Order from Disorder (Pt. 3): Bifurcation Examples

In two earlier blogs on statistical mechanics and its applications to simple systems (e.g. spin systems) we saw how order can emerge naturally out of disorder. What one looks for, say in an electron spin system (or Ising model of larger scale magnetizations) is rapid fluctuations that quickly alter the state from one of disorder to order, or from equilibrium to order - say like a 16 spin magnet system with 8 up and 8 down that evolves to one with 14 up and 2 down, giving a spin excess of 12 up, denoting higher order.

Accessing such simple systems allows us to infer fundamental measures applicable to the systems, for example the "magnetic moment" of a state, as well as the "degeneracy function". Consider an N= 2 model system with either 2 ups (two up arrows) or 2 downs. Then, if m denotes the magnetic permeability we can have:

M = +2m or M = -2m

where the first is the magnetic moment for two spins up particles, and the second for two spins down. One can also, of course, have the mixed state inclusive of one spin up plus one spin down, then:

M = O m or O

Meantime, the degeneracy function computes the number of states having the same value of m (individual spins) or M such that:

g(N,m) = N!/ (½N + m)! (½N - m)! [M_av]


where [M_av] denotes the average value of the total magnetic moment summed over all states (e.g. with m_s)

Accessing these quantities helps to see at which point a system is likely to be subject to what we call "bifurcation" - which was illustrated generically in the last blog on statistical mechanics and order.Let's look at a couple of examples of this, starting out with more or less prosaic systems. An actual example from fluid dynamics is the famous ‘pitchfork bifurcation’, which has nothing to do with demons! It arises by considering the complex interactions of a controlled water channel whose flow continuously recycles. At some point, beyond a critical value of the Reynolds number (R_c), the single flow relinquishes its symmetry and two stable flows result. These are shown in Fig. 1.

The critical value (where the vertical dotted line intersects the abscissa) turns out to be 40.5. As before, with the Ising model, we see that hidden complexities manifest in a kind of order or self-organization for each of the bifurcation paths. Indeed, each one of these paths can be thought of as mirror images of the same Markov process, tending to some new ‘equilibrium’ displaced from the original one.

An interesting but somewhat more complex example from plasma physics is the ‘two stream instability’. In this case we have the plasma dispersion function F(w) which leads to two bifurcation ‘paths’, including a split symmetrical one and a symmetrical one similar to that shown- but in a different direction relative to coordinate axes. This is shown in Fig. 2.

In finding conditions under which it operates, one considers treating a dispersion relation for plasma waves such that, viz.:

F(w) = F(x, y) = (m_e/m_i)/ x² + 1/ (x² - y²)

where (m_e/m_i) denotes the electron to ion mass ratio, and we define the variables x, y as follows:

x = w / w _e

or the ratio of the plasma frequency to electron plasma frequency.

Meanwhile:

y = k V_o/ w _e

or the ratio of the product of the wave number k by the electron thermal velocity (V_o) to the electron plasma frequency.

Plotting the graph on the axes yields a bifurcated graph with 4 roots (Fig. 2). It will always feature a local minimum F_m such that: 0 (less than or equal) F_m (less than or equal) x=y.

When: F(x_m, y) = F_m  < 1 there will be four real roots.

When F_m  > 1 there will be two real and two complex roots with suitable approximations, e.g. k² V_o ²    £  w _e ² then the 2 complex roots are found to be:

w / w _e  =   - ½ + i((Ö3/2) ,   - ½ -  i((Ö3/2)

 

These will give the limits for the instability for when F_m >1