Partition Function
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Partition Function(Z)

Foundation of Equilibrium Statistical Mechanics

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

âś” Central quantity in statistical mechanics
âś” Determines all thermodynamic properties
âś” Different forms for different ensembles
âś” Bridges microscopic physics and macroscopic laws

Problems on Partition Function

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

Solution

\[\begin{align} E &={{\sum{{{n}_{i}}E}}_{i}} \\ \Rightarrow 1000kT &={{n}_{1}}{{E}_{1}}+{{n}_{2}}{{E}_{2}}+{{n}_{3}}{{E}_{3}} \\ & =\frac{N}{Z}{{e}^{-\left( \frac{{{E}_{1}}}{kT} \right)}}{{E}_{1}}+\frac{N}{Z}{{e}^{-\left( \frac{{{E}_{2}}}{kT} \right)}}{{E}_{2}}+\frac{N}{Z}{{e}^{-\left( \frac{{{E}_{3}}}{kT} \right)}}{{E}_{3}} \\ & =\frac{N}{Z}\left( {{e}^{-\left( \frac{{{E}_{1}}}{kT} \right)}}{{E}_{1}}+{{e}^{-\left( \frac{{{E}_{2}}}{kT} \right)}}{{E}_{2}}+{{e}^{-\left( \frac{{{E}_{3}}}{kT} \right)}}{{E}_{3}} \right) \\ & =\frac{N}{Z}\left( {{e}^{-\left( \frac{0}{kT} \right)}}\times 0+{{e}^{-\left( \frac{kT}{kT} \right)}}\times kT+{{e}^{-\left( \frac{2kT}{kT} \right)}}\times 2kT \right) \\ & =\frac{N}{Z}\left( {{e}^{-\left( \frac{kT}{kT} \right)}}\times kT+{{e}^{-\left( \frac{2kT}{kT} \right)}}\times 2kT \right) \\ & =\frac{NkT}{Z}\left( {{e}^{-\left( 1 \right)}}+2{{e}^{-\left( 2 \right)}} \right) \\ & =\frac{NkT}{\sum{{{e}^{-\beta {{E}_{i}}}}}}\left( {{e}^{-\left( 1 \right)}}+2{{e}^{-\left( 2 \right)}} \right) \\ & =\frac{NkT}{{{e}^{-0}}+{{e}^{-\left( 1 \right)}}+{{e}^{-\left( 2 \right)}}}\left( {{e}^{-\left( 1 \right)}}+2{{e}^{-\left( 2 \right)}} \right) \\ & =\frac{NkT}{1+\frac{1}{e}+\frac{1}{{{e}^{2}}}}\left( \frac{1}{e}+2\frac{1}{{{e}^{2}}} \right) \\ & =\frac{NkT\left( 1+2e \right)}{1+e+{{e}^{2}}} \\ \Rightarrow 1000 &=N\left( \frac{1+2e}{1+e+{{e}^{2}}} \right) \\ \Rightarrow N&=2354 \\ \end{align}\]

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

Solution

\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}