Partition Function Calculator

The partition function Z is a normalization sum over all accessible microstates of a system, weighted by their Boltzmann factors exp(-Ei/kBT). It might look like a bookkeeping quantity, but essentially every thermodynamic property of the system (average energy, entropy, free energy, heat capacity) can be derived from Z and its derivatives with respect to temperature.

Z = sum over i of gi * exp(-Ei / (kB*T)), where gi is the degeneracy (number of distinct states) at energy level Ei, kB is the Boltzmann constant, and T is the absolute temperature. Once Z is known, the occupation probability of level i is pi = gi*exp(-Ei/kBT) / Z.

Because the Boltzmann factor exp(-Ei/kBT) decreases as Ei increases, for any finite positive temperature. A lower Ei always produces a larger Boltzmann factor than a higher Ei, so states are exponentially weighted toward the ground state, this is the Boltzmann distribution, and it is why the ground state (or lowest-energy accessible level) always has the highest occupation probability.

As T to 0, kBT to 0, so exp(-Ei/kBT) vanishes for every level except the ground state (Ei=0), whose factor stays exactly 1. In this limit only the ground state is populated (p1 to 1, all others to 0), the system freezes entirely into its lowest-energy configuration.

As T to infinity, kBT becomes much larger than every energy gap, so exp(-Ei/kBT) to 1 for every level regardless of Ei. The occupation probabilities then approach the ratio of degeneracies alone, pi to gi / (sum of all gj), completely independent of the energy differences between levels.

At T=300 K, kBT = 8.617333262e-5 eV/K x 300 K is approximately 0.02585 eV, or about 25.85 meV. This is a widely used reference figure in condensed matter and semiconductor physics: any energy gap much larger than roughly 25.85 meV is only weakly populated at room temperature, while gaps much smaller than that are close to equally populated.

Enter your ground state as Level 1 (E1=0, g1 as needed) and your excited state as Level 2, then set the degeneracy of Level 3 and Level 4 to 0. A degeneracy of 0 removes a level from the sum entirely, so it contributes nothing to Z, the occupation probabilities, or the average energy.

The average energy avgE = sum over i of pi*Ei is the thermally averaged energy of the system at temperature T, weighting each level's energy by how likely the system is to occupy it. It rises from 0 (all population in the E=0 ground state) toward the degeneracy-weighted average of all level energies as T increases.

As T rises, kBT grows, which shrinks the exponent Ei/kBT for any nonzero Ei, making exp(-Ei/kBT) closer to 1 relative to the ground state's factor of exactly 1. The excited state's Boltzmann factor therefore catches up toward the ground state's, so its share of the total population (its occupation probability) increases with temperature, this is exactly the curve plotted on this page.

Not quite. The (canonical) partition function is a Boltzmann-weighted sum over microstates, not a plain count. At infinite temperature it does reduce to a simple count (the total degeneracy, sum of gi), but at any finite temperature lower-energy microstates are weighted more heavily than higher-energy ones.

This tool supports up to 4 discrete levels via the 4 input rows, which is enough for most introductory two-level and few-level statistical mechanics problems (spin systems, simple electronic transitions). For systems with many more levels or a continuous energy spectrum, the sum generalizes to an integral over the density of states, outside the scope of this direct-entry calculator.