I am facing the following question. It is well known that power laws arise in many situations in nature. They arise even in thats physical systems that are completely deterministic (e.g. sand piles). But power law is a probability distribution function and can be thought as generated from a stochastic variable. What is the relation between a power law distributed variable and a deterministic dynamical system? In other words why I can see statistical distribution in a deterministic system?
I don't think that anyone has answered your question yet ( this one "How a probabilistic behavior arise from a deterministic system? Or how happens that we pass from trajectories to frequencies densities?" )
If you can answer this question you have demonstrated that statistical mechanics and so thermodynamics derive from classical mechanics. As far as i know, no one actually have understood why synergetic properties, that is properties that derive from the collective behavior such as entropy, arise. All we can know is that particles "communicate informations" through collision.
The theoretical tool you need to use in order to point out this problem is the "H theorem". One of the consequences of this theorem is that the evolution of a system of many particle provide us a privileged verse of time. Let me explain better. Also computer simulation based on classical mechanics fail (see Orben and Bellmans Physics Letters 24 page 620 1967). They collected a set of particles in a box and they simulate the evolution of each particle with the classical equation of motion: in this way they could trace position, momentum and the value of the H function. But then, they stopped the system and inverted all the vectors: from a classical point of view, that is following newton's laws, this would result in a "specular" motion. Inverting the system after 50 collisions resulted in a situation very similar to the beginning one. But after 100 collisions the value of the H function was lower.
The presence of probability in this description of a physical system is also a philosophical question. Probability is something that we assign to the system since we are ignorant of his exact behavior or it's a property of the system itself? There are proper books which talk about this and it's still an open question.
Umm. After giving it some thought I came to this conclusions.
As you point out, is clear why you can obtain statistical mechanics from stochastic variables. See Monte Carlo for example. All is deterministic each step but the choice of the random number that gives your ensemble.
Another way to put it is adding noise to your system. Like in a Langevin equation.
Yet another one would be Chaos. Is not really statistical Say you got the solution of the differential equation. Now you put an initial condition and operate + iterate. For diferent initial conditions you will get different realizations. Fractals are made this way.
In all the case above written you can encounter power-laws at critical points, or as an answer from the system telling you "hey, I'm rescalable!"
So my guess will be that stochastic behaviour is madness coming from tiny little details as: initial conditions, using random numbers to generate the distribution you are looking for or adding noise.
The answer to your question lies in understanding that probability and information are connected. A statement about the probability of finding a system in a certain set of states (observing an event) is, in fact, a statement about your knowledge about the system.
Even for perfectly deterministic events, we
- may not know the microscopic details that determine their occurance, or
- we do not wish to inspect these details -- it may be tedious, or we may want to generalize.
- And there is also the possibility that cannot make good use of this information, if the system is chaotic -- meaning that even very small errors rapidly become very important.
Naturally it is true that if we do know microscopic details, and the system is deterministic, there is no "probability distribution" -- or rather, it is a single spike or pulse placing 100% probability at the outcome we are certain we will observe.
Now, if we have a system that can have very many starting conditions, we may want to know how it is likely to behave when any starting condition is possible, subject to its own 'prior' probability distribution. It may not be possible to exhaustively simulate the results for all possible starting conditions. Besides, if you can get at it analytically, it may be much faster and does give you more insight too.
But why should it work, to apply stats to deterministic things? Again, it is only a reflection of what we know and what we include in our reasoning. Probability distributions are usually derived from combinatorial arguments -- by considering the number of ways that an event could occur -- and such combinatorial reasoning is valid for deterministic systems.