14.2 Probability and statistical ensembles

Probability and Statistical Ensembles

Statistical ensembles give us a way to connect the microscopic behavior of particles to the macroscopic properties we actually measure, like temperature and pressure. Instead of tracking every single particle (which is impossible for real systems), we consider a huge collection of hypothetical copies of the same system, each in a different possible microstate. By looking at the statistics across this collection, we can predict what the system will do on average.

Concept of Statistical Ensembles

A statistical ensemble is a large collection of identical systems, where each copy sits in one of the system's possible microstates. A microstate is a specific configuration defined by the positions and momenta of all particles in the system.

The central idea is that the ensemble average of any physical quantity (energy, pressure, magnetization, etc.) corresponds to the macroscopic value you'd actually measure in an experiment. So rather than solving equations of motion for $10^{23}$ particles, you calculate a weighted average over microstates. This is what makes statistical mechanics practical.

Probability in Microstates

The probability of finding a system in a particular microstate depends on two things: the energy of that microstate and the temperature of the system.

The Boltzmann distribution gives the probability $p_i$ of a system occupying a microstate with energy $E_i$ at temperature $T$:

$p_i = \frac{e^{-E_i / k_B T}}{\sum_j e^{-E_j / k_B T}}$

• $k_B$ is the Boltzmann constant ($1.38 \times 10^{-23}$ J/K)
• The denominator is the partition function $Z$, which sums over all microstates and ensures the probabilities add up to 1

Two key behaviors follow from this expression:

• Lower energy microstates are more probable than higher energy ones, because the exponential $e^{-E_i / k_B T}$ decreases as $E_i$ increases.
• As temperature rises, the distribution flattens out. At high $T$, the ratio $E_i / k_B T$ becomes small for many states, so the system explores a wider range of microstates more evenly. Think of gas molecules at high temperature spreading across many energy levels, versus at low temperature clustering near the ground state.

Types of Statistical Ensembles

Each ensemble type models a different physical situation, depending on what the system can exchange with its surroundings.

• Microcanonical ensemble (NVE):
  • Models an isolated system with fixed particle number $N$, volume $V$, and total energy $E$.
  • No energy or particle exchange with the environment.
  • All accessible microstates at that energy are equally probable. This is the fundamental postulate of statistical mechanics.
  • Example: an insulated gas container with rigid walls.
• Canonical ensemble (NVT):
  • Models a system in thermal contact with a heat bath at fixed temperature $T$.
  • Particle number $N$ and volume $V$ are fixed, but the system can exchange energy with the bath.
  • Microstate probabilities follow the Boltzmann distribution.
  • Example: a sample of gas in a container submerged in a large water bath.
• Grand canonical ensemble ($\mu$VT):
  • Models a system that exchanges both energy and particles with a reservoir at fixed temperature $T$ and chemical potential $\mu$.
  • Volume $V$ is fixed, but $N$ and $E$ both fluctuate.
  • Example: gas molecules adsorbing onto and desorbing from a surface in contact with a gas reservoir.

Microstate Probability Distributions

Here's a summary of the probability expression for each ensemble, along with its normalization factor:

• Microcanonical ensemble: $p_i = \frac{1}{\Omega(E)}$ where $\Omega(E)$ is the total number of microstates with energy $E$. Every accessible microstate gets the same weight.

• Canonical ensemble: $p_i = \frac{e^{-E_i / k_B T}}{Z}$ where $Z = \sum_j e^{-E_j / k_B T}$ is the canonical partition function. States with lower energy get higher probability, weighted by temperature.

• Grand canonical ensemble: $p_i = \frac{e^{-(E_i - \mu N_i) / k_B T}}{\Xi}$ where $\Xi = \sum_{j,N} e^{-(E_j - \mu N_j) / k_B T}$ is the grand canonical partition function. Notice the extra $\mu N_i$ term: states with more particles are favored when the chemical potential is high, because the reservoir "wants" to push particles into the system.

The partition function in each case plays the same role: it normalizes the probability distribution so that $\sum_i p_i = 1$. But it also encodes all the thermodynamic information about the system. Once you know $Z$ or $\Xi$, you can derive quantities like average energy, entropy, and free energy directly from it.