P340 Lecture 3 “Probability and Statistics”

1. P340 Lecture 3“Probability and Statistics” • Bernoulli Trials • REPEATED, IDENTICAL, INDEPENDENT, RANDOM trials for which there are two outcomes (“success” prob.=p; “failure” prob=q) • EXAMPLES: Coin tosses, queuing problems, radioactive decay, scattering experiments, 1-D random walk,… • A question of fundamental importance in these problems is: Suppose I have N such trials, what is the probability that I have “n” successes? The answer is: P(n) = N!/[n!(N-n)!] pn(1-p)N-n We will spend a bit of time looking at this distribution.

2. Other Probability distributions(Poisson distributions) P(n)=(ln e-l)/n! m=l (mean) s=l1/2 (st. dev.) This is very important in counting expts. http://en.wikipedia.org/wiki/Poisson_distribution

3. Meaning of m, s(Gaussian distributions) http://en.wikipedia.org/wiki/Normal_distribution

4. P340 Lecture 4 Examples Consider a simple model for thin film growth via vapor-phase deposition, in which atoms arrive randomly on a surface and stay where they hit. Model the surface as a simple checkerboard with squares of atomic dimension, and assume that any atom that hits within a square may be considered to occupy that square. After depositing six atomic layers worth of atoms, what fraction of the surface is still bare? What fraction has 6 atoms? What fraction has 3 atoms? In an experiment out at IUCF, I am interested in determining the radioactivity of a certain piece of gold. When placed next to the detector, the gold produces a count rate of roughly 1 count every 3 seconds in the relevant part of the spectrum. For how long should I count to determine the activity level to a precision of 2%? (you may assume that the detector efficiency and experimental geometry are known sufficiently precisely that only fluctuations in the recorded number of counts contributes to the uncertainty in the activity).

5. P340 Lecture 5(Second Law) • THE FUNDAMENTAL POSTULATE • (OF THERMAL PHYSICS) • Any isolated system in thermodynamic equilibrium is equally likely to be in any one of its available microstates • THE SECOND LAW OF THERMODYNAMICS • Any isolated thermodynamic system will, with overwhelming probability, evolve into the macrostate of largest multiplicity (consistent with the constraints imposed on the system) and subsequently will remain in that state.

6. Multiplicity and volume II I Fixed wall Thermal barrier Consider two volumes isolated from the rest of the universe by barriers to particle or heat exchange, and from each other by a thin membrane. Suppose initially one of them is filled with gas (at some temperature and pressure) and the other is empty. What happens to the number of microstates available to the system (i.e. the multiplicity) if the barrier is removed (or say it breaks)?

7. Thermal interactions II I Wall with a thermal conductivity that is changed from zero to non-zero Thermal barrier Consider two systems that are isolated from the universe, and each other, but then we allow them to exchange heat (i.e. energy exchange with no work done by either one on the other). What does the 2nd law tell us about this situation, and how might we analyse the relevant physics quantitatively?

8. Thermal interactions II I Wall with a thermal conductivity that is changed from zero to non-zero Thermal barrier Consider two systems that are isolated from the universe, and each other, but then we allow them to exchange heat (i.e. energy exchange with no work done by either one on the other). What does the 2nd law tell us about this situation, and how might we analyse the relevant physics quantitatively? The system evolves to the macrostate (i.e. division of energy between the two systems, since that is all we are allowing to change) of maximum multiplicity and stays there. (dWtot/dEI)V,E,VI= 0; what is the condition on the individual subsystems that assures this mathematical result?