FlucuationDissipationTheorems - crowlogic/arb4j GitHub Wiki

Fluctuation-dissipation theorems are a set of fundamental principles in statistical mechanics that establish a connection between the fluctuating behavior of a system and its dissipative properties. These theorems relate the response of a system to an external perturbation (fluctuation) with its relaxation or dissipation behavior.

The basic idea behind fluctuation-dissipation theorems is that the random or fluctuating behavior of a system, often described by statistical quantities like correlation functions or response functions, is intimately connected to how the system dissipates energy or relaxes back to equilibrium.

There are several versions of fluctuation-dissipation theorems, each tailored to different physical situations and systems. However, they all share a common essence: they establish a relationship between the fluctuation spectrum of a system and its dissipation spectrum.

One of the most well-known fluctuation-dissipation theorems is the Fluctuation-Dissipation Theorem (FDT) for equilibrium systems. It states that the response of a system to a small perturbation, such as an external force or a change in an external parameter, is related to the system's equilibrium fluctuations through a fluctuation-dissipation relation.

Mathematically, the FDT can be expressed as follows:

$$ C(\omega) = \frac{2 k_B T}{\omega} \text{Im}[G(\omega)] $$

where C(ω) is the correlation function of the system, G(ω) is the response function, k_B is the Boltzmann constant, T is the temperature, and ω is the frequency. This equation relates the Fourier transforms of the correlation and response functions.

The FDT essentially states that the fluctuation spectrum (characterized by the correlation function) and the dissipation spectrum (characterized by the imaginary part of the response function) are proportional to each other, with the temperature acting as a proportionality constant.

Fluctuation-dissipation theorems have important implications in various areas of physics, including condensed matter physics, quantum field theory, and nonequilibrium statistical mechanics. They provide a powerful tool for understanding the behavior of systems in thermal equilibrium and predicting their dynamic response to external perturbations.

In summary, fluctuation-dissipation theorems establish a deep connection between the fluctuating behavior of a system and its dissipative properties, allowing us to relate the statistical fluctuations of a system to its response to perturbations and gain insights into its dynamics.