Quantum Mechanics governs the microscopic behavior of particles and
atoms and their interactions. The results of Classical Mechanics are
true only because they are the statistical averages over the quantum
behavior underlying the system.

Similarly, we can gain a better understanding of Thermodynamics and the
explanations for the macroscopic behavior of systems by first
understanding how systems behave on the microscopic, quantum level.

Before we look at the fundamental assumption of Thermodynamics, we
must first define a few terms that are crucial to understanding what it
says. The term closed system refers to a system that maintains a
constant number of particles, constant energy, constant volume, and is
free from any change in influences external to the system, such as an
oscillating magnetic field. A quantum state is the minimal
collection of information about a system that is maximally informative.
For example, in classical mechanics one need only specify the position
and momentum of a particle to fully describe its behavior for all time;
the collection of this data details the state of the system.

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The Fundamental Assumption

The Fundamental Assumption states that any closed system has an equal
probability to be in any of its possible quantum states.

The Fundamental Assumption is quite simple--there is no reason that a
system would prefer a given state over any other state, provided both
are possible. However, the statement is powerful in that we can now
count the states available to a system and subsequently make statements
about the probability of being in a particular state. We will
investigate this application through a quantum model of spins.

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Binary Systems

Let us suppose that we have a system consisting of N magnets, each of
which is localized and attached to a separate site. Each magnet has a
magnetic moment whose magnitude is *m*. Think of each magnet as a
vector of magnitude *m*. We won't focus on the details of the
Electromagnetism here but on the statistics that rule our system.