Statistical Physics - Notes#

Ensembles#

Microcanonical ensemble#

Canonical ensemble#

Macrocanonical esemble#

Statistics#

Each of the \(N\) components of the system is in an energy level \(i\). Energy level \(i\) has \(g_i\) sublevels with the same energy level.

  • energy levels, \(E_i\) of each component

  • occupation number \(N_i\) of level \(i\)

  • Central role of energy. In a system macroscopically at rest, the energy of a system is the only macroscopic meaningful non-zero mechanical quantity, constant for closed and isolated systems

  • Principle of maximum uncertainty, maximum entropy, minimum information: given a measurement of a macroscopic variable \(V\), describing the macrostate of the system, the feasible un-observed/able microstates of the system are the microstates consistent with it: there’s usually a sharp maximum of in the probability density of the micrsotates.

Given a macrostate, what’s the number of ways \(W(N_i; g_i)\) to get a consistent microstate? Once the expression is found, constrained optimization follows: optimization w.r.t. \(N_i\) is usually performed in the limit of \(N_i \rightarrow +\infty\) (why in Fermi-Dirac distribution, obeying Pauli exclusion principle?), with the values of the macroscopic variables as constraints usually treated with Lagrange multiplier.

Maxwell-Boltzmann#

Statistics of distinguishible components.

Bose-Einstein#

Statistics of undistinguishable components that can be in the same (sub)level. Given the number of elementary components \(\sum_{i} N_i = N\) and the energy \(\sum_{i} N_i E_i = E\),

(1)#\[W_{BE,i} = \frac{(N_i + g_i - 1)!}{N_i! (g_i-1)!} \qquad , \qquad W_{BE} = \prod_i W_{BE,i} \ .\]
Counting microstates

todo write page Combinatorics and add link

Most likely microstate. Instead of maximizing (1), the objective function is \(\ln W_{BE}\), after using Stirling approximation in the limit of large \(N_i\) and \(g_i\), \(N_i! \sim \left(\frac{N_i}{e} \right)^{N_i}\). The approximate occupation number of one of the \(G_i\) sublevels of the \(i^{th}\) level of the most likely microstate is

\[n_i := \frac{N_i}{G_i} = \frac{1}{e^{\alpha + \beta E_i} - 1} \ .\]
Optimization
\[\begin{split}\begin{aligned} J(N_i, \alpha, \beta) & = \ln W_{BE} + \alpha \left( N - \sum_i N_i \right) + \beta \left(E - \sum_i N_i E_i \right) = \\ & = \sum_i \left\{ \ln(N_i + g_i -1)! - \ln N_i! - \ln (g_i-1)! \right\} + \alpha \left( N - \sum_i N_i \right) + \beta \left(E - \sum_i N_i E_i \right) \simeq \\ & \simeq \sum_i \left\{ (N_i+g_i-1)\ln(N_i+g_i-1) - N_i\ln N_i - (g_i-1) \ln (g_i-1) + N_i + g_i -1 - N_i - (g_i-1) \right\} + \alpha \left( N - \sum_i N_i \right) + \beta \left(E - \sum_i N_i E_i \right) = \\ & = \sum_i \left\{ (N_i+g_i-1)\ln(N_i+g_i-1) - N_i\ln N_i - (g_i-1) \ln (g_i-1) \right\} + \alpha \left( N - \sum_i N_i \right) + \beta \left(E - \sum_i N_i E_i \right) \\ \end{aligned}\end{split}\]

Using \(\partial_{n} (n+a) \ln (n+a) = \ln (n+a) + 1\),

\[\begin{aligned} 0 = \partial_{N_k} J & \simeq \left\{ \ln (N_k+g_k-1) - \ln N_k \right\} - \alpha - \beta E_k \ , \end{aligned}\]

and thus

\[\ln \frac{N_k + g_k - 1}{N_k} = \alpha + \beta E_k \ ,\]
\[\frac{N_k + g_k - 1}{N_i} = e^{\alpha + \beta E_k}\]
\[N_k = \frac{g_k - 1}{e^{\alpha + \beta E_k}-1} \simeq \frac{g_k}{e^{\alpha + \beta E_k}-1} \ , \]

