Phys 344 Lecture 5 Jan. 16th 2009 1
Fri. 1/16 2.4, B.2,3 More Probabilities HW5: 13, 16, 18, 21; B.8,11 Mon. 1/20 2.5 Ideal Gas HW6: 26 HW3,4,5 Wed. 1/22 (C 10.3.1) 2.6 Entropy & 2nd Law HW7: 29, 32, 38 Fri. 1/23 (C 10.3.1) 2.6 Entropy & 2nd Law (more)
10_wells_oscillator.py & helix.py (note: helix must be lowercase) BallSpring.mov Statmech.exe Didn’t get through 2.3 last time, so doing it now
2. The Second Law At the End, reassess what homework will be due Monday. Motivation / transition o Combinatorics 2.1 Two-State Systems Microstate = state of the system in terms of microscopic details. Macrostate = state of the system in terms of macroscopic variables Multiplicity = : How many Microstates are consistent with a given Macrostate. Fundamental assumption of statistical mechanics: In an isolated system in internal thermal equilibrium, all accessible microstates are equally probable. Probability: N Fair Coins N! N o N, n n! N n ! n 2.1.1 The Two-State Paramagnet N N! o N, N N N ! N N ! 2.2 The Einstein Model of a Solid o Demo. BallSpring.mov o Demo. 10_wells_oscillator.py N 1 q ! o N, q N 1 ! q ! 2.3 Thermal equilibrium of two blocks To address thermodynamic equilibrium, we need a way of describing two, interacting objects. We’ll take two Einstein Solids. We’ll begin simple, with each “solid” simply being an atom, i.e. 3 oscillators.
Solid A Solid B
U U A B q q A B N N A B
Two single-atom blocks o We’re going to consider these two sharing a total of 4 quanta of energy, so, at any given instant, one of the atoms may have all 4, 3, 2, 1, or none of the quanta. So we’re going to need the… Phys 344 Lecture 5 Jan. 16th 2009 2 o Multiplicities for one atom-solid with 4,3,2,1,0 quanta. . First, picking up where we left off last time, let’s consider the multiplicities for such a single atom (3-oscillators) given 4, 3, 2,1, or 0 quanta of energy. q N 1 ! N 1,q . osc quanta q ! N 1 ! Multiplicity for N=3, q = 4 (3 1,4) 15 Demo:10_wells_oscillator.py . See some of the different ways that 4 quanta can be distributed among the 3 oscillators of an atom. Multiplicity for N = 3, q = 3 (3 1,3) 10 Multiplicity for N = 3, q = 2 (3 1,2) 6 Multiplicity for N = 3, q = 1 (3 1,1) 3 Multiplicity for N = 3, q = 0 (3 1,0) 1 o Note: This last result makes perfect sense – there is only one way for no energy to be added to the system. In our equation we encounter 0! At first blush, you may guess that that is 0. But what works with our intuition for our system is that 0!=1.
o Multiplicity table for 2 solids of 3 oscillators and 4 quanta . Ok. Now say we take two such “solids” and place them in “thermal” contact, i.e., allow that 4 quanta of energy can flow between the two. . Macrostate: how much energy each ‘solid’ has, q1 and q2. . Microstate: how that energy is distributed among the individual oscillators. . Let’s tabulate the possibilities and the corresponding multiplicities.
