In this scenario the equation of motion for an arbitrary system operator Y is called the " quantum Langevin equation " and may be written as:
12.
This differential equation is interpreted as defining the system operator a as the quantum ItM integral of the right hand side, and is equivalent to the Langevin equation ( ).
13.
Another prototypical feature of the Langevin equation is the occurrence of the damping coefficient \ lambda in the correlation function of the random force, a fact also known as Einstein relation.
14.
There is, however, an analogue of the formula that is both fully quantum and relativistic, called the " Abraham Lorentz Dirac Langevin equation " see Johnson and Hu.
15.
Using the Langevin equation to model Brownian motion, the path integral formation can be used to determine an effective action and pre-exponential model to see the effect of dissipation on tunnelling.
16.
One of the simplest forms of the Langevin equation is when its " noise term " is Gaussian; in this case, the Langevin equation is exactly equivalent to the convection diffusion equation.
17.
One of the simplest forms of the Langevin equation is when its " noise term " is Gaussian; in this case, the Langevin equation is exactly equivalent to the convection diffusion equation.
18.
This differential equation is interpreted as defining the system operator a as the quantum Stratonovich integral of the right hand side, and is in the same form as the Langevin equation ( ).
19.
The self-diffusion mechanism itself was not directly investigated, although it had been indirectly investigated by behavior of impurity diffusion in a pure material as shown in the Einstein s Brown theory and the Langevin equation.
20.
The time evolution of the position of the Brownian particle itself is best described using Langevin equation, an equation which involves a random force field representing the effect of the thermal fluctuations of the solvent on the particle.
