21. The Nature of Information

How did life emerge from non-life? It did so through a long succession of processes and events in which more complex structures evolved from simpler ones. Beginning with this post, I shall take you through a fascinating journey of easy comprehension, explaining how complexity science answers such questions. This and the next few posts will introduce some of the jargon and basic concepts in modern complexity science.

The second law of thermodynamics for open systems is the primary organizing principle for all natural phenomena (cf. Part 6). The relentless expansion and cooling of our universe has been creating gradients of various types, which tend to get annulled as the blind forces of Nature take the local systems towards old or new equilibria. New patterns and structures get created when new equilibrium structures arise.

Take any living entity; say the human body, or even a single-celled organism. The amount of information needed for describing the structure of a single biological cell is far more than the information needed to describe, say, an atom or a molecule. The technical term one introduces here is 'complexity'. We say that the biological cell has a much higher DEGREE OF COMPLEXITY than an atom or a molecule.

Let us tentatively define the degree of complexity of any object or system as the amount of information needed for describing its structure and function.

Since the degree of complexity has been defined in terms of 'amount of information', we should be clear about the formal meaning of 'information'. The word 'bit' is commonplace in this IT age. It was introduced by Claude Shannon in 1948, and is the short form for 'binary digit'. A bit has two states: 0 or 1. Shannon took the bit as the unit of information.

One bit is the quantity of information needed (it is the 'missing' or 'not-yet-available' information) for deciding between two equally likely possibilities (for example, whether the toss of a coin will be 'heads' or 'tails'). And the information content of a system is the minimum number of bits needed for a description of the system.

The term ‘missing information’ is assigned a numerical measure by defining it as the uncertainty in the outcome of an experiment yet to be carried out. The uncertainty may be high either because only one of a large number (N_s) of outcomes is possible, or, what is the same thing, the probability of a particular outcome is inherently low.

Suppose we have a special coin with heads on both sides. What is the probability that the result of a spin of the coin will be 'heads'? The answer is 100% or 1; i.e., certainty. Thus the carrying out of this experiment gives us zero information. We were certain of the outcome, and we got that outcome; there was no missing information.

Next, we repeat the experiment with a normal, unbiased, two-sided coin. There are two possible outcomes now (N_s = 2). In this case the actual outcome gives us information which we did not have before the experiment was carried out (i.e., we get the missing information).

Suppose we toss two coins instead of one. Now there are four possible outcomes (N_s = 4). Therefore, any particular experiment here gives us even more information than in the two situations above. Thus: Low probability means high missing information, and vice versa.

To assign a numerical measure to information, we would like the following criteria to be met:

1. Since (missing) information (I) depends on N_s, the definition of information should be such that, if we are dealing with a combination of two or more systems, N_s for the composite system should be correctly accounted for in the definition of information. For example, for the case of two dice tossed together or successively, N_s should be 6 x 6 = 36, and not 6 + 6 = 12.

2. Information I for a composite or 'multivariate' system should be a sum (and not, say, a multiplication) of the information for the components comprising the system.

The following relationship meets these two requirements:

N_s ~ base^I.

Let us see how. Suppose system X has N_x states and the outcome of an experiment gives information I_x. Let N_y and I_y be the corresponding quantities for system Y. For the composite system comprising of X and Y, we get N_xN_y ~ base^Ix base^Iy. Since N_s = N_xN_y, we can write

N_s ~ base^{(Ix + Iy)}.

Taking logarithms of both sides, and writing I_x + I_y = I, we get

logN_s ~ I log(base)

or

I ~ logN_s / log(base).

What kind of logarithm we take (base = 10, 2, or e), and what proportionality constant we select, is a matter of context. All such choices differ only by some scale factor, and units. The important thing is that this approach for the definition of information has given us a correct accounting of the number of states, 'bins', or classes (i.e. by a multiplication (N_xN_y) of the individual states), and a correct accounting of the individual measures of information (i.e. by addition).

All the cases considered above are equiprobability cases: When the die is thrown, the probability P₁ that the face with ‘1’ will show up is 1/6, as are the probabilities P₂, P₃, .. P₆ that '2', '3', .. '6' will show up. For such examples, the constant probability P is simply the reciprocal of the number of possible outcomes, classes, or bins; i.e., N_s:

P = 1 / N_s.

Substituting this in the above relation, we get

I ~ log (1/P) / log (base).

Introducing a suitable proportionality constant c, we can write

I = c log (1/P).

This is close to the SHANNON FORMULA for missing information. Even more pertinently, this equation is similar to the famous Boltzmann equation for entropy:

S = k log W.

