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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.
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
Meaning of m, s
(Gaussian distributions)
http://en.wikipedia.org/wiki/Normal_distribution
P340 Lecture 4
Examples
1. 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?
2. 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).
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.
Multiplicity and volume
Thermal
barrier
I
II
Fixed wall
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)?
Thermal interactions
Thermal
barrier
I
II
Wall with a
thermal
conductivity that
is changed from
zero to nonzero
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?
Thermal interactions
Thermal
barrier
I
II
Wall with a
thermal
conductivity that
is changed from
zero to nonzero
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?