Demo: PowerPoint Visuals of Multiplicitie
Constraints: Qtot = 4 NA = 3, NB = 3
Possibilities & Multiplicities qA qB A(NA-1,qA) B(NB-1,qB) A&B = A× B 0 4 1 15 15 1 3 3 10 30 2 2 6 6 36 3 1 10 3 30 4 0 15 1 15 Total possible microstates: 126
o Multiplicities multiply . Ex. If I have 10 shirts and 10 pants, then there would be 100 possible outfits (all be it, some are really atrocious). . So, if there are 6 ways of arranging 2 quanta in solid A and 6 ways of arranging 2 quanta in solid B, then there are a total of 6×6=36 ways of doing both. o Multiplicity Plot Phys 344 Lecture 5 Jan. 16th 2009 3
40
30
20 A B 10 A&B
0 0 1 2 3 4 =qA 4 3 2 1 0 =qB . Noteworthy: The whole system’s multiplicity is maximized when the multiplicities of the two solid’s are balanced, not individually maximized. o Multiplicities and Probabilities . If the microstates that we have defined are indeed each equally probable, then the probability of a macrostate is proportional to the number of microstates compatible with it. . So, the macrostate with the greatest multiplicity is the most probable and the macrostate with the smallest multiplicity is the least probable. . Ex. 36 out of 126 possible microstates correspond to an equal split of energy among equal “solids,” or 36/126 = 29% of the time or a probability of 0.29 that when I look at the system I find the energy evenly split. o Peak width. Though there clearly is a peak, and it’s where we would intuitively imagine it to be, it isn’t very sharp – only 29% of the time is the energy evenly split. 15% of the time, atom 1 has all the energy. This is a fairly broad peak. o Equilibrium & Probability. Now, recall, we said that two objects in thermal contact exchange energy until they achieve thermal equilibrium. Looking at this system, energy can be swapped back and forth and the system will generally tend toward the most probable state, though with only a 29% chance, it is far form inevitable that the system will be there. We can identify the most probable state with that of equilibrium. Phys 344 Lecture 5 Jan. 16th 2009 4
You will be asked in the homework to use Excel to tabulate the possibilities for the same system with 6 quanta of energy. While it could be done by hand, later we’ll be considering much larger systems, and those, you don’t want to do by hand. Here’s a start. At a given instant our total system looks like this
Set up for Problems 9 and 10: Two 3-oscillator solids share 6 quanta. Model with Excell.
A B C D E F
total 1 Two Einstein Solids 2 NA= 3 NB= 3 q_total= 6 3 qA A qB B Total 4 0 1 6 28 28 5 1 3 5 21 63 6 2 6 4 15 90
7 3 10 3 10 100 qA
8 4 15 2 6 90 6 6 1 9 5 21 1 3 63 total possible = 462= 10 6 28 0 1 28 6
What math is executed in cell B4?
q A N A 1 (q A N A 1)! N A , q A q A q A!(N A 1)! where NA is 3 (found in cell B2) and qA is 0 (found in cell A4).
What must be the code for cell B4? =COMBIN($B$2-1+A4,A4)
What math is executed in cell C4? Calculates the number of ways of distributing 0 energy units in body A AND 6 in body B. It should be the product of the number of ways of doing each individually:
N A , q A N B , qB
What must be the code for cell C4? =B4*D4
Phys 344 Lecture 5 Jan. 16th 2009 5 o Average / Distribution for large sets . Increasing the numbers of members and quanta by two orders of magnitude has a significant effect on the sharpness of the peak.
V. Lab 5.2 Two Einstein Solids. In StatMech.exe, start with Na = 1 (one atom, thus three oscillators), Nb=1, and q=6. See the same table, same plot. Increase particles & offset (Na>Nb) and quantum # to Na 300, Nb 200,q 100. See a much stronger peak
o Irreversibility . Scenario Imagine the following, in a system of 300 members of A, 200 members of B, and 100 energy quanta, the most probable state (60 quanta in A) is 1033 times more probable than is the least probable state (no energy in solid A). Say then that you started with the system in this most probable state, then checked up on it periodically hoping to find no more than 10 quanta in A – almost all the energy spontaneously shifted to B. Collectively, these states have a probability around 10-20, or you’d have to check about 1020 times, or 100 times a second for the lifetime of the universe, to stand a good chance of finding it in such a state. Then again, if you started with the system in such a state – say B got heated, then touched A, it wouldn’t be long before the energy redistributed itself, and though only 7% of the time you’d find it split 60 – 40, the vast majority of the time you’d find it near this. . Conclusion It is extremely likely that a system will progress from any initial state to the vicinity of its most probable state, but it is prohibitively improbable that it will progress away – the approach of equilibrium is ‘irreversible.’ V. Lab 4.1 Conduction(equilib.exe)– just play around & see how the probabilities play out. Note, that on a short time scale, not all distributions are equally probable, owing to the time it takes for energy to be conducted from one local to the next. Also note that the energy flows into the macrostate with the greatest number of microstates: evenly distributed.
Heating is a probabilistic phenomenon. Energy flows from hotter to cooler until equilibrium is reached because equilibrium is the most probable state, not because any specific mechanism drives or requires it. . This observation is of fundamental importance in thermodynamics. It is the physics content of what’s known as the Second Law of Thermodynamics. This law actually gets phrased many different ways, depending on the application, but the underlying content is always the same. Second Law of Thermodynamics: the spontaneous flow of energy stops when a system is very near its most likely macrostate, that is, the one with the greatest multiplicity. i.e. Heat flow maximizes multiplicity. Phys 344 Lecture 5 Jan. 16th 2009 6 Temperature: Recall that the book said Temperature quantified the tendency of bodies to spontaneously give up energy. Temperature, energy change, and multiplicity change are inextricably enter twined.