Entropy has the same meaning as missing information, or uncertainty. More on this next time.

Just One God More

20. The Anthropic Principle

Even slightly different values for some of the fundamental constants of Nature would have led to entirely different histories of the cosmos, making our emergence and existence impossible. Why do these parameters have the values they have? According to a ‘weak’ version of the anthropic principle:

The parameters and the laws of physics can be taken as fixed; it is simply that we humans have appeared in the universe to ask such questions at a time when the conditions are just right for our life. 

Life as we know it exists only on planet Earth. Here is a partial list of necessary conditions for its existence:

1. Availability of liquid water is one of the preconditions for our kind of life. Around a typical star like our Sun, there is an optimum zone (popularly called the ‘Goldilocks zone’), neither so hot that water would evaporate, nor so cold that water would freeze, such that planets orbiting in that zone can sustain liquid water. Our Earth is one such planet.

2. This optimum orbital zone should be circular or nearly circular. Once again, our Earth fulfils that requirement. A highly elliptical orbit would take the planet sometimes too close to the Sun, and sometimes too far, during its cycle. That would result in periods when water either evaporates or freezes. Our kind of life needs liquid water all the time.

3. The location of the planet Jupiter in our Solar system is such that it acts like a ‘massive gravitational vacuum cleaner’, intercepting asteroids that would have been otherwise lethal to our survival.

4. Planet Earth has a single relatively large Moon, which serves to stabilize its axis of rotation.

5. Our Sun is not a binary star. Binary stars can have planets, but their orbits can get messed up in all sorts of ways, entailing unstable or varying conditions, inimical for life to survive and evolve.

It is not only that the planet we live on is conducive to our existence; even the universe we live in (with its operative set of laws of physics) is so. The 'cosmological' or 'strong' version of the anthropic principle says that:

Our universe has the fundamental constants and the laws of physics that are compatible with our existence; had they been different (i.e. inimical to our existence), we would not be here, discussing the principle.

The chemical elements needed for life were forged in certain stars, and then flung far into space through supernova explosions (cf. Part 18). This required a certain amount of time. Therefore the universe cannot be younger than the lifetime of the stars. The universe cannot be too old either, because then all the stars would be ‘dead’. Thus, according to the cosmological anthropic principle, life exists only when the universe has the age that we humans have measured it to be, and has the physical constants that we measure them to be.

Rees (1999), in the book Just Six Numbers, listed six fundamental constants which together determine the universe we see. Their mutual values are such that even a slightly different set of these six numbers would have been inimical to our emergence and existence. Consideration of just one of these constants, namely the strength of the strong nuclear interaction (which determines the binding energies of nuclei), is enough to make the point. It can be roughly defined as that fraction of the mass of an atom of hydrogen which is released as energy when two hydrogen atoms fuse to form an atom of helium. Its value is 0.007 (in certain units), which is just right (give or take a small acceptable range) for any known chemistry to exist; and no chemistry means no life:

Our chemistry is based on reactions among the 90-odd elements. Hydrogen is the simplest among them. Many of the other elements in our universe got synthesised by fusion of hydrogen atoms. This nuclear fusion depends on the strength of the strong nuclear interaction, and also on the ability of a system to overcome the intense Coulomb repulsion between the fusing nuclei. Existence of intense temperatures is one way of overcoming the Coulomb repulsion. A small star like our Sun has a temperature high enough for the production of only helium from hydrogen. As explained in Part 18, the other elements in the periodic table have been made in the much hotter interiors of stars larger than our Sun. The value 0.007 for the strong interaction determined the upper limit on the mass number of the elements we have here on Earth and elsewhere in our universe. A value of, say, 0.006, would mean that the universe would contain nothing but hydrogen, making impossible any chemistry whatsoever. And if it were too large, say 0.008, all the hydrogen would have disappeared by fusing into heavier elements. No hydrogen would mean no life as we know it; in particular there would be no water without hydrogen.

Similarly for the other fundamental constants of our universe.

But why? Why does the universe have these values for the fundamental constants, and not some other set of values? A fallout of Hawking’s model for our universe (cf. Part 1 and Part 19) is that even the strong anthropic principle acquires validity, provided it is stated properly and in the context provided by the M-theory. The new statement of the strong version goes something like this:

Out of the various possible universes, our universe just happens to have the fundamental constants and physical laws it has; other universes (which we cannot observe) have different laws of physics and different values for the fundamental constants. Our existence in our universe has been possible because it is compatible with our laws of physics and fundamental constants; other universes may or may not be conducive to life of any kind.

Watch this great  video for visualizing the 11 dimensions, and for getting a feel for much else in modern cosmology: