Saturday, April 2, 2011
Solutions to Stat Mech test for Fundie Blowhards
Well, it's clear by now the Fundies can't or won't answer the simple questions on order and disorder to do with simple spin systems that I gave in my little test. So, I guess the kind thing to do is help them out! Below I give the questions, and the answers in turn. Hey blabbermouth whiner fundies! You taking notes?
1)Consider the simple spin system shown in the top diagram. (Where t could be in billions of years or hundreds of seconds, depending on the system).
a) What system would most likely have t in billions of years?
Ans. The system of the cosmos, which from red shift measurements we estimate to be ~ 13.8 billion years old
b) Which system would most likely have t in seconds?
A polymer, molecular or simple ferrite (e.g. magnetized iron) system.
Give the reasons for your choice in each case, (a) and (b).
The extremely long time interval for (a) favors a very large and complex system, which could only be the expanding universe. By contrast, a system which is subject to times on the order of 100s to achieve order must be very small, compact, even microscopic.
c) By examining the orientation of the spin magnets, compute the "spin excess" in each case, for time t(o) and time: t(o) + t.
As shown in the diagram, the spin excess for time t(o) is 0 (8 ups - 8 downs) and for time t(o) + t is 12 ups (14 ups - 2 downs = 12 ups)
Is the direction of change from order to disorder, or the converse? Explain.
Since t(o) is the earlier time, and t(o) + t is later, and the later time exhibits the greater order, then the change is from greater disorder to greater order (since the larger spin excess always indicates higher order, organization).
d)In terms of the measure of "spin excess", give the difference in terms of the order. Account for this direction of change in the direction given for time.
Since the spin excess (+12 or 12 up) is in the direction of higher order but the arrow of time defines increasing disorder in the direction of increasing proper time, then it must be that the change of the system from t(o) to t(o) + t is a result of a spontaneous orgnization of the spin magnets.
e) Quantify the magnetic energy for the system at time t(o) compared to time t(o) + t, if the magnetic energy of one spin magnet can be written:
M = -uB cos Θ
where u is the magnetic moment (-eL/2m, L = 1) and assume Θ = +/- π, and B = 1G (gauss).
at time t(o) since the spin excess = 0 then M = -uB cos Θ = 0
at time t(o) + t, the spin excess is 12+.
Then, assuming Θ = - π, M(t) = 12uB = 12(eL/2m)= 6eL/m
Does this agree with your results from (c) and (d) or not? Explain.
Yes, because we expect greater magnetic moment for the more ordered system and this is what we find.
f) Based on your results, at which time does the spin system have the greatest entropy and WHY?
The greatest entropy is always associated with the greatest disorder, so the time for greatest entropy is at t(o) when spin excess = 0 and M = 0.
g) What do you conclude from (f) if the change in spin magnet system is spontaneous?
One must conclude a spontaneous transition occurred to re-arrange the spin magnets from their equal and opposite configuration at time t(o) to one in which 4 more elements-magnets acquired higher spin level energies.
2. The diagram for Fig. 2 shows a multiple bifurcation process. In general this will refer to the energy and order changes in a dynamic system over time, wherein both stable (equilibrium) and chaotic conditions can be produced. Chaos and order, in other words, can be differentiated on the basis of where they appear in relation to the original bifurcation.
a) Which region of the diagram would have the greatest degree of order? Why?
Segement or Band E has the greatest degree of order as it occupies a bifurcation 'band' with higher periodicity then preceding bands in the dynamic process. Also, increasing 'lambda' parameter coinciding with white bands in the diagram correlates with higher order.
b)Which region would have the greatest degree of disorder? Why?
F, for the opposite reasons given above, i.e. the region is within a darekened region correspponding to much greater chaos in the process.
c) Say that S = log (g) determines the entropy for a statistical mechanical system, where g denotes the number of accessible states. Then, would region E or F have a larger entropy? Would region B or C have a larger entropy? Would region A or E have the larger entropy?
As already noted, between E and F, E has the higher order, hence F must have the larger entropy. Between B and C regions, B occupies the larger cycle area in an "island" of order, hence must have lower entropy than C, therefore C is the answer. Between A and E regions, A has the larger entropy than E.