However, the Langevin equation is used to describe the motion of a " macroscopic " particle at a much longer time scale, and in this limit the \ delta-correlation and the Langevin equation become exact.
22.
However, the Langevin equation is used to describe the motion of a " macroscopic " particle at a much longer time scale, and in this limit the \ delta-correlation and the Langevin equation become exact.
23.
A Langevin equation is a coarse-grained version of a more microscopic model; depending on the problem in consideration, Stratonovich or It?interpretation or even more exotic interpretations such as the isothermal interpretation, are appropriate.
24.
The time evolution of the position of the Brownian particle itself can be described approximately by a Langevin equation, an equation which involves a random force field representing the effect of the thermal fluctuations of the solvent on the Brownian particle.
25.
However, in the case of multiplicative noise, the Langevin equation is not a well-defined entity on its own, and it must be specified whether the Langevin equation should be interpreted as an It?SDE or a Stratonovich SDE.
26.
However, in the case of multiplicative noise, the Langevin equation is not a well-defined entity on its own, and it must be specified whether the Langevin equation should be interpreted as an It?SDE or a Stratonovich SDE.
27.
The Wong Zakai theorem states that physical systems with non-white noise spectrum characterized by a finite noise correlation time ? can be approximated by a Langevin equations with white noise in Stratonovich interpretation in the limit where ? tends to zero.
28.
Note that inertial effects have to be considered in the Langevin equation, otherwise the equation becomes singular . so that simply removing the inertia term from this equation would not yield an exact description, but rather a singular behavior in which the particle doesn't move at all.
29.
Additive noise is the simpler of the two cases; in that situation the Langevin equation has only one natural notion of solution and, in the linear case, this solution has an explicit expression found using the ordinary rules of calculus as if the \ eta _ m were functions.
30.
As a result, for isotropic material ( where K _ \ text { an } = 0 ) ), presented form of anisotropic anhysteretic magnetization M _ \ text { an } ^ \ text { aniso } is not coherent with isotropic anhysteretic magnetization M _ \ text { an } ^ \ text { iso } given by Langevin equation.
