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Phys 344 Lecture 5 Jan. 16 2009 1 10 Wells Oscillator.Py & Helix.Py

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Phys 344 Lecture 5 Jan. 16 2009 1 10 Wells Oscillator.Py & Helix.Py 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.
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