Thus, in the limit of \(g_k \gg 1\), the occupation number of the \(k\) level is

\[N_k = \frac{G_k}{e^{\alpha + \beta E_k} - 1} \ ,\]

and the average occupation number of one of the \(g_k\) sublevels in the \(k\) level is

\[n_k := \frac{N_k}{G_k} = \frac{1}{e^{\alpha + \beta E_k} - 1}\]
Meaning of \(\alpha\), \(\beta\)

Example 1 (Black-body radiation: Planck, Wien, and Stefan-Boltzmann laws)

Planck’s law. Energy density w.r.t. frequency

\[u_{f}(f, T) = \frac{8 \pi h f^3}{c^3} \frac{1}{e^{\frac{hf}{k_B T}} - 1}\]
Planck’s law in a cubic box

Planck’s law uses:

  • relation between pulsation and wave vector, or frequency and wave number and the speed of light \(c\) for light waves

    \[c = \frac{\omega}{|\vec{k}|} = \lambda f\]
    \[f = \frac{\omega}{2\pi} = \frac{c |\vec{k}|}{2 \pi}\]

  • Planck assumption that the minimum non-zero energy of a mode with frequency \(f\) is \(E = h f\), and all the possible values of the energy of the mode is

    \[E_m = m h f \quad , \quad m \in \mathbb{N} \ .\]

Taking a cubic box with sides \(L_x = L_y = L_z = L\), the possibile modes have (todo why? Which boundary condition? Periodic? Some physical? Just fictitious discretization?) in each direction wave-lengths \(\lambda_n = \frac{L}{|\vec{n}|} = \frac{2 \pi}{|\vec{k}|}\),

\[\vec{k} = \frac{2 \pi}{L} \vec{n} \ .\]

Mode density in \(\vec{n}\)-domain is 2 mode per each volume of unit length (2 polarization), and thus the number of modes \(d N\) in an elementary volume is

\[d N = 2 \, d^3 \vec{n} \ ,\]

Changing variables, it’s possible to find the mode density w.r.t. wave vector \(\vec{k}\),

\[d N = 2 \, d^3 \vec{n} = 2 \, \frac{L^3}{(2 \pi)^3} \, d^3 \vec{k} \ ,\]

or with its absolute value, exploiting the isotropy of the density function - and writing the elementary volume using “spherical coordinates” \(d^3 \vec{k} = 4 \pi \left| \vec{k} \right|^2 d \, \left| \vec{k} \right|\),

\[\begin{split}\begin{aligned} d N & = \frac{V}{(2 \pi)^3} 8 \pi \left| \vec{k} \right|^2 d \left| \vec{k} \right| = \\ & = \frac{V}{(2 \pi)^3} 8 \pi \frac{8 \pi^3}{c^3} f^2 d f = \\ & = V \frac{8 \pi}{c^3} f^2 df =: V g(f) df \ . \end{aligned}\end{split}\]
Average energy of a mode

Using Boltzmann distribution (why?) for the energy distribution in a single mode,

\[P(E_r) = \frac{e^{-\beta E_r}}{Z} \ ,\]

with \(E_r = r h f\), and the partition function

\[Z = \sum_{s} e^{- \beta E_s} = \sum_s e^{-\beta h f s} = \frac{1}{1 - e^{-\beta h f}} \ .\]

The average energy of the mode reads

\[\begin{split}\begin{aligned} \langle E \rangle & = \sum_r E_r P(E_r) = \\ & = \sum_r r h f \frac{e^{- \beta h f r}}{Z} = \\ & = h f (1-e^{-\beta h f}) \sum_r r e^{- \beta h f r} = \\ & = h f (1-e^{-\beta h f}) \frac{e^{- \beta h f}}{(1-e^{-\beta h f})^2} = \\ & = \frac{h f}{e^{\beta h f} - 1} \ . \end{aligned}\end{split}\]