Summing up Thermodynamic systems are characterized by macroscopic variables, i.e., we can determine their macrostates; however the fundamental physics is down on the microscopic level and determines the microstates. If we assume that all microstates are equally probable, then the probability of a macrostate depends simply on the number of microstates which it encompasses – its multiplicity, and the total number of possible microstates consistent with whatever constraints we have on our system. For example, last time, we considered two Einstein solids in thermal contact. We constrained the system by saying there’s only so much energy to go around, qtotal units. We then counted the multiplicities of each macrostate (solid A has all the energy, solid B has all the energy,…) and thus determined the probability of each state. Irriversibility…
If you’ve ever encountered the 2nd Law of Thermodynamics before, it was probably in terms of temperature or entropy. So we’ve got a little work ahead of us relating multiplicities to these properties. The first step, which we’ll focus on today, is extending our multiplicities to easily handle systems of realistic sizes (huge numbers of particles). Agenda suggestions 1. So, what was important from today’s reading? What do you get asked to do in the homework? If they don’t know, take a moment to let them look over the homework questions. 2. Part of what we’ll do today is see quantitatively where irreversibility comes form – why systems evolve in one direction and not another, and when that behavior should and shouldn’t be expected. We already have a qualitative sense, but this is physics, so we have to back it up quantitatively.
StatMech.exe Through this intro, have them look at small, medium, and large systems.
2.4 Large Systems Very Small. To construct our statistical tools, we first considered a very small system: three coins, with two possible states each. This was few enough that we Phys 344 Lecture 5 Jan. 16th 2009 7 could count micro and macrostates by hand and thus directly determine the multiplicities. Small. We developed the tools so that we could calculate, rather than count, the multiplicities. This allowed us to get quickly through, say 4 oscillators with infinite states but only 3 energy units to share. o Tell the program that you want each Einstein solid to have 3 members and there to be 6 units of energy. Note the breadth of the peak. Medium. In the homework, you asked a computer to calculate the multiplicities for a system of 200 or so oscillators. This could be extended to 500, 1000, 10,000, 100,000, 1,000,000 oscillators. o Tell the program that you want each Einstein solid to have 30 members and there to be 60 units of energy. Note the breadth of the peak. o Tell the program that you want each Einstein solid to have 300 members and there to be 600 units of energy. Note the breadth of the peak. large. But what about 1010, 1023, you know, the actual number of particles in a typical thermodynamic system? It won’t do to ask a computer to handle this many particles. Today, we’ll evolve our tools to handle them. The main thrust will be approximating our factorials (well defined, but too discrete for doing much math with) in terms of more analytical functions – ones that can be integrated rather than summed. o Pay off. You can fairly imagine, as we consider more and more members in our systems, the multiplicities, and thus probabilities, get more and more sharply defined. When we consider a very large number of members, we get such a sharp peak that the few states it indicates, while not inevitable, are terribly probable. For example, we can say, with no fear of contradiction, that two identical solids, brought into thermal contact, will come to share equal amounts of energy – that state (and it’s near neighbors) is overwhelmingly more probable than any other. 2.4.1 Very Large Numbers Very Large Numbers o If you have a system of a large number of particles, the multiplicities 23 become very large, as in 10 10 ; and that’s huge. Logs and Very Large Numbers o A simple device to make such numbers a tad more manageable is to take their log. We’ll be doing this a bit, so it’s worth remembering and confirming the basic properties of the natural log. So off we go into Log- Math-Land for a little while.
Phys 344 Lecture 5 Jan. 16th 2009 8 Example: 2.12 The natural logarithm function, ln is defined so that elnx = x for any positive number x. a. Sketch a graph of the natural logarithm function. A few particular values can be found by asking “e raised to the what gives Do this 1?”: ln (1) = 0, “e raised to the what gives e?”: ln (e ) = 1, “e raised to the one what gives 0?”: ln (0) = - . Plotting these out then gives
1
x 0 x
ln 1 2 3
b Just b. Prove the identities ln(ab) = ln(a) + ln(b) and ln(a ) = b ln(a). remind Appealing to the defining relation, eln ab) ab , but a e ln(a) and them of b eln b , so ab eln a eln b eln a ln b this identity b b Similarly, eln a ab eln a e ln a b
d 1 c. Prove that ln x . dx x Just Again, we’ll appeal to the defining relation eln x x remind d d them of e ln x x dx dx this d d df (x) identity ln x e ln x 1 using e f (x) e f (x) dx dx dx d 1 1 ln x dx e ln x x Where the last step again appeals to the definition.
