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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 ).
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 ...
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 ...
Fick's first law relates the diffusive flux to the gradient of the concentration. It postulates that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient (spatial derivative), or in simplistic terms the concept that a solute will move from a region of high concentration to a region of low ...
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.
The narrow escape problem is a ubiquitous problem in biology, biophysics and cellular biology which has the following formulation: a Brownian particle (ion, molecule, or protein) is confined to a bounded domain (a compartment or a cell) by a reflecting boundary, except for a small window through which it can escape. The narrow escape problem is ...
The convection–diffusion equation describes the flow of heat, particles, or other physical quantities in situations where there is both diffusion and convection or advection. For information about the equation, its derivation, and its conceptual importance and consequences, see the main article convection–diffusion equation. This article ...