The study of thermodynamics is the study of systems that are too large
to understand by mechanics alone. For many years, thermodynamics was
understood vaguely, and many of the results had been determined
only experimentally. Some results posed great theoretical challenges
to physicists, who offered many unsuccessful attempts at explaining the
origins of the formulas.

With the advent of quantum mechanics came the explanations for the
results. The mechanics of individual particles is still too
complicated, however. For this reason, statistical physics plays a
significant role in the basis of thermodynamics. Instead of worrying
about the exact values of properties for each particle in a system, we
look at the average values statistically over quantum probabilities.
Even fundamental concepts like the energy of a system are derived as
averages.

New concepts arise as we talk about large systems, such as entropy
and temperature. Defining these carefully from quantum mechanics
allows us to make sense of the "3 Laws of Thermodynamics".

There is great symmetry in the structure of thermodynamics. The six
variables we look at repeatedly parallel each other in formulations of
the energy. We can use a mathematical tool known as the Legendre
Transform to posit alternate definitions of energy. This symmetry
allows us to derive numerous relationships between the variables, and
the multiple definitions of energy greatly simplify problem solving throughout
all of thermodynamics.

We can form the Partition Function as a measure of the total weighted
probabilities of the various states of a system, and relate this quantum
counting result to the energy of a the system. The spectrum of blackbody
radiation is derived directly from this counting. For systems in thermal and
diffusive contact with a reservoir, the Gibbs Sum replaces the Partition
Function.

With the few tools developed up to that point, the entire ideal gas problem
can be solved, including the derivation of expressions for all of the
interesting variables that describe the gas. In the non-classical regime, an
ideal gas behaves quite differently depending on the nature of its constituents
. A gas comprised of fermions exhibits a regime of total occupation and a
regime of zero occupation, while a gas comprised of bosons can form an
Einstein condensate by crowding into the ground orbital of the system.

Heat engines and other devices were the historical motivation for the
development of thermodynamics as a science. The devices can be well explained
using the framework already developed, and illustrative diagrams can be drawn
to make plain the energy and entropy flow involved. Real engines undergo
repeated cycles to achieve their purpose. We look at a simplified model known
as the Carnot cycle, and discuss different processes and how they relate to
the various energies defined.