11.2 Partition functions and their applications

Partition functions are the backbone of statistical mechanics, bridging the gap between microscopic and macroscopic properties. They help us calculate crucial thermodynamic quantities like energy, entropy, and free energy, giving us a deeper understanding of how systems behave at equilibrium.

By summing over all possible microstates, partition functions allow us to predict a system's behavior without knowing its exact state. This powerful tool is essential for studying everything from ideal gases to complex quantum systems, making it a cornerstone of modern physics.

Partition function for systems

Definition and properties of the partition function

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• The partition function, denoted as Z, is a fundamental quantity in statistical mechanics that encodes the statistical properties of a system in thermodynamic equilibrium
• It is defined as the sum of the Boltzmann factors over all possible microstates of the system: $Z = \Sigma_i \exp(-\beta*E_i)$, where $\beta = 1/(k_B*T)$, $k_B$ is the Boltzmann constant, $T$ is the absolute temperature, and $E_i$ is the energy of the $i$-th microstate
• The partition function is a dimensionless quantity that depends on the temperature and the energy levels of the system, but not on the specific microstate the system is in
• It serves as a bridge between the microscopic properties (energy levels and degeneracies) and the macroscopic thermodynamic quantities of a system

Partition function for discrete energy levels

• For a system with discrete energy levels, the partition function can be expressed as $Z = \Sigma_i g_i * \exp(-\beta*E_i)$, where $g_i$ is the degeneracy of the $i$-th energy level, representing the number of microstates with the same energy $E_i$
  • Example: A two-level system with energy levels $E_0 = 0$ and $E_1 = \epsilon$ and degeneracies $g_0 = 1$ and $g_1 = 2$ has a partition function $Z = 1 + 2\exp(-\beta\epsilon)$
• The degeneracy factor accounts for the multiplicity of microstates with the same energy, which contributes to the statistical weight of each energy level in the partition function
  • Example: In a system of $N$ distinguishable particles, each with two possible states (e.g., spin-up and spin-down), the degeneracy of a macrostate with $k$ particles in the excited state is given by the binomial coefficient $\binom{N}{k}$

Partition function and thermodynamics

Relation between partition function and thermodynamic quantities

• The partition function allows the calculation of various thermodynamic quantities by taking appropriate derivatives or logarithms
• The internal energy $U$ can be obtained from the partition function as $U = -\partial(\ln Z)/\partial\beta$, where the partial derivative is taken at constant volume
• The entropy $S$ is related to the partition function through the Gibbs entropy formula: $S = k_B * [\ln Z + \beta * \partial(\ln Z)/\partial\beta]$, where the partial derivative is taken at constant volume
• The Helmholtz free energy $A$ is directly related to the partition function by $A = -k_B*T * \ln Z$, which provides a connection between the microscopic and macroscopic descriptions of a system

Deriving other thermodynamic quantities from the partition function

• Other thermodynamic quantities, such as pressure, specific heat, and chemical potential, can also be derived from the partition function by taking appropriate derivatives
  • Pressure: $P = k_B*T * \partial(\ln Z)/\partial V$, where the partial derivative is taken at constant temperature
  • Specific heat at constant volume: $C_V = (\partial U/\partial T)_V = k_B*\beta^2 * \partial^2(\ln Z)/\partial\beta^2$
  • Chemical potential: $\mu = -k_B*T * \partial(\ln Z)/\partial N$, where $N$ is the number of particles and the partial derivative is taken at constant temperature and volume

Partition function calculations

Ideal gas partition function

• For an ideal gas of $N$ non-interacting particles, the partition function factorizes into a product of single-particle partition functions: $Z = (Z_1)^N/N!$, where $Z_1$ is the single-particle partition function given by $Z_1 = (V/\Lambda^3)$, with $V$ being the volume and $\Lambda$ the thermal de Broglie wavelength
  • The thermal de Broglie wavelength is given by $\Lambda = h/\sqrt{2\pi m k_B T}$, where $h$ is Planck's constant and $m$ is the mass of a particle
• The factorization of the partition function for an ideal gas is a consequence of the absence of interactions between particles, allowing the system to be treated as a collection of independent single-particle systems
• The factor of $1/N!$ accounts for the indistinguishability of the particles, as permutations of identical particles do not lead to distinct microstates

Harmonic oscillator partition function

• For a quantum harmonic oscillator with frequency $\omega$, the energy levels are given by $E_n = (n + 1/2) * \hbar\omega$, where $n = 0, 1, 2, ...$
• The partition function is a geometric series: $Z = \Sigma_n \exp(-\beta*E_n) = 1/[2*\sinh(\beta\hbar\omega/2)]$, where $\sinh$ is the hyperbolic sine function
  • The hyperbolic sine function is defined as $\sinh(x) = (\exp(x) - \exp(-x))/2$
• The partition function of a system of $N$ independent harmonic oscillators is given by the product of single-oscillator partition functions: $Z = [1/2*\sinh(\beta\hbar\omega/2)]^N$
  • Example: The vibrational modes of a diatomic molecule can be modeled as a system of independent harmonic oscillators, each with its own characteristic frequency

Equilibrium properties from partition function

Calculating average quantities

• Once the partition function is known, it can be used to calculate various equilibrium properties of the system
• The average energy $\langle E \rangle$ can be calculated as $\langle E \rangle = -\partial(\ln Z)/\partial\beta = \Sigma_i (E_i * \exp(-\beta*E_i))/Z$, which represents the weighted average of the energy levels, with the Boltzmann factors as weights
• The heat capacity $C_V$ can be obtained from the fluctuations in energy: $C_V = (\langle E^2 \rangle - \langle E \rangle^2)/(k_B*T^2)$, where $\langle E^2 \rangle$ is the average of the squared energy, calculated using the partition function
  • The fluctuation-dissipation theorem relates the fluctuations in energy to the system's response to temperature changes, as measured by the heat capacity

Equilibrium occupation probabilities and phase transitions

• The equilibrium occupation probabilities of different energy levels can be determined using the Boltzmann distribution: $P_i = \exp(-\beta*E_i)/Z$, where $P_i$ is the probability of finding the system in the $i$-th energy level
  • Example: In a two-level system with energy levels $E_0 = 0$ and $E_1 = \epsilon$, the equilibrium occupation probabilities are $P_0 = 1/Z$ and $P_1 = \exp(-\beta\epsilon)/Z$, where $Z = 1 + \exp(-\beta\epsilon)$
• The partition function can be used to study phase transitions by analyzing its behavior as a function of temperature or other external parameters, such as pressure or magnetic field
  • Example: The Ising model, a simple model for ferromagnetism, exhibits a phase transition from a paramagnetic to a ferromagnetic state as the temperature is lowered below a critical value, which can be studied using the partition function