units

3. Algorithmic entropy or Kolmogorov complexity – measured also in bits, but deals with the compactness of a representation. A file that can be compressed substantially has low algorithmic entropy, whereas files which can’t be compressed evidence high algorithmic entropy (Kolmogorov complexity). Both Shannon entropy and algorithmic entropies are within the realm of information theory, but by default, unless otherwise stated, most people associate Shannon entropy as

theentropy in information theory.4. disorder in the popular sense – no real units assigned, often not precise enough to be of scientific or engineering use. I’ll argue the term “disorder” is a misleading way to conceptualize entropy. Unfortunately, the word “disorder” is used even in university science books. I will argue mathematically why this is so…

The reason the word *entropy* is used in the disciplines of Thermodynamics, Statistical Mechanics and Information Theory is that there are strong mathematical analogies. The evolution of the notion of entropy began with Clausius who also coined the term for thermodynamics, then Boltzmann and Gibbs related Clausius’s notions of entropy to Newtonian (Classical) Mechanics, then Shannon took Boltzmann’s math and adapted it to information theory, and then Landauer brought things back full circle by tying thermodynamics to information theory.

How entropy became equated with disorder, I do not know, but the purpose of these essays is to walk through actual calculations of entropy and allow the reader to decide for himself whether disorder can be equated with entropy. My personal view is that Shannon entropy and Thermodynamic entropy cannot be equated with disorder, even though the lesser-known algorithmic entropy can. So in general entropy should not be equated with disorder. Further, the problem of organization (which goes beyond simple notions of order and entropy) needs a little more exploration. Organization sort of stands out as a quality that seems difficult to assign numbers to.

The calculations that follow are to give an illustration how I arrived at some my conclusions.

First I begin with calculating Shannon entropy for simple cases. Thermodynamic entropy will be covered in the Part II.

Bill Dembski actually alludes to Shannon entropy in his latest offering on Conservation of Information Made Simple

In the information-theory literature, information is usually characterized as the negative logarithm to the base two of a probability (or some logarithmic average of probabilities, often referred to as

entropy).William Dembski

Conservation of Information Made Simple

To elaborate on what Bill said, if we have a fair coin, it can exist in two microstates: heads (call it microstate 1) or tails (call it microstate 2).

After a coin flip, the probability of the coin emerging in microstate 1 (heads) is 1/2. Similarly the probability of the coin emerging in microstate 2 (tails) is 1/2. So let me tediously summarize the facts:

N = Ω(N) = Ω = Number of microstates of a 1-coin system = 2

x

_{1}= microstate 1 = heads

x_{2}= microstate 2 = tailsP(x

_{1}) = P(microstate 1)= P(heads) = probability of heads = 1/2

P(x_{2}) = P(microstate 2)= P(tails) = probability of tails = 1/2

Here is the process for calculating the Shannon Entropy of a 1-coin information system starting with Shannon’s famous formula:

where I is the Shannon entropy (or measure of information).

This method seems a rather torturous way to calculate the Shannon entropy of a single coin. A slightly simpler method exists if we take advantage of the fact that each microstate of the coin (heads or tails) is equiprobable, and thus conforms to the fundamental postulate of statistical mechanics, and thus we can calculate the number of bits by simply taking the logarithm of the number of microstates as is done in statistical mechanics.

Now compare this equation of the Shannon entropy in information theory

to Boltzmann entropy from statistical mechanics and thermodynamics

and even more so using different units whereby k_{b}=1

The similarities are not an accident. Shannon’s ideas of information theory are a descendant of Boltzmann’s ideas from statistical mechanics and thermodynamics.

To explore Shannon entropy further, let us suppose we have a system of 3 distinct coins. The Shannon entropy relates the amount of information that will be gained by observing the collective state (microstate) of the 3 coins.

