Introduction
The partition function is the central quantity in statistical mechanics.
It connects microscopic energy states of a system to its macroscopic
thermodynamic properties.
Once the partition function is known, all thermodynamic quantities
can be derived systematically.
\( P_i = \dfrac{e^{-\beta E_i}}{Z} \)
where \( \beta = \dfrac{1}{k_B T} \) and \( Z \) is the partition function.
Canonical Partition Function
For a system in the canonical ensemble (constant \(N,V,T\)):
\( Z = \sum_i e^{-\beta E_i}\)
For a classical system with continuous phase space:
\( Z = \dfrac{1}{h^{3N}N!} \int e^{-\beta H(p,q)} d^{3N}p\, d^{3N}q\)
The Hamiltonian \(H(p,q)\) determines the energy of the microstates.
Statistical Mechanics Partition Functions
1. Microcanonical Ensemble (N, V, E)
Physical State: An isolated system with fixed particles, volume, and energy. It assumes all accessible microstates are equally probable.
Partition Function (Multiplicity): Ω(E)
Ω(E, V, N) = ∑ δ(E - Ei)
Thermodynamic Link: Entropy (S) = kB ln Ω
2. Canonical Ensemble (N, V, T)
Physical State: A system in thermal equilibrium with a heat bath at temperature T. Energy can fluctuate.
Partition Function: Z (or Q)
Z(N, V, T) = ∑ exp(-βEi)
Where β = 1 / (kBT)
Thermodynamic Link: Helmholtz Free Energy (F) = -kBT ln Z
3. Grand Canonical Ensemble \( (\mu;, V, T) \)
Physical State: An open system that can exchange both energy and particles with a reservoir at chemical potential \( \mu;\).
Partition Function:\( \Xi; (Grand Partition Function) \)
\( \Xi;(\mu;, V, T) = \sum;N exp(\beta;\mu;N) Z(N, V, T)\)
Thermodynamic Link: Grand Potential \( (&\hi;G) = -kBT ln &\i; = -PV \)
Summary Comparison
| Ensemble |
Constants |
Function |
Potential |
| Microcanonical |
N, V, E |
Ω |
Entropy (S) |
| Canonical |
N, V, T |
Z |
Free Energy (F) |
| Grand Canonical |
μ, V, T |
Ξ |
Grand Potential (Φ) |
Thermodynamic Quantities from Z
| Quantity |
Expression |
| Helmholtz Free Energy |
\(F = -k_B T \ln Z\) |
| Internal Energy |
\(U = -\dfrac{\partial}{\partial \beta} \ln Z\) |
| Entropy |
\(S = -\left(\dfrac{\partial F}{\partial T}\right)_V\) |
| Pressure |
\(P = k_B T \left(\dfrac{\partial \ln Z}{\partial V}\right)_T\) |
Single Particle Partition Function
For non-interacting identical particles:
\( Z = \dfrac{Z_1^N}{N!}\)
For a free particle in volume \(V\):
\( Z_1 = \dfrac{V}{\lambda^3} \)
\(\lambda = \dfrac{h}{\sqrt{2\pi m k_B T}}\)
\(\lambda\) is the thermal de Broglie wavelength.
Grand Canonical Partition Function
When particle number is variable, we use the grand partition function:
\(\Xi = \sum_{N=0}^{\infty} e^{\beta \mu N} Z_N\)
\(\Phi = -k_B T \ln \Xi\)
Here \( \mu \) is the chemical potential and \( \Phi \) is the grand potential.
Physical Significance
- Connects microscopic states to macroscopic observables
- Determines thermodynamic potentials
- Basis of Bose–Einstein and Fermi–Dirac statistics
- Essential in studying phase transitions
The partition function acts as a generating function for equilibrium thermodynamics.
