Ergodicity

Relation to Kubo transport coeffecients

The Kubo formulas are a set of equations that describe linear response theory, which is a way to explain the behavior of a physical system when it's gently stimulated or slightly adjusted from its equilibrium state. This theory is very important in quantum mechanics and statistical mechanics, and is used to calculate various transport coefficients, such as electrical conductivity, thermal conductivity, and susceptibility.

The Kubo transport coefficients are derived from the correlation functions of quantum mechanical operators that represent the observables in the system. These correlation functions can be calculated using the methods of statistical mechanics and are then Fourier transformed to give the transport coefficients.

The KMS (Kubo-Martin-Schwinger) theorem, on the other hand, is a result in quantum statistical mechanics that provides a connection between quantum systems at thermal equilibrium and analytic properties of certain functions associated with the system. It essentially states that for a system in thermal equilibrium, the correlation functions satisfy a certain condition, known as the KMS condition.

The KMS condition plays a crucial role in deriving the Kubo formulas. It can be viewed as a foundational principle of quantum statistical mechanics that ensures that the system's response to a mild adjustment is determined by the properties of the system in equilibrium. In this manner, the KMS theorem supports the use of Kubo formulas to calculate transport coefficients.

Therefore, the Kubo transport coefficients offer us a practical means to understand the behavior of many-body quantum systems, while the KMS theorem provides the foundational principle underlying this approach.