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

Order Arising from Disorder (Pt 2)

In the previous blog on order and disorder in statistical mechanics, I focused on some basic principles, and simple illustrations, including the basis for a Markov process, and the Ehrenfest effect. We confined the basis to very simple systems while now we want to turn our attention to actual spin systems, such as occur in quantum mechanics. In conjunction with this, some readers may find it useful to go back to the blogs dealing with electron spin to refresh memories, e.g.  http://brane-space.blogspot.com/2010/07/more-quantum-mechanics-electron-spin.html

and http://brane-space.blogspot.com/2010/07/spin-orbit-coupling-in-quantum.html


Consider two systems of atomic magnets possessing different ‘spins’. We look at what happens when they undergo changes, and what is observed at the quantum level, as well as macroscopic scales. A fact of 20th century physics is that all electrons have a property called spin that’s either up (+½) or down (-½).

The numbers are "spin quantum numbers" and the main thing to remember about them is that they refer to different directions in the atom, and differing energy states. (The plus state slightly higher than the minus). An atom can assume a net spin as a result of the spins of the constituent electrons. Helium, for example, has two electrons which are forbidden (by the Pauli Exclusion Principle) to have the same spin numbers at the same time and so the net spin of the Helium atom cancels out: (-½) + (+½) = 0.

At a more detailed level, spin is assessed in atoms for particular (x, y, z) orientations. That level of detail is omitted here to keep the presentation simple. Also, the details of the experiment by Stern and Gerlach (1921) to infer electron spin using the magnetic moment are omitted, though these were given in a much earlier blog. Basically, they passed a narrow beam of silver atoms through an unequal magnetic field and found it split into two beams: one for the ½ spin, the other for minus ½ spin. Historically, it was Wolfgang Pauli who introduced electron spin into quantum mechanics in 1927, after S. A. Goudsmit and G. E. Uhlenbeck first postulated it in 1925.

Observationally speaking, spin is inferred from fine splitting of atomic spectral lines, cf. Fig. 1, wherein the unperturbed line splits into two separate lines of slightly differing energy. The appropriate inferences are derived by applying a set of quantum rules for combining angular momentum vectors and their quantum numbers in what is known as ‘L-S coupling’. A spin-orbit interaction results with a total angular momentum J’ = L’ + S’, where L’ denotes total angular momentum = L1 + L2 etc. = {l’ (l’ + 1)}^1/2  ħ , and S’ the total spin angular momentum = S1 + S2 etc. = {s’ (s’ + 1)}^1/2 ħ, then:

 J’ = {j’ (j’ + 1)}^1/2  ħ.

And recall here  ħ = h/ 2π , where h is the Planck constant.  One applies these to a particular atomic configuration, say: 2p 3d then goes through a procedure of taking differences, assigning values etc. based on selection rules. (See prior blogs on quantum mechanics)

Let's now look at an elementary, statistical mechanical spin system. For starters, let's say that there is a system consisting of 16 atomic magnets. These atomic magnets exist temporarily in the state S(1) as shown in Fig. 2 (i.e. each arrow denotes the net spin of the atom based on the sum of electron orientations within it). We want to find the degree of order applicable to the system, say at time t(o) and do the appropriate counting of "spin ups" and spin downs" as shown in the left side of Fig. 2 below the system at t(o). We find on doing so (which the reader can verify) that we get 8 spin ups - 8 spin downs = 0 net spin, or in other words the system is at equilibrium.

The degree of order, as well as information, for the simple spin system shown is determined from what is called "the spin excess", or the net spin difference (up minus down or vice versa). The larger this number, the greater the degree of order, and the lower the entropy of the system. Obviously, since 0 denotes an extremely low number, we can deduce large entropy.

 

Consider then another system but at a different time, for which we behold the right side orientations of the elementary spin magnets. Here we get: 14 spin ups - 2 spin downs = 12 spin ups, or in other words the spin excess = 12. This system, call it S(2), has much higher degree of order (less entropy) than the system S(1). (We should add here that higher entropy - as in S(1) - corresponds to the most probable state, defined by the minimal spin excess of zero.

