Statistical Mechanics Basics

Microstates and Macrostates

A microstate is a complete specification of the positions and momenta of every particle in a system. A macrostate is described by macroscopic variables like temperature, pressure, and volume. Many different microstates can correspond to the same macrostate. The number of microstates W corresponding to a given macrostate is called the multiplicity, and it is the key quantity in statistical mechanics.

For a system in thermal equilibrium, the fundamental postulate of statistical mechanics states that all accessible microstates are equally probable. This means the system spends equal time in each microstate, and the macroscopic properties are averages over all microstates. The macrostate that is observed is overwhelmingly likely to be the one with the largest number of microstates, because the probability ratios between macrostates scale exponentially with particle number.

Boltzmann entropy S = k ln W directly connects multiplicity to the thermodynamic quantity entropy. The second law of thermodynamics, which states that entropy tends to increase, is revealed by statistical mechanics to be a statistical tendency: systems evolve toward macrostates with more microstates because those states are overwhelmingly more probable. For macroscopic systems with 10²³ particles, this tendency is so strong that spontaneous entropy decreases are never observed.

The Boltzmann Distribution

When a system is in thermal equilibrium at temperature T, the probability of finding it in a microstate with energy E is proportional to exp(-E/kT), where k is Boltzmann constant. This is the Boltzmann distribution, the most important probability distribution in statistical mechanics. It tells us that lower-energy states are more probable than higher-energy states, with the ratio depending on temperature.

At low temperatures (kT much less than the energy spacing), almost all particles occupy the lowest energy state. At high temperatures (kT much greater than the energy spacing), particles are spread more evenly across many energy levels. The partition function Z = sum of exp(-E{sub}i{/sub}/kT) over all microstates normalizes the probabilities and encodes all thermodynamic information about the system. Once Z is known, every thermodynamic quantity (energy, entropy, free energy, pressure) can be calculated from it.

The Boltzmann distribution explains a vast range of phenomena. The barometric formula for atmospheric pressure (exponential decrease with altitude) follows directly from the Boltzmann distribution applied to air molecules in a gravitational field. The population ratios of atomic energy levels in stellar atmospheres determine spectral line intensities. Chemical reaction rates depend exponentially on activation energy through the Boltzmann factor.

Ensembles and Averages

An ensemble is an imaginary collection of many copies of a system, each in a different microstate but sharing the same macroscopic constraints. The microcanonical ensemble describes isolated systems with fixed energy, volume, and particle number. The canonical ensemble describes systems in thermal contact with a heat bath at fixed temperature, volume, and particle number. The grand canonical ensemble allows both energy and particle exchange.

Thermodynamic quantities are ensemble averages. The internal energy is the average energy over all microstates, weighted by their Boltzmann probabilities. Pressure is the average force per unit area exerted by particles on the container walls. Temperature is related to the average kinetic energy per degree of freedom: (1/2)kT per quadratic degree of freedom (the equipartition theorem). These averages become exact in the thermodynamic limit of large particle numbers.

The fluctuations around average values decrease relative to the average as the system size increases. For N particles, relative fluctuations scale as 1/sqrt(N). For a mole of gas (N approximately 6 x 10²³), relative fluctuations in energy are about 10^-12, which is undetectably small. This is why macroscopic thermodynamics works so well: the averages predicted by statistical mechanics are essentially exact for real-world systems.

Quantum Statistical Mechanics

At low temperatures or high densities, quantum effects become important and the classical Boltzmann distribution must be replaced. Fermions (particles with half-integer spin, like electrons, protons, and neutrons) obey Fermi-Dirac statistics, which forbids two identical fermions from occupying the same quantum state (the Pauli exclusion principle). Bosons (particles with integer spin, like photons and helium-4 atoms) obey Bose-Einstein statistics, which allows unlimited occupation of any quantum state.

Fermi-Dirac statistics explains the electronic properties of metals, semiconductors, and insulators. At absolute zero, electrons fill energy states up to the Fermi energy, above which all states are empty. The existence of this sharp boundary, a purely quantum effect, determines electrical conductivity, heat capacity, and many other properties of solid materials. The free electron model of metals, based on Fermi-Dirac statistics, was one of the first great successes of quantum statistical mechanics.

Bose-Einstein statistics leads to remarkable collective phenomena. At sufficiently low temperatures, a macroscopic fraction of bosons condenses into the lowest energy state, forming a Bose-Einstein condensate (BEC). First achieved experimentally in 1995 by Eric Cornell and Carl Wieman using rubidium atoms cooled to nanokelvin temperatures, BECs exhibit quantum behavior on macroscopic scales. Superfluidity in liquid helium-4 and the laser (stimulated emission of photons) are also consequences of Bose-Einstein statistics.