1.9 KiB
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.