PHYS 359 Winter 2025 - Final Exam Review
Prof. Alan Jamison
1
, I. THE BIG PICTURE
Statistical mechanics studies the properties of systems with many parts. Problems involv-
ing one or two particles are common in other parts of physics. What we find in statistical
mechanics is that when the number of particles in a system becomes enormous, we care
little about what individual particles are doing. Instead, we are concerned with averaged
quantities (such as pressure or energy per particle). The averaging process becomes more
and more precise as the number of particles increases. We find ourselves in the situation that
we cannot possibly solve the microscopic dynamics for a system of 3 or more particles, but
we can calculate averaged quantities of interest that improve as we move to more particles.
The standard example is that of a gas. The motion of any individual gas particle would
be both intractable and uninteresting. However, if we can calculate the pressure and heat
capacity from the underlying microscopic physics, we have valuable tools for designing en-
gines, which was the original goal of thermodynamics. We’ve seen in this course that we
can understand the behaviours of white dwarves and astrophysical plasmas, magnetizations
and heat capacities in condensed matter systems, and blood oxygenation through the use of
statistical techniques.
The key formula for finding averages is
X
x̄ = x(s)p(s),
states,s
where p(s) is the probability of realizing state s. Recall that the probability of some event
A is the number of outcomes represented by A divided by the total number of possible
outcomes.
The starting point for understanding these systems with many particles is the consider-
ation of an isolated system.
A. The Microcanonical Ensemble
The microcanonical ensemble is the formal description for an isolated system. An isolated
system has no contact with the outside world, so all of its macroscopic variables (energy U ,
number N , volume V , etc.) are fixed. Given those fixed values, there will be a large number
of accessible states, i.e., states that match the fixed values of macroscopic variables.
2
Prof. Alan Jamison
1
, I. THE BIG PICTURE
Statistical mechanics studies the properties of systems with many parts. Problems involv-
ing one or two particles are common in other parts of physics. What we find in statistical
mechanics is that when the number of particles in a system becomes enormous, we care
little about what individual particles are doing. Instead, we are concerned with averaged
quantities (such as pressure or energy per particle). The averaging process becomes more
and more precise as the number of particles increases. We find ourselves in the situation that
we cannot possibly solve the microscopic dynamics for a system of 3 or more particles, but
we can calculate averaged quantities of interest that improve as we move to more particles.
The standard example is that of a gas. The motion of any individual gas particle would
be both intractable and uninteresting. However, if we can calculate the pressure and heat
capacity from the underlying microscopic physics, we have valuable tools for designing en-
gines, which was the original goal of thermodynamics. We’ve seen in this course that we
can understand the behaviours of white dwarves and astrophysical plasmas, magnetizations
and heat capacities in condensed matter systems, and blood oxygenation through the use of
statistical techniques.
The key formula for finding averages is
X
x̄ = x(s)p(s),
states,s
where p(s) is the probability of realizing state s. Recall that the probability of some event
A is the number of outcomes represented by A divided by the total number of possible
outcomes.
The starting point for understanding these systems with many particles is the consider-
ation of an isolated system.
A. The Microcanonical Ensemble
The microcanonical ensemble is the formal description for an isolated system. An isolated
system has no contact with the outside world, so all of its macroscopic variables (energy U ,
number N , volume V , etc.) are fixed. Given those fixed values, there will be a large number
of accessible states, i.e., states that match the fixed values of macroscopic variables.
2
