1 Motivating the Boltzmann Factor In class we looked at a system of three particles with total energy E = 10u, under the assumption that the energy can be divided only into discrete (quantized) pieces of size u. That is, each particle can have energy Ei, equal to an integer multiple of u (including 0). We are interested in calculating the probability that a given particle, say the first one, has a particular energy P(E1 = nu) for 0 < n < 10. A further assumption is all specific configurations, or “microstates,” (e.g. E1 = 5u, E2 = 2u, E3 = 3u) are equally likely. In class we enumerated a few of these states and showed that P(E1 = 10u) < P(E1 = 9u) < P(E1 = 8u); that is P(E1) increases as E1 decreases. The key point is to recognize that, as E1 gets smaller, the system has more ways to distribute the remaining energy among the other particles and hence more microstates that it might choose.
1.a Three Particles Complete the three-particle calculation and make a plot of normalized probability P(E1 = nu) versus n
1.b Four Particles The result in (a) actually does not resemble very closely the exponential dependence expected from the Boltzmann factor. The exponential form arisses in the limit of large number of particles. To see if the trend is in the right direction, repeat the above calculation and plot P(E1 = nu) for a system of four particles with total energy E = 10u.
2 Nelson “Your Turn” 3K
2.a Air molecules Compare the average kinetic energy 3 2 kBTr of air molecules to the difference in gravitational potential energies ∆U at the top and the bottom of a room. Assume that the height of the ceiling is z = 3m. Why doesn’t the air in the room fall to the floor? What could you do to make it fall?
2.b Dirt Repeat
(a), but this time calculate the appropriate energies for a dirt particle. Suppose that the particle weighs about as much as a 50µm cube of water. Why does dirt fall to the floor?
3 Nelson “Your Turn” 3L
Find the most probable value of the speed u. Find the mean speed hui. Looking at the graph you drew in Your Turn 3F (or the related function in figure 3.3), explain geometrically why these are/aren’t the same
4 Thickness of the Atmosphere
In problem 2, you showed that the difference in gravitational potential energy of an air molecule near the floor of a room versus near the ceiling is negligible. However, due to the Boltzmann factor, larger differences in height are important. Taking the kinetic energy of air molecules to be indepednent of elevation (i.e. taking the temperature to be independent of elevation) so that the difference in average energy between two molecules at different heights is simply their difference in gravitational potential energy, find the elevation at which the probability of finding an air molecule, and hence the density of air molecules, is 1/e that at sea level. How does this elevation compare with the thickness of the atmosphere?
5 Nelson “Your Turn” 4A
You can’t do the preceding calculation on a calculator. You could do it with a compter-algebnra package, but now is a good time to learn a handy tool: Stirling 6 formula gives an approximation for the factorial M! of a large number M as
ln M! ≈ M ln M − M + 1 2 ln(2πM)
Work out for yourself the result of P0 just quoted, using this formula.
6 Application of the Boltzmann Factor
The nuclei of atoms have a magnetic dipole moment, or “spin”. Manipulation of these spins within an external magnetic field is the basis of magnetic resonance imaging (MRI). Assume the dipole moments of a set of nuclei can point in only one of two directions, either parallel to the external field B (“spin up”) or antiparallel to B (“spin down”). The energy of a nucleus with spin up is −µB and with spin down is +µB.
6.a Ratio Find an expression for the ratio of the number of nuclei with spin up to the number with spin down at temperature T.
6.b MRI Machine The field strength in an MRI machine is typically of order B = 1Tesla. Taking µ = 1.4 ∗ 10−26J/Tesla, find the temperature required to make 90% of the nuclei in a sample spin up.
6.c Room Temperature Using the numbers in (b), what is the fraction of nuclei aligned spin up by an MRI machine operating at room temperature?
7 Nelson 4.2 The genome of the HIV-1 virus, like any genome, is a string of “letters” (basepairs) in an “alphabet” containing only four letters. The message for HIV is rather short, just n ≈ 104 letters in all. Because any of the letters can mutate to any of the three other choies, there’s a total of 30000 posssible distinct one-letter mutations. In 1995, A. Perelson and D. Ho found that every day about 1010 new virus particles are formed in an asymptomatic HIV patient. They further estimated that about 1% of these virus particles proceed to infect new white blood cells. It was already known that the error rate in duplicating the HIV genome was about one error for every 3 ∗ 104 “letters” copied. Thus the number of newly infected white cells receiving a copy of the viral genome with one mutation is roughly
1010 ∗ 0.01 ∗ (104 /(3 ∗ 104 )) ≈ 3 ∗ 107 per day. This number is much larger than the total 30000 possible 1-letter mutations, so every possible mutation will be generated many times per day.
7.b Probability for Two Mutations You can work out the probability P2 that a given viral particle has two bases copied inaccurately from the previous generation by using the sum and product rules of probability. Let P = 1/(3 ∗ 104 ) be the probability that any given base is copied incorrectly. Then the probability of exactly two errors is P 2 , times the probability that the remaining 9998 letters don’t get copied inaccurately, times the number of distinct ways to choose which two letters get copied inaccurately. Find P2.
7.c Infections Per Day Find the expected number of two-letter mutant viruses infecting new white cells per day and compare to your answer to (a).
7.d Three-Letter Mutations Repeat (a-c) for three independent mutations.
7.e Antiviral Drug Suppose that an antiviral drug attacks some part of HIV but that the virus can evade the drug’s effects by making one particular, single-base mutation. According to the preceding information, the virus will very quickly stumble upon the right mutation - the drug isn’t effective for very long. Why do you suppose an effective HIV therapy involves a combination of three different antiviral drugs administered simultaneously?
8 Random Walk Consider a random walk along the x-direction. Each ∆t the walker makes a step by −L, 0, or L with equal probability. The position after N steps is denoted xN . There are no correlations between successive steps.
8.b Diffusion Coefficient
8.c Another Walk