These notes were written for the class PHY 801 at Michigan State University in Spring 2026.
These notes are very closely related to Statistical Mechanics. However, there are some notable notation differences between the two texts. Some of these differences include: total energy $U$ instead of $E$, multiplicity function $W$ instead of $\Omega$ and many others. The texts are self consistent, but cross comparison should be done carefully.
Definition 1.1.1 Energy denoted $U$ with SI unit Joules (J) is a conserved quantity that is transferred between systems by work and heat.
Definition 1.1.2 Work is energy transferred to a system by macroscopic forces.
Definition 1.1.3 Heat is energy transferred to a system by microscopic forces.
Definition 1.1.4 The multiplicity function $W$ of a system is the number of possible microstates for a given macrostate.
Definition 1.1.5 The Boltzmann constant denoted $k_B$ is the the proportionality factor fixed at exactly $k_B=1.380649\times 10^{-23}\text{J}/\text{K}$ that defines temperature as it is related to the statistical probability of energy states in a system.
Definition 1.1.6 The entropy $S$ of a system is the Boltzmann constant times the natural log of the multiplicity function.
\[S = k_B\log W\]
Definition 1.1.7 The temperature $T$ in units of kelvin (K) and thermodynamic temperature $\beta$ in units of joules (J) of a system are defined in terms of the derivative of energy $U$ with respect to entropy $S$.
\[T = \frac{\partial U}{\partial S} = \frac{1}{k_B\beta},\quad\beta = \frac{1}{k_B}\frac{\partial S}{\partial U} = \frac{1}{k_B T}\]
Definition 1.1.8 The microcanonical ensemble is the ensemble of statistical mechanics where the macrostates are described by the total energy $U$, volumes $\mathbf{V}$ and particle numbers $\mathbf{N}$. The probability of each possible microstate $\mathscr{p}_i$ is assumed to be the same, so it is simply one over the multiplicity function $W$, which can be written exactly as the number of states that match the macrostate. The canonical ensemble and the grand canonical ensemble can be derived by considering a system inside a large reservoir in the microcanonical ensemble.
\[\mathscr{p}_i = \frac{1}{W(U,\mathbf{V},\mathbf{N})}\]
Definition 1.1.9 The canonical ensemble is the ensemble of statistical mechanics where the macrostates are described by the temperature $T$, volumes $\mathbf{V}$ and particles numbers $\mathbf{N}$. The probability of a particular microstate $i$ is written in terms of the energy of the microstate $E_i$, the thermodynamic temperature $\beta$ and the partition function $z$.
\[\mathscr{p}_i = \frac{1}{W(T,\mathbf{V},\mathbf{N})}=\frac{e^{-\beta E_i}}{\sum_{j}{e^{-\beta E_j}}} = \frac{e^{-\beta E_i}}{z}\]\[z = \sum_{j}{e^{-\beta E_j}} = \sum_{j}{e^{-E_j/(k_BT)}}\]
Definition 1.1.10 The grand-canonical ensemble is the ensemble of statistical mechanics where the macrostates are described by the temperature $T$, volumes $\mathbf{V}$, and chemical potentials $\mathbf{\mu}$. The probability of a particular microstate $i$ is written in terms of the energy of the microstate $E_i$, the particle numbers of the microstate $\mathbf{N}_i$, the thermodynamic temperature $\beta$, the chemical potentials $\mathbf{\mu}$ and the grand partition function $\mathscr{z}$.
\[\mathscr{p}_i = \frac{1}{W(\mathbf{T},\mathbf{V},\mathbf{\mu})}=\frac{e^{-\beta(E_i-\mathbf{\mu}\cdot\mathbf{N}_i)}}{\sum_\mathbf{N}{\sum_j{e^{-\beta(E_j-\mathbf{\mu}\cdot\mathbf{N})}}}}=\frac{e^{-\beta(E_i-\mathbf{\mu}\cdot\mathbf{N}_i)}}{\mathscr{z}}\]\[\mathscr{z} = \sum_\mathbf{N}{\sum_j{e^{-\beta(E_j-\mathbf{\mu}\cdot\mathbf{N})}}}\]
Definition 1.2.1 Energy denoted $U$ with SI unit Joules (J) is a conserved quantity that is transferred between systems by work and heat.
Definition 1.2.2 The heat capacity denoted $C$ is the derivative of total energy $U$ in terms of temperature $T$ of a system.
\[C = \frac{\partial U}{\partial T}\]
Definition 1.2.3 The heat capacity at constant pressure denoted $C_P$ is the derivative of total energy $U$ in terms of temperature $T$ of a system while pressure $P$ is held constant.
\[C_P = \left(\frac{\partial U}{\partial T}\right)_P\]
Definition 1.2.4 The heat capacity at constant volume denoted $C_V$ is the derivative of total energy $U$ in terms of temperature $T$ of a system while pressure $V$ is held constant.
\[C_V = \left(\frac{\partial U}{\partial T}\right)_V\]
Definition 1.2.5 The coefficient of thermal expansion denoted $\alpha$ is the derivative of volume $V$ in terms of temperature $T$ of a system while pressure $P$ is held constant divided by the volume of the system.
\[\alpha = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P>0\]
Definition 1.2.6 The isothermal compressibility denoted $\kappa$ is the negative derivative of volume $V$ in terms of pressure $P$ of a system while temperature $T$ is held constant divided by the volume of the system.
\[\kappa = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T>0\]
Result 1.2.7 Mayer's relation states that the heat capacity at constant pressure $C_P$ and heat capacity at constant volume $C_V$ must differ by a positive nonzero value determined by the temperature $T$, volume $V$, coefficient of thermal expansion $\alpha$ and isothermal compressibility $\kappa$ of the system.
\[C_P-C_V = \frac{TV\alpha^2}{\kappa}= -T\left(\frac{\partial V}{\partial T}\right)^2_P\left(\frac{\partial P}{\partial V}\right)_T>0\]
Definition 1.2.8 An intensive quantity is a variable of a system that does not scale with the size of the system.,
Definition 1.2.9 An extensive quantity is a variable of a system that does scale with the size of the system.
Definition 1.2.10 The specific heat denoted $c$ is the derivative of total energy $U$ in terms of temperature $T$ scaled per unit mass $M$ of the system such that it is an intensive quantity.
\[c = \frac{C}{M} = \frac{1}{M}\frac{\partial U}{\partial T}\]
Definition 1.2.11 The molar specific heat denoted $c_{mol}$ is the derivative of total energy $U$ in terms of temperature $T$ scaled to the number of moles $N_{mol}$ in the system such that it is an intensive quantity. It can also be defined in terms of the specific heat $c$ and molar mass $m_{mol}$.
\[c_{mol} = \frac{C}{N_{mol}} = \frac{c}{m_{mol}} = \frac{1}{N_{mol}}\frac{\partial U}{\partial T}\]
Definition 1.2.12
Result 1.2.13
Law 1.2.14 The Dulong Petit law states that the molar specific heat for most bulk materials is a constant at high temperatures.
\[C = N_{atoms}\frac{\partial u}{\partial T} = 3k_BN_{atoms}\]\[c_{mol}=3k_BN_{mol} = 3R\]
Definition 1.3.1 wao
Definition 1.4.1
Definition 1.5.1
Definition 1.5.2
Definition 1.5.3
Section 2.1
Section 2.2
Section 2.3
Section 2.4
Section 3.1
Section 3.2
Section 3.3
Section 3.4
Section 4.1
Section 4.2
Section 4.3
Section 5.1
Section 5.2
Section 5.3