Putting together the mode number density and the average energy of a mode, the energy density per unit volume, per frequency reads

\[\begin{split}\begin{aligned} u(f, T) & = \langle E \rangle(f) \, g(f) = \\ & = \frac{hf}{e^{\beta h f} - 1} \frac{8 \pi}{c^3} f^2 = \\ & = \frac{8 \pi h f^3}{c^3} \frac{1}{e^{\beta h f} - 1} \ . \end{aligned}\end{split}\]
Property of the series
\[\sum_{n=0}^{+\infty} n x^n = \frac{x}{(1-x)^2}\]

Proof. If the series is convergent (is this the required condition?)

\[\frac{d}{d x} \sum_{n=0}^{+\infty} x^n = \frac{d}{dx} \frac{1}{1 - x} = \frac{1}{(1-x)^2}\]
\[\frac{d}{d x} \sum_{n=0}^{+\infty} x^n = \sum_{n=0}^{+\infty} n x^{n-1}\]
\[x \frac{d}{d x} \sum_{n=0}^{+\infty} x^n = \sum_{n=0}^{+\infty} n x^n = \frac{x}{(1-x)^2}\]

Sperctral radiance, \(B_{f}\), so that an infinitesimal amount of power radiated by a surface … is \(d P = B_f(f,T) \cos \theta \, dA \, d\Omega \, d f\)

\[B_{f}(f, T) = \frac{2 h f^3}{c^2}\frac{1}{e^{\frac{hf}{k_B T}} - 1} \ .\]

This expression is obtained1 assuming homogeneous radiation from a small hole cut into a wall of the box. Only half of the energy radiates through the hole - so factor \(\frac{1}{2}\) in front of the energy density - through a solid angle \(2 \pi\) - and thus this process give the same result as a radiation of all the energy density in all the space directions, just providing the same factor \(\frac{1}{4 \pi}\). The flux of energy “has velocity” \(c\) and thus

\[B_{f}(f, T) = \frac{1}{4 \pi} u_{f}(f,T) c \ .\]

Wien’s law. Wien’s law tells that the frequency \(f^*\) corresponding to the maximum of the spectral radiance of a black-body radiation described by Planck’s law is proportional to its temperature.

From direct evaluation of the derivative of the spectral radiance as a function of \(f\),

\[\begin{split}\begin{aligned} \partial_f B_{f}(f,T) & = \frac{2 h}{c^2} \left[ 3 f^2 \frac{1}{e^{\frac{hf}{k_B T}}-1} + f^3 \left(-\frac{\frac{h}{k_B T} e^{\frac{hf}{k_B T}}}{\left( e^{\frac{hf}{k_B T}} - 1 \right)^2} \right) \right] = \\ & = \frac{2 h f^2 e^{\frac{hf}{k_B T}}}{c^2 \left( e^{\frac{hf}{k_B T}} - 1 \right)^2} \left[ 3 \left( 1 - e^{-\frac{hf}{k_B T}} \right) - \frac{h f}{k_B T} \right] \ . \end{aligned}\end{split}\]

Now, if \(\partial_f B_{f}(f,T) = 0\) the frequency is either \(f = 0\), or the solution of the nonlinear algebraic equation

\[0 = 3 \left(1 - e^{-\frac{h f}{k_B T}} \right) - \frac{hf}{k_B T} \ .\]

Defining \(x := \frac{h f}{k_B T}\), this equation becomes

\[0 = 3 (1 - e^x) - x \ ,\]

whose solution \(x^* \approx 2.82\) can be easily evaluated with an iterative method (or expressed in term of the Lambert’s function \(W\), so loved at Stanford and on Youtube: they’d probaly like to look at tabulated values, or pose). Once the solution \(x^*\) of this non-dimensional equation is found, the frequency where maximum energy density occurs reads

\[f^* = \frac{k_B T}{h} x^* \simeq 2.82 \frac{k_B}{h} T \ .\]

