Transcript Document

TA office hours:
Switch to Thursdays 5 – 6 PM.
Homework # 3 assigned today.
Due next wed.
The Canonical (Boltzmann) probability distribution
“If we know the temperature of a system and the values of its external
parameters, how can we estimate its physical properties, such as
energy, pressure, magnetic moment, and distribution of molecular
velocities? The question is answered {..} by deriving the canonical
probability distribution…”
R. Baierlein, Thermal Physics
Meaning of probabilities in thermal physics
Probabilities enter into thermal physics because the available
data are insufficient to determine the individual properties of
1020 molecules. (Or, because of quantum mechanics, even the
individual properties of a single molecule are expressed in terms
of probabilities.) Even if all necessary data were available, no
person or computer could cope with it.
Meaning of probabilities in thermal physics
Probabilities enter into thermal physics because the available
data are insufficient to determine the individual properties of
1020 molecules. (Or, because of quantum mechanics, even the
individual properties of a single molecule are expressed in terms
of probabilities.) Even if all necessary data were available, no
person or computer could cope with it.
Probabilities always arise in a context. For example, the probability
of 2 appearing if we role a die once is 1/6.
Meaning of probabilities in thermal physics
Probabilities enter into thermal physics because the available
data are insufficient to determine the individual properties of
1020 molecules. (Or, because of quantum mechanics, even the
individual properties of a single molecule are expressed in terms
of probabilities.) Even if all necessary data were available, no
person or computer could cope with it.
Probabilities always arise in a context. For example, the probability
of 2 appearing if we role a die once is 1/6.
Two schools of thought on what probability means in thermal
physics.
Meaning of probability
1. Frequency meaning. A probability is a relative frequency in
the long run, i.e.
probability = (number of successes)/(number of tries)
OR,
probability = (number of objects with property X)/ (total number of objects)
e.g. objects = 1020 gas of sodium atoms
X = Na atom in first electronic excited state
if this is true for 1016 atoms:
Probability = 1016/1020 = 10-4
Meaning of probability
2. Degree of belief meaning. Probability is the rational degree
of belief in the correctness of proposition A given the context B.
For example:
If
A = “The first sodium atom we examine will be in the first
excited electronic state.”
and
B = There exist 1020 Na gas atoms, 1016 of which are in the first
excited state.
Then, the rational degree of belief we can assign this statement A,
given the context B, is 10-4.
Why does this matter? (It does and it doesn’t.)
“In thermal physics one often wants to estimate the behavior
of a specific system, e.g. a lithium fluoride crystal grown last
week in the lab which resides in the core of the labs only
superconducting magnet. The degree of belief interpretation
permits ‘one of a kind’ applications of probability theory.
The frequency interpretation would require that one must
imagine a large number of replicas of last week’s crystal
and consider a relative frequency in that collection, or doing
the same experiment on the same crystal again and again.
Some physicists find such constructions artificial.”
“Almost all calculations in thermal physics come out the same,
Regardless of which interpretation of probability one adopts.”
R. Baierlein
Probabilities when the temperature is fixed
How do we describe, in probabilistic terms, a system whose
temperature is fixed?
An example system from low temperature physics:
Copper
disk
Cerium magnesium
nitrate
Cu cooled with liquid He which was pumped on for additional
Evaporative cooling until all He evaporated. All isolated from enviroment
In a cryostate (double walled vessel).