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The fluctuation–dissipation theorem is a general result of statistical thermodynamics that quantifies the relation between the fluctuations in a system that obeys detailed balance and the response of the system to applied perturbations.
Atomic diffusion on the surface of a crystal. The shaking of the atoms is an example of thermal fluctuations. Likewise, thermal fluctuations provide the energy necessary for the atoms to occasionally hop from one site to a neighboring one. For simplicity, the thermal fluctuations of the blue atoms are not shown.
This random motion is described by a differential equation, known as the diffusion equation. The diffuson is the Green's function of the diffusion equation. [1] The diffuson plays an important role in the theory of electron transport in disordered systems, especially for phase coherent effects such as universal conductance fluctuations. [3]
As mentioned above, chemical molar flux of a component A in an isothermal, isobaric system is defined in Fick's law of diffusion as: = where the nabla symbol ∇ denotes the gradient operator, D AB is the diffusion coefficient (m 2 ·s −1) of component A diffusing through component B, c A is the concentration (mol/m 3) of component A. [9]
In physics, a Langevin equation (named after Paul Langevin) is a stochastic differential equation describing how a system evolves when subjected to a combination of deterministic and fluctuating ("random") forces. The dependent variables in a Langevin equation typically are collective (macroscopic) variables changing only slowly in comparison ...
The general equation can then be written as [6] = + + (),. where the "force" term corresponds to the forces exerted on the particles by an external influence (not by the particles themselves), the "diff" term represents the diffusion of particles, and "coll" is the collision term – accounting for the forces acting between particles in collisions.
The conservation of mass is expressed locally by the fact that the flow of mass density satisfies the continuity equation: + =, where is the mass flux vector. The formulation of energy conservation is generally not in the form of a continuity equation because it includes contributions both from the macroscopic mechanical energy of the fluid flow and of the microscopic internal energy.
A turbulent event is a series of turbulent fluctuations that contain more energy than the average flow turbulence. [ 11 ] [ 12 ] The turbulent events are associated with coherent flow structures such as eddies and turbulent bursting, and they play a critical role in terms of sediment scour, accretion and transport in rivers as well as ...