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

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

The second derivative measures canonical energy fluctuations:

\[\left\langle(\Delta E)^2\right\rangle =\frac{\partial^2\ln Z}{\partial\beta^2} =k_BT^2C_V.\]

For an extensive system $C_V\propto N$, while $U\propto N$, so $\sqrt{\langle(\Delta E)^2\rangle}/U\propto N^{-1/2}$. This is why canonical and microcanonical predictions agree for macroscopic matter even though energy fluctuates in the canonical ensemble.

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.

Solved Problems

1. Characteristic molecular speeds

Find the most probable, mean, and root-mean-square speeds of nitrogen molecules at $300\ \mathrm K$. Use $m_{N_2}=28.0134\ \mathrm u$.

The molecular mass is

\[m=(28.0134)(1.66053906660\times10^{-27}) =4.65173\times10^{-26}\ \mathrm{kg}.\]

Moments of the normalized Maxwell distribution give

\[v_{mp}=\sqrt{\frac{2k_BT}{m}},\qquad \bar v=\sqrt{\frac{8k_BT}{\pi m}},\qquad v_{rms}=\sqrt{\frac{3k_BT}{m}}.\]

At $300\ \mathrm K$,

\[v_{mp}=421.997\ \mathrm{m\,s^{-1}},\] \[\bar v=476.173\ \mathrm{m\,s^{-1}},\qquad v_{rms}=516.839\ \mathrm{m\,s^{-1}}.\]

Their order $v_{mp}<\bar v<v_{rms}$ reflects the long high-speed tail. Each expression has units $\sqrt{\mathrm{J/kg}}=\mathrm{m\,s^{-1}}$.

2. Testing the classical-gas condition

Helium gas at $300\ \mathrm K$ has number density $n=2.45\times10^{25}\ \mathrm{m^{-3}}$. For $m=4.002602\ \mathrm u$, calculate $\lambda_T$ and $n\lambda_T^3$.

The translational thermal wavelength is

\[\lambda_T=\frac{h}{\sqrt{2\pi mk_BT}} =5.03811\times10^{-11}\ \mathrm m=0.0503811\ \mathrm{nm}.\]

Thus

\[n\lambda_T^3=(2.45\times10^{25}) (5.03811\times10^{-11})^3 =3.13307\times10^{-6}.\]

This dimensionless parameter is much smaller than unity, so wave packets overlap negligibly and Maxwell-Boltzmann statistics is self-consistent. The classical limit is approached as $T$ increases or $n$ decreases.

Descriptive Questions

1. Derive the canonical probability distribution by treating the heat reservoir multiplicity to first order in the subsystem energy.
2. Starting from $W=N!\prod_i g_i^{n_i}/n_i!$, obtain the Maxwell-Boltzmann occupation law under fixed-$N$ and fixed-energy constraints.
3. Explain the roles of $h^{3N}$ and $N!$ in classical phase-space counting and show how indistinguishability restores extensive entropy.
4. Compare the microcanonical, canonical, and grand-canonical ensembles in terms of fixed variables and permitted exchanges; identify the appropriate equilibrium extremum in each case: maximum entropy for an isolated system, minimum Helmholtz free energy at fixed $N,V,T$, and minimum grand potential at fixed $\mu,V,T$.

Numerical Problems

1. Find $U$ and $H$ for $2.00\ \mathrm{mol}$ of a monatomic ideal gas at $400\ \mathrm K$.

   Answer: $U=9.97736\ \mathrm{kJ}$ and $H=16.6289\ \mathrm{kJ}$.

2. One mole of ideal gas expands isothermally and reversibly from $V$ to $2V$. Find its entropy change.

   Answer: $\Delta S=R\ln2=5.76315\ \mathrm{J\,K^{-1}}$.

3. For a monatomic ideal gas containing $N=1.00\times10^{20}$ particles, find the canonical relative rms energy fluctuation $\sigma_E/U$.

   Answer: $\sigma_E/U=\sqrt{2/(3N)}=8.16497\times10^{-11}$.

4. One mole of a rigid diatomic ideal gas is heated through $100\ \mathrm K$ while vibration remains frozen. Find the heat absorbed at constant volume.

   Answer: $Q_V=(5/2)R\Delta T=2.07862\ \mathrm{kJ}$.

Maxima verification worksheet

References

1. Maxwell-Boltzmann statistics — Wikipedia
2. F. Reif, Fundamentals of Statistical and Thermal Physics, chapters 6–9 on ensembles, canonical distributions, and ideal gases.
3. R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed., chapters 1, 3, and 4 on ensembles and classical gases.
4. K. Huang, Statistical Mechanics, 2nd ed., chapters 6–9 on classical ensembles, ideal gases, and the Gibbs paradox.