First we have to compute the number of microstates or ways the system of coins can be configured. I will lay them out specifically.

microstate 1 = H H H

microstate 2 = H H T

microstate 3 = H T H

microstate 4 = H T T

microstate 5 = T H H

microstate 6 = T H T

microstate 7 = T T H

microstate 8 = T T TN = Ω(N) = Ω = Number of microstates of a 3-coin system = 8

So there are 8 microstates or outcomes the system can realize. The Shannon entropy can be calculated in the torturous way:

or simply taking the logarithm of the number of microstates:

It can be shown that for the Shannon entropy of a system of N distinct coins is equal to N bits. That is, a system with 1 coin has 1 bit of Shannon entropy, a system with 2 coins has 2 bits of Shannon entropy, a system of 3 coins has 3 bits of Shannon entropy, etc.

Notice, the more microstates there are, the more uncertainty exists that the system will be found in any given microstate. Equivalently, the more microstates there are, the more improbable the system will be found in a given microstate. Hence, sometimes entropy is described in terms of improbability or uncertainty or unpredictability. But we must be careful here, uncertainty is not the same thing as disorder. That is subtle but important distinction.

So what is the Shannon Entropy of a system of 500 distinct coins? Answer: 500 bits, or the Universal Probability Bound.

By way of extension, if we wanted to build an operating system like Windows-7 that requires gigabits of storage, we would require the computer memory to contain gigabits of Shannon entropy. This illustrates the principle that more complex designs require larger Shannon entropy to support the design. It cannot be otherwise. Design requires the presence of entropy, not absence of it.

Suppose we found that a system of 500 coins were all heads, what is the Shannon entropy of this 500-coin system? Answer: 500 bits. No matter what configuration the system is in, whether ordered (like all heads) or disordered, the Shannon entropy remains the same.

Now suppose a small tornado went through the room where the 500 coins resided (with all heads before the tornado), what is the Shannon entropy after the tornado? Same as before, 500-bits! What may arguably change is the algorithmic entropy (Kolmogorov complexity). The algorithmic entropy may go up, which simply means we can’t represent the configuration of the coins in a compact sort of way like saying “all heads” or in the Kleene notation as H*.

Amusingly, if in the aftermath of the tornado’s rampage, the room got cooler, the thermodynamic entropy of the coins would actually go down! Hence the order or disorder of the coins is independent not only of the Shannon entropy but also the thermodynamic entropy.

Let me summarize the before and after of the tornado going through the room with the 500 coins:

BEFORE : 500 coins all heads, Temperature 80 degrees

Shannon Entropy : 500 bits

Algorithmic Entropy (Kolmogorov complexity): low

Thermodynamic Entropy : some finite starting valueAFTER : 500 coins disordered

Shannon Entropy : 500 bits

Algorithmic Entropy (Kolmogorov complexity): high

Thermodynamic Entropy : lower if the temperature is lower, higher if the temperature is higher

Now to help disentangle concepts a little further consider three 3 computer files:

File_A : 1 gigabit of binary numbers randomly generated

File_B : 1 gigabit of all 1’s

File_C : 1 gigabit encrypted JPEG

Here are the characteristics of each file:

File_A : 1 gigabit of binary numbers randomly generated

Shannon Entropy: 1 gigabit

Algorithmic Entropy (Kolmogorov Complexity): high

Thermodynamic Entropy: N/A

Organizational characteristics: highly disorganized

inference : not designedFile_B : 1 gigabit of all 1’s

Shannon Entropy: 1 gigabit

Algorithmic Entropy (Kolmogorov Complexity): low

Thermodynamic Entropy: N/A

Organizational characteristics: highly organized

inference : designed (with qualification, see note below)File_C : 1 gigabit encrypted JPEG

Shannon Entropy: 1 gigabit

Algorithmic Entropy (Kolmogorov complexity): high

Thermodynamic Entropy: N/A

Organizational characteristics: highly organized

inference : extremely designed

Notice, one cannot ascribe high levels of improbable design based on the Shannon entropy or algorithmic entropy without some qualification. Existence of improbable design depends on the existence of high Shannon entropy, but is somewhat independent of algorithmic entropy. Further, to my knowledge, there is not really a metric for organization that is separate from Kolmogorov complexity, but this definition needs a little more exploration and is beyond my knowledge base.

