# Grand Canonical Partition Function

## Table of Contents

The grand canonical partition function, denoted by \( \small \Xi\ \), is a fundamental concept in statistical mechanics, particularly in the grand canonical ensemble. It represents the sum of the canonical partition functions weighted by the Boltzmann factors over all possible microstates with different numbers of particles.

$$ \Xi = \sum_{i} e^{\beta ( \mu N_i - E_i)} $$

Where:

- \( \small \Xi \) is the grand canonical partition function,
- \( \small \beta = \frac{1}{k_B T} \) is the thermodynamic beta with \( \small k_B\) being the Boltzmann constant and \( \small T \) being the absolute temperature,
- \( \small \mu \) is the chemical potential of the system,
- \( \small N_i \) is the number of particles in the \( \small i \)-th microstate, and
- \( \small E_i \) is the energy of the \( \small i \)-th microstate.