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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.
The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion , resulting from the random movements and collisions of the particles (see Fick's laws of diffusion ).
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]
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 ...
Another method to describe the motion of a Brownian particle was described by Langevin, now known for its namesake as the Langevin equation.) (,) = (,), given the initial condition (, =) = (); where () is the position of the particle at some given time, is the tagged particle's initial position, and is the diffusion constant with the S.I. units ...
Assuming that N particles start from the origin at the initial time t = 0, the diffusion equation has the solution (,) = (). This expression (which is a normal distribution with the mean μ = 0 {\displaystyle \mu =0} and variance σ 2 = 2 D t {\displaystyle \sigma ^{2}=2Dt} usually called Brownian motion B t {\displaystyle B_{t}} ) allowed ...
The convection–diffusion equation can be derived in a straightforward way [4] from the continuity equation, which states that the rate of change for a scalar quantity in a differential control volume is given by flow and diffusion into and out of that part of the system along with any generation or consumption inside the control volume: + =, where j is the total flux and R is a net ...
The Fokker–Planck equation for this particle is the Smoluchowski diffusion equation: (, |,) = [(()) (, |,)] Where is the diffusion constant and =. The importance of this equation is it allows for both the inclusion of the effect of temperature on the system of particles and a spatially dependent diffusion constant.