Only in rare cases will high Shannon entropy and low algorithmic entropy (Kolmogorov complexity) result in a design inference. One such example is 500 coins all heads. The general method to infer design (including man-made designs), is that the object:

1. has High Shannon Entropy (high improbability)

2. conforms to an independent (non-postdictive) specification

In contrast to the design of coins being all heads where the Shannon entropy is high but the algorithmic entropy is low, in cases like software or encrypted JPEG files, the design exists in an object that has both high Shannon entropy and high algorithmic entropy. Hence, the issues of entropy are surely nuanced, but on balance entropy is good for design, not always bad for it. In fact, if an object evidences low Shannon entropy, we will not be able to infer design reliably.

The reader might be disturbed at my final conclusion in as much as it grates against popular notions of entropy and creationist notions of entropy. But well, I’m no stranger to this controversy. I explored Shannon entropy in this thread because it is conceptually easier than its ancestor concept of thermodynamic entropy.

In the Part II (which will take a long time to write) I’ll explore thermodynamic entropy and its relationship (or lack thereof) to intelligent design. But in brief, a parallel situation often arises: the more complex a design, the higher its thermodynamic entropy. Why? The simple reason is that more complex designs involve more parts (molecules) and more molecules in general imply higher thermodynamic (as well as Shannon) entropy. So the question of Earth being an open system is a bit beside the point since entropy is essential for intelligent designs to exist in the first place.

[UPDATE: the sequel to this thread is in Part 2]

Acknowledgements (both supporters and critics):

1. Elizabeth Liddle for hosting my discussions on the 2nd Law at TheSkepticalZone

2. physicist Olegt who offered generous amounts of time in plugging the holes in my knowledge, particularly regarding the Liouville Theorem and Configurational Entropy

3. retired physicist Mike Elzinga for his pedagogical examples and historic anecdotes. HT: the relationship of more weight to more entropy

4. An un-named theoretical physicist who spent many hours teaching his students the principles of Statistical Mechanics and Thermodynamics

5. physicists Andy Jones and Rob Sheldon

6. Neil Rickert for helping me with Latex

7. Several others that have gone unnamed

NOTE:

[UPDATE and correction: gpuccio was kind enough to point out that in the case of File_B, the design inference isn’t necessarily warranted. It’s possible an accident or programming error or some other reason could make all the bits 1. It would only be designed if that was the designer’s intention.]

[UPDATE 9/7/2012]

Boltzmann

“In order to explain the fact that the calculations based on this assumption [“…that by far the largest number of possible states have the characteristic properties of the Maxwell distribution…”] correspond to actually observable processes, one must assume that an enormously complicated mechanical system represents a good picture of the world, and that all or at least most of the parts of it surrounding us are initially in a very ordered — and therefore very improbable — state. When this is the case, then whenever two of more small parts of it come into interaction with each other, the system formed by these parts is also initially in an ordered state and when left to itself it rapidly proceeds to the disordered most probable state.” (Final paragraph of #87, p. 443.)

That slight, innocent paragraph of a sincere man — but before modern understanding of q(rev)/T via knowledge of molecular behavior (Boltzmann believed that molecules perhaps could occupy only an infinitesimal volume of space), or quantum mechanics, or the Third Law — that paragraph and its similar nearby words are the foundation of all dependence on “entropy is a measure of disorder”. Because of it, uncountable thousands of scientists and non-scientists have spent endless hours in thought and argument involving ‘disorder’and entropy in the past century. Apparently never having read its astonishingly overly-simplistic basis, they believed that somewhere there was some profound base. Somewhere. There isn’t. Boltzmann was the source and no one bothered to challenge him. Why should they?

Boltzmann’s concept of entropy change was accepted for a century primarily because skilled physicists and thermodynamicists focused on the fascinating relationships and powerful theoretical and practical conclusions arising from entropy’s relation to the behavior of matter. They were not concerned with conceptual, non-mathematical answers to the question, “What is entropy, really?” that their students occasionally had the courage to ask. Their response, because it was what had been taught to them, was “Learn how to calculate changes in entropy. Then you will understand what entropy ‘really is’.”

There is no basis in physical science for interpreting entropy change as involving order and disorder.