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The fluctuation–dissipation theorem (FDT) or fluctuation–dissipation relation (FDR) is a powerful tool in statistical physics for predicting the behavior of systems that obey detailed balance.
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 ...
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 ).
D is the diffusion coefficient; μ is the "mobility", or the ratio of the particle's terminal drift velocity to an applied force, μ = v d /F; k B is the Boltzmann constant; T is the absolute temperature. This equation is an early example of a fluctuation-dissipation relation. [7]
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 ...
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 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.
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]