One of the main goals of statistical mechanics is to find equilibrium states of systems of many interacting particles and expected values of relevant random variables, for example magnetization in the Ising model. At low temperatures, equilibrium states may be constructed as small perturbations of ground states. Here we will discuss constructions of non-equilibrium stationary states in two simple models: gene self-regulation and Kawasaki dynamics in ion channels.

• Mathematical cell: Stochastic model of gene expression
• Intermission: Markov jump processes and detailed balance
• Main actor: Self-repressing gene
• Mean-field approximation
• Expansions
  • around detailed balance
  • around adiabatic regime
• Kawasaki dynamics in ion channels
• Gibbs states versus non-equilibrium stationary states
• Ground states versus stochastically stable states