could someone shed some light on difference between chaos and complexity ? What is the difference both ontological and epistemological between complexity science and statistical mechanics ?
As far as I understand it, I think chaos theory basically says:
There are some functions that depend so strongly on their initial conditions, that a very small difference in the initial conditions can lead to a very different outcome.
For instance, take the trajectory of a ball on a rectangle billiard table. If you shift the initial conditions (the initial position and speed of the ball) by a very small margin, then you should expect the resulting trajectory to also be changed by a small margin. As a very concrete example, if you shift the initial position of the ball by 1 cm in any direction, and don’t change its initial speed, then the new trajectory will be entirely parallel to the old trajectory, and shifted by exactly 1 cm. So, we can say that there is no chaos on a rectangle billiard table with a single ball, and that trajectory of the ball is easy to predict.
However, if you do the same thing on a billiard table with a round obstacle in the centre, then even a very small shift in the initial conditions can result in a widely different trajectory after the ball has hit the round obstacle. So, with a round billiard table or a round obstacle, there is chaos.
Billiard with round obstacle, from chaos-math.org: Chaos V - Billiards
Chaos without complexity
This doesn’t mean that the trajectory of the ball cannot be predicted when there is a round obstacle. The math with the round obstacle is a bit more complex than without the round obstacle, but most importantly, the trajectory is a lot more sensitive to the initial conditions. Geometrically, given perfect information about the initial conditions, a middle-school student should have the tools to solve the problem. So in that sense, chaos is not synonymous with complexity. However, if you account for measurement errors in the initial conditions, like physicists do, then the problem becomes much more complex and the small incertitude on the initial measurements translate into an incertitude so huge in the final position of the ball that the prediction is useless.
Similarly, when you roll a die, you can pretty much assume that the result is fully random. In a sense, it is not, and if you had all the data describing the initial conditions, then it should be possible to predict which face the die would land on. However, the math is very complex, and the result is so sensible to the initial conditions that you’d need extremely precise data; not just the initial position and speed of the die, but also its initial 3d rotation on itself, its elasticity, etc.
Now, the math behind the trajectory of a real die is extremely complex. But look at pseudo-random generators in computers. Modern "cryptocraphically-secure" random number generators can be complex, but more primitive random number generator numbers are as simple as an iterative sequence given by an arithmetic formula such as:
next_number = (previous_number * 16807) % 2147483647
/ is integer division. This sequence is entirely deterministic and the formula is relatively simple, but it is chaotic enough for many purposes. So we can definitely have chaos without complexity.
Complexity without chaos
Conversely, we can have very complex systems that are not chaotic.
Consider any application of statistics and the law of big numbers.
One given molecule in a gas will display erratic, random, chaotic behaviour, but one cubic meter of air contains 10^25 molecules, and together they form a gas that behave in a very orderly and predictable fashion.
One given human being can display wild emotions, passions, and make chaotic decisions. But take 10 million citizens in one country, and together they will act like a predictable body consumers and voters.
So in the case of gas particles and humans, one individual is chaotic, but a complex system of millions or more of individuals becomes easily predictable.
Chaos and complexity together
Weather forecasts become very inaccurate if we try to predict the weather more than a few days in the future, because the weather is so sensible to the initial conditions, and the initial conditions include so much data: pressure, temperature, wind speed, etc, at every point of the globe all around the world. This is an example where a system is both extremely complex and chaotic.
A basic concept in this field is the concept of “deterministic chaos”.
The well-known Mandelbrot set shows those complex numbers c where the iteration of the simple quadratic function f(z):= z^2+c does not converge to infinity for the start value z=0, see Mandelbrot set. The behaviour shows a sensible dependency on the parameter c. Parameter values nearby show quite a different behaviour.
This dependency exemplifies chaotic behaviour. But the complexity of the behaviour is generated by a simple deterministic rule.
For a short introduction on a more academic level see Complexity, Chaos, and Fractals.
Chaos denotes confusion, randomness and the apparent absence of any kind of order. Complexity suggests multiple effects and components interacting in a way that is difficult or impossible to fathom- the individual interactions can be orderly and readily understood, but their combined effect may be anything but. In summary... Complexity can lead to chaos, but need not. Chaos can exist without complexity.
Nonlinear dynamics is a field of mathematics with applications to physical science which is concerned with predicting the behavior of equations which exhibit large changes in the output values for small changes in input parameters.
In the physical sciences the equations can represent physical systems with the appropriate choice of equations and parameters
Chaos describes a nonlinear dynamical system which does not converge to a long-lasting state in which small changes in input parameters no longer result in large changes in the output values.
Complexity describes systems composed of many dissimilar interacting parts. Again these may either be purely mathematical objects or mathematical models of real objects. Complex systems usually but do not always comprise and/or constitute nonlinear dynamical systems. As a rule: complex systems are systems that look more and more different the closer you zoom in.
Statistical mechanics is a field of mathematics with applications to physical science which applies probability theory to very large assemblies of interacting parts. When these parts are similar and can interact only in similar ways, statistical mechanical systems are not complex. When these parts are dissimilar and can interact in dissimilar ways, statistical mechanical systems are complex. Both kinds of systems can be, but need not be, nonlinear dynamical systems.