Summary
✔ Central quantity in statistical mechanics
✔ Determines all thermodynamic properties
✔ Different forms for different ensembles
✔ Bridges microscopic physics and macroscopic laws
N particles are distributed among 3 nondegenerate energy levels having energies\(
E_
=0,E_2
=kT,E_3
=2kT.\) If the total internal energy of the system is 1000kT,Find the value of N
\[\begin{align}
E &={{\sum{{{n}_{i}}E}}_{i}} \\
\Rightarrow 1000kT &={{n}_{1}}{{E}_{1}}+{{n}_{2}}{{E}_{2}}+{{n}_{3}}{{E}_{3}} \\
& =\frac{N}{Z}{{e}^{-\left( \frac{{{E}_{1}}}{kT} \right)}}{{E}_{1}}+\frac{N}{Z}{{e}^{-\left( \frac{{{E}_{2}}}{kT} \right)}}{{E}_{2}}+\frac{N}{Z}{{e}^{-\left( \frac{{{E}_{3}}}{kT} \right)}}{{E}_{3}} \\
& =\frac{N}{Z}\left( {{e}^{-\left( \frac{{{E}_{1}}}{kT} \right)}}{{E}_{1}}+{{e}^{-\left( \frac{{{E}_{2}}}{kT} \right)}}{{E}_{2}}+{{e}^{-\left( \frac{{{E}_{3}}}{kT} \right)}}{{E}_{3}} \right) \\
& =\frac{N}{Z}\left( {{e}^{-\left( \frac{0}{kT} \right)}}\times 0+{{e}^{-\left( \frac{kT}{kT} \right)}}\times kT+{{e}^{-\left( \frac{2kT}{kT} \right)}}\times 2kT \right) \\
& =\frac{N}{Z}\left( {{e}^{-\left( \frac{kT}{kT} \right)}}\times kT+{{e}^{-\left( \frac{2kT}{kT} \right)}}\times 2kT \right) \\
& =\frac{NkT}{Z}\left( {{e}^{-\left( 1 \right)}}+2{{e}^{-\left( 2 \right)}} \right) \\
& =\frac{NkT}{\sum{{{e}^{-\beta {{E}_{i}}}}}}\left( {{e}^{-\left( 1 \right)}}+2{{e}^{-\left( 2 \right)}} \right) \\
& =\frac{NkT}{{{e}^{-0}}+{{e}^{-\left( 1 \right)}}+{{e}^{-\left( 2 \right)}}}\left( {{e}^{-\left( 1 \right)}}+2{{e}^{-\left( 2 \right)}} \right) \\
& =\frac{NkT}{1+\frac{1}{e}+\frac{1}{{{e}^{2}}}}\left( \frac{1}{e}+2\frac{1}{{{e}^{2}}} \right) \\
& =\frac{NkT\left( 1+2e \right)}{1+e+{{e}^{2}}} \\
\Rightarrow 1000 &=N\left( \frac{1+2e}{1+e+{{e}^{2}}} \right) \\
\Rightarrow N&=2354 \\
\end{align}\]
The lowest level of Oxygen is three fold degenerate. The next level is doubly degenerate
and lies 0.97eV above the lowest level. If the lowest level has an energy 0 .Calculate the
Partition function at Temperature 1000K and 3000K
\begin{align}
\text{Partition function at temperature } T \text{ is :} \quad
Z &= \sum_i g_i e^{-\beta E_i} \\
&= g_1 e^{-\beta E_1} + g_2 e^{-\beta E_2} \\
&= 3 e^{-\frac{E_1}{kT}} + 2 e^{-\frac{E_2}{kT}} \\
&= 3 e^{-\frac{0}{kT}} + 2 e^{-\frac{0.97}{kT}} \\
&= 3 + 2 e^{-\frac{0.97}{kT}}
\end{align}
\begin{align}
\text{At } T = 1000\,\text{K:} \quad
Z_{(1000\text{K})} &= 3 + 2 e^{-\frac{0.97}{1000k}} \\
\text{At } T = 3000\,\text{K:} \quad
Z_{(3000\text{K})} &= 3 + 2 e^{-\frac{0.97}{3000k}}
\end{align}
\begin{align}
\text{Since} k=8.617\times 10^{-5}eV/Kelvin \quad
Z_{(1000\text{K})} &= 3.000026 \\
Z_{(3000\text{K})} &= 3.047
\end{align}
