# Chapter 1 - The Second Law

## Section 1.1 - Two-State Systems

**Definition**. Given a system defined by a test statistic $XS$ and positive integer $N$, an ordered tuple with length $N$ and elements in the range of $X$ is known as a *microstate*. An unordered tuple with the same length $N$ and elements in the range of $X$ is known as a *macrostate*.

**Definition**. The *multiplicity* of a macrostate is the number of possible microstates that, when when written as an unordered tuple, produce said macrostate. In this work, we will define $\Omega(Macrostate) = Multiplicity$.

Note that if the test statistic $X$ is a uniform and discrete test statistic, the probability of generating a given macrostate $m$ can be written as $P(m) = \frac{\Omega{m}}{\sum_\M \Omega M}$.

**Recall**. From Statistics, $C(n, k) = \binom{n}{k}$, or *$n$ choose $k$*, is the number of unordered pairs of length $k$ that can be generated from a list of $n$ distinct elements.

**Definition**. A *paramagnet* is a material whose molecular magnetic moments do not align unless in the presence of an external magnetic field.

**Definition**. A *ferromagnet* is a material whose molecular magnetic moments will be aligned in the presence of an external magnetic field and retain their alignment in its absence.

**Definition**. The individual magnetic particles in a material are referred to as *dipoles*, as each contains a unique magnetic vector.

**Definition**. In a *two-state paramagnet*, when exposed to a magnetic field, each dipole may only be parallel or antiparallel to the applied field. We denote $N = N_\uparrow + N_\downarrow$ to represent the number of dipoles pointing up or down.

Assuming the external magnetic field points up, we note that an up-dipole contains less energy than a down-dipole. The total energy of a system is determined by $N_\uparrow$ and $N_\downarrow$, so the macrostate of this system can be used to determine the total energy.