28 May 2025

Classical Statistics

Ensembles, phase space, entropy, Maxwell-Boltzmann statistics, ideal-gas thermodynamics, Gibbs paradox, and equipartition.

bsc semester-vi statistical-mechanics classical-statistics partition-function

A macrostate specifies macroscopic constraints such as $N,V,E$; a microstate specifies every microscopic coordinate and momentum compatible with them. For $N$ classical particles, a microstate is a point in $6N$-dimensional phase space $(\mathbf q_1,\ldots,\mathbf q_N;\mathbf p_1,\ldots,\mathbf p_N)$. The dimensionless number of states in an element is counted as

\[d\Gamma=\frac{d^{3N}q\,d^{3N}p}{N!h^{3N}},\]

where $h^{3N}$ sets the elementary phase-space cell and $N!$ removes permutations of identical particles.

Ensembles and entropy

The microcanonical ensemble fixes $N,V,E$; the canonical ensemble fixes $N,V,T$ and permits energy exchange; the grand canonical ensemble fixes $V,T,\mu$ and permits energy and particle exchange. If a macrostate has multiplicity $\Omega$,

\[S=k_B\ln\Omega.\]

For independent systems, $\Omega_{AB}=\Omega_A\Omega_B$, hence $S_{AB}=S_A+S_B$.

In the canonical ensemble, maximizing total reservoir-plus-system entropy gives

\[p_r=\frac{e^{-\beta E_r}}{Z},\qquad Z=\sum_re^{-\beta E_r},\qquad \beta=\frac1{k_BT}.\]

Normalization determines $Z$. Differentiation yields

\[\frac{\partial\ln Z}{\partial\beta} =\frac1Z\sum_r(-E_r)e^{-\beta E_r} =-U,\]

so

\[U=-\frac{\partial\ln Z}{\partial\beta},\qquad F=-k_BT\ln Z,qquad S=-\left(\frac{\partial F}{\partial T}\right)_{V,N}.\]

Maxwell-Boltzmann distribution

Let $n_i$ particles occupy a one-particle level $\epsilon_i$ of degeneracy $g_i$. For distinguishable placements with $n_i\ll g_i$, the multiplicity is proportional to

\[W=N!\prod_i\frac{g_i^{n_i}}{n_i!}.\]

Using $\ln n!\simeq n\ln n-n$ and maximizing $\ln W$ subject to $\sum_i n_i=N$ and $\sum_i n_i\epsilon_i=E$ gives

\[\delta\left[\ln W-\alpha\sum_i n_i-\beta\sum_i n_i\epsilon_i\right]=0,\]

and therefore

\[\ln g_i-\ln n_i-\alpha-\beta\epsilon_i=0.\]

Thus

\[\boxed{n_i=g_ie^{-\alpha}e^{-\beta\epsilon_i}},\]

with $\beta=1/(k_BT)$ and $e^{-\alpha}$ fixed by $N$.

Ideal-gas partition function

For one nonrelativistic particle in volume $V$,

\[z_1=\frac{V}{h^3}\int e^{-\beta p^2/2m}d^3p =\frac{V}{h^3}\left(\int_{-\infty}^{\infty}e^{-\beta p_x^2/2m}dp_x\right)^3.\]

Using $\int e^{-ax^2}dx=\sqrt{\pi/a}$,

\[z_1=\frac{V}{\lambda_T^3},\qquad \lambda_T=\frac{h}{\sqrt{2\pi mk_BT}}.\]

Indistinguishability gives

\[Z_N=\frac{z_1^N}{N!}=\frac1{N!}\left(\frac{V}{\lambda_T^3}\right)^N.\]

Then

\[F=-k_BT\ln Z_N,\qquad U=-\frac{\partial\ln Z_N}{\partial\beta}=\frac32Nk_BT,\] \[P=k_BT\left(\frac{\partial\ln Z_N}{\partial V}\right)_{T,N}=\frac{Nk_BT}{V}.\]

Since $H=U+PV$ and $\mu=(\partial F/\partial N)_{T,V}$, Stirling’s approximation gives

\[H=\frac52Nk_BT,\qquad \mu=-k_BT\ln\left(\frac{V}{N\lambda_T^3}\right).\]

Using Stirling’s approximation in $S=k_B(\ln Z_N+\beta U)$ gives the Sackur-Tetrode expression

\[S=Nk_B\left[\ln\left(\frac{V}{N}\left(\frac{2\pi mk_BT}{h^2}\right)^{3/2}\right)+\frac52\right].\]

Without the factor $1/N!$, entropy would not be extensive and mixing identical gases would spuriously increase it. Dividing by permutations of identical particles resolves the Gibbs paradox.

Maxwell speed distribution

The one-particle momentum probability is proportional to $e^{-\beta p^2/2m}$. Normalizing its three Cartesian Gaussian factors and using $p=mv$ gives

\[f_{\mathbf v}(\mathbf v) =\left(\frac{m}{2\pi k_BT}\right)^{3/2} e^{-mv^2/(2k_BT)}.\]

All directions with speeds between $v$ and $v+dv$ occupy the shell $d^3v=4\pi v^2dv$, hence

\[f(v)=4\pi\left(\frac{m}{2\pi k_BT}\right)^{3/2} v^2e^{-mv^2/(2k_BT)},\qquad \int_0^\infty f(v)\,dv=1.\]

Differentiating $\ln f=2\ln v-mv^2/(2k_BT)+\text{constant}$ gives the most probable speed:

\[\frac{d\ln f}{dv}=\frac2v-\frac{mv}{k_BT}=0 \quad\Longrightarrow\quad v_{mp}=\sqrt{\frac{2k_BT}{m}}.\]

The normalized Maxwell speed distribution $f(v)=4\pi(m/2\pi k_BT)^{3/2}v^2e^{-mv^2/2k_BT}$ at three temperatures; each curve is generated from the displayed equation and has unit area.

Equipartition and heat capacity

For a quadratic coordinate $x$ contributing $\epsilon=ax^2$, its canonical mean is

\[\langle ax^2\rangle =-\frac{\partial}{\partial\beta}\ln\int_{-\infty}^{\infty}e^{-\beta ax^2}dx =-\frac{\partial}{\partial\beta}\ln\left(\sqrt{\frac{\pi}{\beta a}}\right) =\frac1{2\beta}=\frac12k_BT.\]

Each independent quadratic term contributes $k_BT/2$. A monatomic ideal gas has three translational momentum terms, hence $U=3Nk_BT/2$ and $C_V=3Nk_B/2$. A classical rigid diatomic molecule adds two rotational terms, predicting $C_V=5Nk_B/2$; a fully excited vibrational mode adds two more quadratic terms and contributes $Nk_B$. The law fails when level spacings are not small compared with $k_BT$, because quantum coordinates then cannot explore energy continuously.

Maxima verification worksheet