KPZ-Equation
Encyclopedia
The KPZ-equation is a non-linear, stochastic
, partial differential equation
. It describes the temporal change of the height
at place 
and time 
. It is given by
where
is white
Gaussian noise
with average
and second moment 
. 
, 
, and 
are parameters of the model and 
is the dimension.
By use of renormalization group
techniques it can be shown that the KPZ-equation is the field theory of many surface growth models, such as the Eden model
, ballistic deposition, and the SOS model.
Langevin equation
In statistical physics, a Langevin equation is a stochastic differential equation describing the time evolution of a subset of the degrees of freedom. These degrees of freedom typically are collective variables changing only slowly in comparison to the other variables of the system...
, partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
. It describes the temporal change of the height
at place and time . It is given by
where
is whiteWhite noise
White noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...
Gaussian noise
Gaussian noise
Gaussian noise is statistical noise that has its probability density function equal to that of the normal distribution, which is also known as the Gaussian distribution. In other words, the values that the noise can take on are Gaussian-distributed. A special case is white Gaussian noise, in which...
with average
and second moment . , , and are parameters of the model and is the dimension.By use of renormalization group
Renormalization group
In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...
techniques it can be shown that the KPZ-equation is the field theory of many surface growth models, such as the Eden model
Eden growth model
The Eden growth model describes the growth of specific types of clusters such as bacterial colonies and deposition of materials. These clusters grow by random accumulation of material on their boundary. These are also an example of a surface fractal....
, ballistic deposition, and the SOS model.
Sources
- Mehran KardarMehran KardarMehran Kardar is a prominent Iranian born physicist, full Professor of Physics at the Massachusetts Institute of Technology, and co-faculty at the New England Complex Systems Institute. He received his B.A...
, Giorgio ParisiGiorgio ParisiGiorgio Parisi is an Italian theoretical physicist. He is best known for his works concerning statistical mechanics, quantum field theory and various aspects of physics, mathematics and science in general....
, and Yi-Cheng Zhang, Dynamic Scaling of Growing Interfaces, Physical Review Letters, Vol. 56, 889 - 892 (1986). APS - A.-L.Barabási and H.E.Stanley, Fractal concepts in surface growth (Cambridge University Press, 1995)