Statistical mechanics, structure of matter

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• ConceptPartition function

  The partition function $p(n)$ gives the number of integer partitions of $n$. It is a classical object of additive number theory, with a product generating function and deep congruences. The node partition function gives the precise definition.

• ConceptGibbs measure

  A Gibbs measure is a probability measure in which the weight of a configuration is proportional to $e^{-\text{energy}}$. It comes from statistical mechanics, where it describes a system in thermal equilibrium. In dynamics, the thermodynamic formalism uses Gibbs measures on shift spaces. There the measure of a cylinder of length $n$ is comparable to the exponential of a Birkhoff sum of a potential. For a Hölder potential on a mixing subshift of finite type the Gibbs measure is unique and is the equilibrium state of the potential.

• ConceptGross–Pitaevskii equation

  The Gross–Pitaevskii equation describes the ground state and the dynamics of a Bose–Einstein condensate. It is a nonlinear Schrödinger equation with a cubic term, which models the interaction between the particles, and an external trapping potential. Its unknown is a single complex wave function for the whole condensate. It can be derived from the many-body quantum mechanics of bosons in a suitable limit of many weakly interacting particles.

• ExampleIsing model

  Let $G = (V, E)$ be a finite graph, and let $J > 0$, $h \in \R$ and $\beta \ge 0$ be real numbers. A spin configuration is a map $\sigma \colon V \to \set{-1, +1}$. Its energy is $$H(\sigma) = -J \sum_{\set{u, v} \in E} \sigma(u) \sigma(v) - h \sum_{v \in V} \sigma(v).$$ The Ising model on $G$ at inverse temperature $\beta$ is the probability measure on $\set{-1, +1}^V$ given by $$\mu_\beta(\sigma) = \frac{e^{-\beta H(\sigma)}}{Z_\beta}, \qquad Z_\beta = \sum_{\tau \in \set{-1, +1}^V} e^{-\beta H(\tau)}.$$ Check: each weight $e^{-\beta H(\sigma)}$ is positive and there are finitely many configurations. So $Z_\beta > 0$, and the values of $\mu_\beta$ are positive and sum to $1$.

• ConceptBoltzmann equation

  The Boltzmann equation describes a dilute gas through the density $f(t, x, v)$ of particles at position $x$ with velocity $v$. Particles move freely in straight lines and change velocity only in binary collisions, which the collision operator accounts for. It links the motion of individual molecules with the equations of fluid dynamics. Boltzmann's H-theorem shows that a certain entropy cannot decrease, which explains the approach to equilibrium.

• ConceptDetailed balance

  Detailed balance is a property of a distribution $\pi$ and a Markov chain. It says that, in equilibrium, the probability flow from each state $i$ to each state $j$ equals the flow from $j$ back to $i$. Detailed balance implies that $\pi$ is a stationary distribution, and a chain with such a $\pi$ is called reversible. It is the usual design principle of Markov chain Monte Carlo algorithms.