The Boltzmann factor

Systems in contact with a heat bath

Many systems are not isolated. Instead, they exchange energy with a large environment at fixed temperature. This environment is called a heat bath or thermal reservoir.

A small system in thermal equilibrium with a heat bath can occupy different energy states. The probabilities of those states are not usually equal. Lower-energy states are more likely, but higher-energy states can still occur.

The Boltzmann factor

For a state with energy $E$, the probability is proportional to the Boltzmann factor:

$e^{-E/(k_BT)}$

More precisely,

$P(E)propto e^{-E/(k_BT)}$

This factor is central to statistical mechanics. It expresses the competition between energy and temperature.

At low temperature, high-energy states are strongly suppressed. At high temperature, energy differences matter less, and more states become significantly populated.

Relative probabilities

For two states with energies $E_1$ and $E_2$, the ratio of probabilities is

rac{P_2}{P_1}=e^{-(E_2-E_1)/(k_BT)}

If $E_2>E_1$, state 2 is less likely. The size of the difference depends on $Delta E/(k_BT)$.

When $Delta Egg k_BT$, the higher state is very unlikely. When $Delta Ell k_BT$, the two states have similar probabilities.

Meaning of $k_BT$

The quantity $k_BT$ is an energy scale. It tells us roughly how much thermal energy is available per microscopic degree of freedom. At room temperature, $k_BT$ is small on everyday energy scales but important for atoms and molecules.

Processes with energy costs comparable to $k_BT$ occur frequently. Processes with energy costs much larger than $k_BT$ are rare unless driven externally.

Degeneracy

If multiple states have the same energy, their combined probability depends on degeneracy. If energy level $E$ has degeneracy $g$, then its total weight is

$g e^{-E/(k_BT)}$

A higher energy level can be significantly populated if it has many available states. Entropy and energy compete.

This idea becomes important in chemical reactions, magnetism, phase transitions, and molecular conformations.

Two-level system

Consider a system with a ground state energy $0$ and excited state energy $epsilon$. The relative probability of the excited state is

rac{P_{excited}}{P_{ground}}=e^{-epsilon/(k_BT)}

At low temperature, almost all systems are in the ground state. At high temperature, the excited state becomes more populated.

This model applies approximately to spins, atoms, molecules, and many simplified systems.

Connection to thermodynamics

The Boltzmann factor shows how temperature controls microscopic occupation. It also explains why systems tend to lower energy but do not always occupy only the lowest-energy state. Thermal fluctuations allow access to higher-energy states.

Macroscopic behavior results from averaging over these probabilities.

The big idea

The Boltzmann factor gives the thermal weight of an energy state. It says that probability decreases exponentially with energy, but the strength of that decrease depends on temperature. This simple factor is the gateway to partition functions, free energy, chemical equilibrium, magnetism, and phase behavior.