Stefan-Boltzmann law.

\[\begin{split}\begin{aligned} \frac{P}{A} & = \int B_{f}(f,T) \cos \phi \, df \, d\Omega = \\ & = \int_{f=0}^{+\infty} \int_{\phi = 0}^{\frac{\pi}{2}} \int_{\theta=0}^{2\pi} B_{f}(f,T) \cos \phi \sin \phi \, df \, d\phi \, d \theta = \\ & = \pi \int_{f=0}^{+\infty} B_{f}(f,T) \, d f = \\ & = \frac{2 \pi h}{c^2} \int_{f=0}^{+\infty} \frac{f^3}{e^{\frac{hf}{k_B T}} - 1} \, d f = \\ & = \frac{2 \pi h}{c^2} \left( \frac{k_B T}{h} \right)^4 \int_{u=0}^{+\infty} \frac{u^3}{e^u - 1} \, d u \ . \end{aligned}\end{split}\]

The value of the integral is \(\frac{\pi^4}{15}\) and thus

\[\frac{P}{A} = \sigma T^4 \qquad , \qquad \sigma = \frac{2 \pi^5 k_B^4}{15 c^2 h^3} \ .\]

Example 2 (Energy density and radiance)

Radiance. The radiance \(L_{e,\Omega}\) of a surface is the flux of energy per unit solid angle, per unit projected area of the source.

Spectral radiance in frequency is the radiance per unit frequency, \(L_{e, \Omega, f} = \frac{\partial L_{e,\Omega}}{\partial f}\).

Fermi-Dirac#

Statistics of undistinguishable components that can’t be in the same (sub)level, obeying to the Pauli exclusion principle. Given the number of elementary components \(\sum_{i} N_i = N\) and the energy \(\sum_{i} N_i E_i = E\),

(2)#\[W_{FD,i} = \frac{G_i!}{(G_i-N_i)! N_i!} \qquad , \qquad W_{FD} = \prod_i W_{FD,i} \ .\]
Counting microstates

todo write page Combinatorics and add link

Most likely microstate. The approximate occupation number of the \(i^{th}\) level of the most likely microstate is

\[n_i := \frac{N_i}{G_i} = \frac{1}{1 + e^{\alpha + \beta E_i}} \ .\]
Optimization
\[\begin{split}\begin{aligned} J(N_i, \alpha, \beta) & = \ln W_{FD} + \alpha \left( N - \sum_i N_i \right) + \beta \left(E - \sum_i N_i E_i \right) = \\ & = \sum_i \left\{ \ln G_i! - \ln(G_i - N_i)! - \ln N_i! \right\} + \alpha \left( N - \sum_i N_i \right) + \beta \left(E - \sum_i N_i E_i \right) = \\ & = \sum_i \left\{ G_i \ln G_i - (G_i - N_i) \ln(G_i - N_i) - N_i \ln N_i \right\} + \alpha \left( N - \sum_i N_i \right) + \beta \left(E - \sum_i N_i E_i \right) = \\ \end{aligned}\end{split}\]

Using \(\partial_{n} (n+a) \ln (n+a) = \ln (n+a) + 1\),

\[\begin{aligned} 0 = \partial_{N_k} J & \simeq \left\{ \ln (G_k - N_k) - \ln N_k \right\} - \alpha - \beta E_k \ , \end{aligned}\]

and thus

\[\ln \frac{G_k - N_k}{N_k} = \alpha + \beta E_k \ ,\]
\[\frac{G_k}{N_k} - 1 = e^{\alpha + \beta E_k} \ \]

The occupation number of the \(k\) level is

\[N_k = \frac{G_k}{1 + e^{\alpha + \beta E_k}} \ .\]

The average occupation of the \(G_k\) sublevels of the \(k\) level is

\[n_k := \frac{N_k}{G_k} = \frac{1}{1 + e^{\alpha + \beta E_k}} \ .\]
Meaning of \(\alpha\), \(\beta\)
1

Derivation of Planck’s Law.