Partition function and free energy

The central object of statistical mechanics

The partition function is the sum of Boltzmann weights over all accessible states:

$Z=sum_i e^{-E_i/(k_BT)}$

For systems with continuous states, the sum becomes an integral. The partition function normalizes probabilities and contains thermodynamic information.

The probability of state $i$ is

P_i=rac{e^{-E_i/(k_BT)}}{Z}

This ensures

$sum_i P_i=1$

Why $Z$ matters

The partition function counts thermally accessible states, weighted by energy. Low-energy states contribute strongly. High-energy states contribute weakly unless temperature is high or there are many such states.

Once $Z$ is known, many macroscopic quantities can be derived. This is why statistical mechanics often focuses on calculating or approximating partition functions.

Average energy

The average energy is

angle=sum_i P_iE_i$$ Using the partition function, it can be written compactly as $$langle E angle=-rac{partial}{partial eta}ln Z$$ where $$eta=rac{1}{k_BT}$$ This notation is common in statistical mechanics. ## Helmholtz free energy The Helmholtz free energy is $$F=-k_BTln Z$$ It connects microscopic probabilities to thermodynamics. For a system at fixed temperature and volume, equilibrium corresponds to minimizing Helmholtz free energy. Thermodynamically, $$F=U-TS$$ This shows free energy balances internal energy and entropy. A system may choose a higher-energy macrostate if it has much higher entropy, because the $-TS$ term can lower $F$. ## Work and free energy At constant temperature and volume, the maximum useful work obtainable from a system is related to a decrease in Helmholtz free energy. Roughly, $$W_{max}=-Delta F$$ for reversible processes under appropriate conditions. Real processes produce entropy and yield less useful work than this ideal maximum. ## Two-level partition function For a two-level system with energies $0$ and $epsilon$, $$Z=1+e^{-epsilon/(k_BT)}$$ The excited-state probability is $$P_{excited}=rac{e^{-epsilon/(k_BT)}}{1+e^{-epsilon/(k_BT)}}$$ At low temperature, $P_{excited}$ is near zero. At high temperature, the two states become nearly equally populated if they have equal degeneracy. ## Entropy from free energy Free energy also gives entropy: $$S=-left(rac{partial F}{partial T} ight)_V$$ This relationship shows that entropy can be derived from how free energy changes with temperature. ## The big idea The partition function sums the thermal weights of all accessible states and normalizes the probability distribution. From $Z$, one can calculate average energy, free energy, entropy, and other thermodynamic quantities. Helmholtz free energy links microscopic state counting to equilibrium and maximum useful work.