Now, it is also important to note that one can dramatically generalize what's depicted in Fig. 2 via the "Ising Model" of statistical mechanics. In the 2-dimensional Ising Model for ferromagnetic matter, for example, one finds magnetic domains or regions which can, after a time, undergo a change such as that shown from S(1)-> S(2) (looking at the same system!) Here the Ising model system, by virtue of undergoing spontaneous magnetization, discloses a higher degree of order at the later time t(0) + t, where t could be in billions of years or hundreds of seconds. (Depending on the system).

The reader should also note that each arrow can be generalized to represent a region of indefinite scale size. For example, the depiction given in Fig. 2 might conceivably represent the evolution of an (essentially) two-dimensional current sheet in a highly compressed magnetic region near a sunspot. The arrows would represent plasma flows and current densities that follow magnetic field lines.

Chaos theory, spurred on by a number of notable successes, sheds new light on the numbers. Particular offshoots of Chaos theory disclose entities such as ‘strange attractors’ that can actually spur a system toward an enhanced degree of self-organization. We call such a turning point a bifurcation. Bifurcations represent sudden, abrupt changes toward higher degrees of organization of a system. These include the sudden alteration of system development from equilibrium (maximum entropy) to non-equilibrium.

In the thermodynamic study of chaos, similar approaches to equilibrium can be examined but permitting far from equilibrium changes. Given the right fluctuations, it’s even possible to generate self-organization from pre-existing chaos. The ultimate example is the formation of matter in the universe. From amidst a sea of pure energy, the ambient temperature cooled, re-setting the so-called Higgs Field H(f) from a zero to a non-zero value. This re-setting of the cosmic thermostat enabled the production of material particles in a sudden and spontaneous breakdown of symmetry.

One of the simplest examples of a non-equilibrium system is a simple chemical reaction of the form:



X + 2Y <-> 3Y

X <-> W


where X, Y, W are arbitrary reactants that meet the conditions and the symbol (<->) denotes a two-way equilibrium process. We note here that at equilibrium detailed balance requires (with k1, k2, k3 and k4 the respective concentrations):

k1 xy² =  k2 y³

k3 y = k4 w

Similarly, a stationary state ‘far from equilibrium’ is disclosed by the concentration equation:

k1 xy² + k3 y = k2 y³ + k4 w
Or:

k1 xy² + k3 y - k2 y³ - k4 w = 0

 
Note that in such a state, far from equilibrium, it isn’t necessary for each separate chemical reaction to balance in both directions. Clearly, non-equilibrium discloses aspects of the system that are inhibited when it is at equilibrium. To assist in identification of far from equilibrium states, bifurcation diagrams can be of use. A typical form or template for such a diagram is depicted in Fig. 3, where F is some state function, and lambda (Greek letter) is called a control parameter.

Note also that the equilibrium solution associated with the above is valid up to some critical value of the control parameter. Beyond this value there are multiple solutions, viz. the two separated or bifurcated ones. Meanwhile, the use of the classical (Newtonian) order paradigm for the cosmos is now recognized as the byproduct of a bygone era in which astronomers didn’t have access to the wide range of instrumentation available today. It also hearkens to an era in which the assumption prevailed that what one saw was what was actually there.

 

The prevalence of dark matter and dark energy, of course, resoundingly refutes this. We now know formidable problems occur when one postulates a supernatural agent, in this case an allegedly "eternal" deity to create order. Logically, such an entity must be outside all temporal reference frames. However, the action of transforming disorder to order would require this being to exist within time. This inconsistency also intrudes on its being ‘all-knowing’ or ‘omnipresent’. We have then a logical contradiction. The point is that any being postulated to exist outside time (eternal) can exhibit no activity which determines natural laws or processes.

In th next instalment I will give examples of systems undergoing bifurcation and subject to spontaneous emergence of order from chaos.