Saturday, April 21, 2012

24. Attractors in Phase Space

The concept of 'phase space' is a very powerful way of depicting the time-evolution of dynamical systems. Imagine a system of N particles. At any instant of time, any particle is at a particular point in space, so we can specify its location in terms of three coordinates, say (x, y, z). At that time the particle also has some momentum. The momentum, being a vector, can be specified in terms of its three components, say (px, py, pz). Thus six parameters (x, y, z, px, py, pz) are needed to specify the position and momentum of a particle at any instant of time. Therefore, for N particles, we need to specify 6N parameters for a complete description of the system. For real systems like molecules in a gas, the number N can be very large, being typically of the order of the Avogadro number (~1023).

So this is a very messy, in fact impossible, way of depicting such a system graphically. The concept of phase space solves this problem. Imagine a 6-dimensional 'hyperspace' in which three of the axes are for specifying the position coordinates of a particle, and the other three are for specifying the momentum components of the same particle. In this space the position and momentum of a particle at any instant of time can be represented by a single point. Similarly, for representing simultaneously the configurations of N particles, we can imagine a 6N-dimensional hyperspace (called phase space or state space). A point in this space represents the state of the entire system of N particles at an instant of time. As time progresses, this 'representative point' traces a trajectory, called the phase-space trajectory. Such a trajectory records the time-evolution of the dynamical system (in classical mechanics).

The figure below illustrates this. In it I have introduced the simplification that, for depiction purposes, all the position coordinates (3N in number) are given the generic symbol q, and only one axis is drawn to denote all the 3N axes. Similarly, all the 3N momentum components are given a representative symbol p, and only one axis is taken to represent all of them. In reality there are a total of 6N axes, 3N for the position components, and 3N for the momentum components.

Some variations of the concept of such an imaginary phase space or state space are: representation space; search space; configuration space; solution space; etc. The basic idea is the same. One imagines an appropriate number of axes, one for each 'degree of freedom'.

Next, let us consider a simple pendulum (a vertical string fixed at the top, and having a weight attached to its lower end). Suppose I pull the weight horizontally by a small distance x0 along the x-axis, and then release it. The weight starts performing an oscillatory motion around the point x = 0. At the moment I released the weight it was at rest, so its momentum was zero, and it had only potential energy. On releasing it the potential energy starts decreasing as the weight moves towards the point x = 0, and its momentum starts increasing. This goes on till the point x = 0 is reached. At this moment the potential energy is zero (it got fully converted to kinetic energy corresponding to the momentum -px).

Because of this momentum, the weight now overshoots the point x = 0 and moves in the opposite direction. When it has moved a distance -x0 it stops, having spent all its kinetic energy for acquiring an equivalent amount of potential energy.

Then it starts moving towards the point x = 0. At this point it has acquired the maximum (but oppositely directed) momentum px. And so on.

What is the phase-space trajectory for this system? It is a circle in a plane defined by the x-axis and the px-axis (Figure (a) below). The weight successively and repeatedly passes through a whole continuum of points in phase space, including the points (-x, 0), (0, px), (x, 0), (0, -px).

If there is no dissipation of energy, the phase-space trajectory in this experiment is a closed loop because the particle repeatedly passes through all the allowed (i.e. energy-conserving) position-momentum combinations again and again.

But in reality, dissipative forces like friction are always present, and in due course all the energy I expended in displacing the weight from its initial equilibrium position will be dissipated as heat. As the total energy decreases, the maximum value of the x-coordinate during the trajectory cycle, as also the maximum value of px, would decrease, implying that the area enclosed by the trajectory in phase space will progressively decrease, till the particle finally comes to a state of rest or zero momentum.

This final configuration corresponds to an ATTRACTOR in phase space: It is as if the dissipative dynamics of the system is 'attracted' by the point (0, 0, 0, 0, 0, 0) as the energy gets dissipated. Thus, because of the gradual dissipation of energy, the phase-space trajectory spirals towards a state of zero area (Figure (b) above).

This is like a particle set rolling in a bowl, spiralling towards the bottom of the bowl; the bowl thus acts as a basin of attraction. The phase-space region around the attractor (0, 0, 0, 0, 0, 0) is the basin of attraction for the oscillator problem I have considered here.

In the above experiment if I move the weight only by a small amount, the restorative force is linearly proportional to the displacement. If we plot this force fx as a function of x, we get a straight line (which is a linear curve).

But if the displacement is too large, the restorative force is not linearly proportional to the displacement x, and we are then dealing with a NONLINEAR DYNAMICAL SYSTEM. All complex systems are governed by nonlinear dynamics, and this makes their detailed analytical investigation very difficult, if not impossible.