QuantumInfo.StatMech.IdealGas

Let $n \in \mathbb{N}$ be the number of particles and $V \in \mathbb{R}_{>0}$ be the volume of a cubic box. The phase space is defined by the dimension $6n$ (representing three spatial components and three momentum components for each particle). For a configuration in phase space consisting of positions $q_{i, ax}$ and momenta $p_{i, ax}$ (where $i \in \{1, \dots, n\}$ and $ax \in \{1, 2, 3\}$), the Hamiltonian $H$ is defined as follows: If all particles are located within the cube of side-length $L = V^{1/3}$ centered at the origin, such that $|q_{i, ax}| \leq \frac{V^{1/3}}{2}$ for all $i$ and $ax$, the energy is the sum of the kinetic energies $\sum_{i=1}^{n} \sum_{ax=1}^{3} \frac{p_{i, ax}^2}{2}$ (assuming normalized mass $m=1$). If any particle is outside this volume, the energy is infinite ($\infty$).

For an ideal gas consisting of $n$ particles in a volume $V$ at an inverse temperature $\beta$, if $V > 0$ and $\beta > 0$, then the partition function $Z$ is given by $$Z(n, V, \beta) = V^n \left( \frac{2\pi}{\beta} \right)^{\frac{3n}{2}}.$$

For an ideal gas consisting of $n$ particles in a volume $V$ at temperature $T$, given that $V > 0$ and $T > 0$, the Helmholtz free energy $A$ is given by: $$A(n, V, T) = -n T \left( \log V + \frac{3}{2} \log (2 \pi T) \right)$$ where $\log$ denotes the natural logarithm.

For an ideal gas consisting of $n$ particles in a volume $V$ at inverse temperature $\beta$, if $V > 0$ and $\beta > 0$, then the partition function integral is convergent (i.e., the system is $Z$-integrable).

For an ideal gas with volume $V > 0$ and temperature $T > 0$, let $P$ be the pressure of the gas at state $(n, V)$ and temperature $T$, where $n$ is the number of particles. Given the gas constant $R = 1$ in the dimensionless unit system, the ideal gas law holds: $PV = nRT$.