Statistical Mechanics

In 1902 Josiah Willard Gibbs, a reserved professor at Yale who had spent decades on thermodynamics, published a general method. Forget the individual molecules. Imagine every state the whole system could be in, and weight each by the factor $e^{-E/kT}$, so that states of lower energy are more likely at lower temperature. Every thermodynamic quantity can then be computed from a single sum over states. Gibbs cautioned that the approach could not explain everything: some predictions, like the heat stored in gases, disagreed with experiment. The disagreements were the first signs of quantum mechanics.

When quantum theory arrived, statistical mechanics needed only new rules for counting. In 1924 Satyendra Nath Bose, a young physicist in Dacca, derived Planck's law for light by treating photons as truly indistinguishable. His paper had reportedly been rejected, and he sent it to Einstein, who translated it himself and applied the idea to atoms. He predicted that very cold atoms would pile into a single state, a Bose–Einstein condensate. In 1926 Enrico Fermi and Paul Dirac found the rules for particles that refuse to share, such as electrons. Those two kinds of counting explain lasers and superfluids on one side, and metals, chemistry and white dwarf stars on the other.

The condensate took seventy years to make. In 1995 Eric Cornell and Carl Wieman cooled rubidium atoms to 170 billionths of a degree above absolute zero and saw them condense. Wolfgang Ketterle did the same with sodium months later.

Divide a box into two equal halves and put $N$ molecules in it, each equally likely to be in either half. What is the chance that all of them are in the left half at a given moment?

Each molecule is on the left with probability $\tfrac12$, so all $N$ are there with probability $2^{-N}$:

With ten molecules it would happen regularly. With a hundred, you would wait far longer than the age of the universe, about $4 \times 10^{17}$ seconds. With a mole, about 24 litres of gas at room temperature and pressure, the number $10^{1.8 \times 10^{23}}$ has $1.8 \times 10^{23}$ digits. Written out at one digit per millimetre, it would stretch about 19,000 light-years. Nothing forbids the gas from gathering on one side. It is just that the arrangements where it does are a vanishingly small fraction of all arrangements.

This is Boltzmann's second law. When a gas spreads from one half of a box into the whole box, each molecule has twice as many places to be, so the number of arrangements $W$ is multiplied by $2^N$. The entropy increases by

which for one mole is $6.02 \times 10^{23} \times 1.38 \times 10^{-23} \times 0.693 \approx 5.76$ J/K. That is exactly the value thermodynamics gives for this expansion, measured with heat and thermometers. The macroscopic law and the microscopic count agree.

Statistical mechanics now reaches well beyond gases. Its most spectacular successes concern systems where the particles act together, in phase transitions such as boiling and magnetisation. Its methods have spread into biology, economics and machine learning, wherever many interacting parts must be understood without following each one. Its hardest open problems are about systems that never reach equilibrium, from glasses that take longer than the age of the universe to settle to the non-equilibrium physics of living cells.