d. Derive the useful approximation ln(1 x) x for |x| << 1 This approximation will of course only be good for small x. Let’s look at Actually go a plot of this function for small x over this one F(x) = x 1 F(x) = ln(1+x)
0 x 1 2 3
Phys 344 Lecture 5 Jan. 16th 2009 9 df In the spirit of a Taylor series, f (x) f (0) x ... dx x 0 at x = 0, the function = 0, approximating it with a straight line in that vicinity, with intercept 0 and the same slope as our function: d 1 ln(1 x) 1, so F(x) = ln(x+1) is approximated by F(x) = x: dx 0 1 x 0 ln(x 1) x for small x.
2.4.2 Stirling’s Approximation The reason we get very large numbers is that in calculating multiplicities, we take the factorials of large numbers. N N For Large N, N! 2 N or, just a little less precise, but fine for large N, e N N N! e Why Bother? o Calculators will balk at N! for large enough N. o More importantly, the relationship on the right hand side, though messier, is continuous instead of discrete – you can take its derivative, integrate it, or do whatever math you desire to it. How Good? o For N = 1 , there’s 7.7% error (and that’s a decidedly small number) o For N = 10 there’s 0.81% error (note: the difference between the numbers is merely Medium sized, while the numbers themselves are rather Large, so % error is small) o For N = 100 there’s only 0.083% error o By the time we’re actually using large numbers for N, the error is a negligible percent.
Why True? o Roughly: B.3 Stirling’s Approximation Derivation 1. n! n n 1 (n 2) (n 3) ... 1 o ln n! ln n n 1 (n 2) (n 3) ... 1 PowerPoint x n x n , or can ln n! ln n ln(n 1) ln(n 2) ... ln(1) ln(x) ln(x) x they do? x 1 x 1 . The last step is a free-be because x = 1 (the size of each step in x) x n x n o ln n! lim ln(x) x ln(x)dx x ln x x n n ln n n (1ln 1 1) n ln n n 1 x 0 1 x 1 x 1
. The first approximation is because we’re pretending that a demonstrably discrete sum is a continuous one, i.e., x is much smaller than x. If n is large, then that’s true for most of the range of the integral. o Error Phys 344 Lecture 5 Jan. 16th 2009 10 . Graphically, the difference is between integrating the area under this smooth curve and adding the area in these discrete boxes. lnx
x
. Now for another approximation. Again, in the case that n is fairly large, the 1 is negligible, so we’ll neglect it. ln n! nln n n 1 nln n n n o n n n! e n ln n n eln n n n n e n e o More Precisely . You can choose better limits on the integral and get a better approximation (homework B.10) . Derivation 2 in Appendix B.3. This gives you the factor of 2 n (to fully appreciate it, you may need to look at Appendix B.1 also) Phys 344 Lecture 5 Jan. 16th 2009 11
2.4.3 Multiplicity of a Large Einstein Solid Now let’s put this approximation to use in a very large Einstein solid. N 1 q ! o The exact formula: N, q N 1 ! q ! o First approximation N q ! . N >> 1 so N, q N ! q ! o Apply Stirling’s N q N q N q 1 N q N q ! e N q N q N q . N,q e N ! q ! N q N q 1 1 N q N q N q eN eq N q e e “High-Temperature” Limit: N << q (enough energy available that the average particle has much more than just one unit – well above ground state) eq N o The book arrived at N, q q>>N high.T N o You do something very similar on Homework 2.19.
“Low-Temperature: Limit: N >> q (not nearly enough energy to go around, the vast majority of particles are down in the ground state, without an additional quantum of energy.) o Picking up where the general case left off” q N q N e q N q N q N N,q q q N N q N Based on q e N e q N their notes, Since q<
Phys 344 Lecture 5 Jan. 16th 2009 13 2.4.4 Multiplicity of a Large Einstein Solid “High-Temperature” Limit: N << q (enough energy available that the average particle has much more than just one unit – well above ground state) eq N o The book arrived at N, q q>>N high.T N “Low-Temperature” Limit: N>>q (most oscillators are in the ground state) q eN o N,q low.T q
This Time Put main topics and headings on board What you’ll be expected to do on homework: . Pr. 22 Alternative approach for estimating multiplicity peak width . Pr. 26 Find Multiplicity for 2-D Ideal Gas
max Width
q/2
Thermodynamic Limit: When the science of thermodynamics is absolutely valid. That science treats systems as if energies, pressures, volumes, temperatures, particle counts, etc. don’t randomly fluctuate. It would really screw things up if half the air particles in the room gave all their kinetic energy to the other half. We’d seen for Einstein solids of just one or two atoms, it wasn’t at all unlikely that this kind of thing would happen. As we added more and more atoms to the solids, that kind of thing became less and less probable. For really large solids, that kind of thing is negligibly improbable. In the thermodynamic limit, none of those shenanigans happen, the most probable state is the only state.
Of course, that is just a limit, an ideal never achieved. To know whether or not we can get away with approximating a real system as being in this limit, whether or not the system can be successfully modeled with the tools of thermodynamics, it’s therefore important to assess the error of the approximation – how wide is the peak that we’re approximating as having no width?
2.4.5 Sharpness of the Multiplicity Function According to the Fundamental assumption of statistical mechanics: In an isolated system in thermal equilibrium, all accessible microstates are equally probable. Phys 344 Lecture 5 Jan. 16th 2009 14 So, we have been finding how many of these equally probable microstates are associated with individual macrostates – thus how probable those macrostates are. Now that we know how to express the probability of an individual macrostate, we can plot out the distribution of multiplicities // probabilities. The reason we bother is that we’ve got something at stake, essentially all of thermodynamics, on there being a very sharply peaked distribution – there being one narrow window of macrostates that is far more popular than all the rest, allowing us to, to first order, pretend that the system’s always in that single most popular state – always has pressure P, energy E, particle density N/V,… Even though all the air on the left side of the room is free to freeze while all that is free to heat up, we can say instead that the energy essentially must be evenly distributed, the temperature must be uniform. Thermodynamic Limit. o With the two Einstein solids, you saw how the probability distribution got progressively tighter with each larger system. We will now quantify that relationship between N and the width of the peak. In the limit that N is absolutely huge, the peak becomes, for all intents a single spike of no width – there is no likelihood of the system perceptibly varying from the most likely state. This is the Thermodynamic limit. Derivation See PowerPoint o Consider two interacting Einstein solids with very many members, each with N. In the hot limit, more than enough energy to go around, q >> N . Multiplicity: N N eq eq e 2N (q , q ) q q A B (q q ) N A B A B N N N A B
Or in terms of q = qA + qB, e 2N . q q q N N A A . What’s most probable qA? By maximizing (take the derivative and set equal to 0), or by symmetry arguments, it’s clear that this is maximized at q q q q , i.e., with ½ the energy in each solid. A 2 B 2 . How high? Plugging this back into the multiplicity relation says that e 2N q 2N the peaked multiplicity is . max N 2 . How Wide? A typical measure of the width of a peak is its “full width at ½ max.” We’ll call the full width q, then the value at which the Ask them to do much of q q curve drops to ½ max is q A . Setting the this. Perhaps tell them to 2 2 get to 1-… and then use multiplicity equal to ½ that of the peak and substituting this ln(1+x) ~ x. in for qA, we can solve for q – the width of the peak. Phys 344 Lecture 5 Jan. 16th 2009 15 e 2N q q q N 1 N A A 2 max N e 2N q q q q 1 eq 2N q N 2 2 2 2 2 N2 N q q q q 1 q 2N 2 2 2 2 2 2
2 2 1 2 q q 1 N q 2 2 2 2 2 1 q 1 N 1 2q 2
2 1 q 1 N ln 1 ln 2q 2
2 q 1 ln(2) 2q N
q ln 2 2 q N 1 o Classic result The bigger the system, the narrower the peak. N You’ll get this result for any system with random fluctuations around a mean, for example a set of data from a single experiment that has been repeated and repeated. o Example numbers . Say we have 100 particles per solid, then the fractional width is q 0.166: 17% q . Say we have 10,000, then it’s 0.0166: 1.7% . Say we have 1023 (one mole), then it’s only 5.3×10-12: 5.3×10-10% That’s getting near our uncertainty in the values of some fundamental constants! o Conclusion . Specifically: if you have as much as two moles of Einstein solids interacting, you won’t see the energy anyway but evenly distributed. Generally: For a large enough system, there exists a macrostate which is so much more probable than any other perceptibly different state, that we can ignore fluctuations. When this is the case, we can blissfully apply the laws of thermodynamics with never a care that our values may fluctuate: This is the thermodynamic limit.