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| The KPZ-equation<ref>[[M. Kardar]], [[G. Parisi]], and Y.-C. Zhang, ''Dynamic Scaling of Growing Interfaces'', Physical Review Letters, Vol. '''56''', 889 - 892 (1986). [http://prl.aps.org/abstract/PRL/v56/i9/p889_1 APS]</ref> (named after its creators [[Mehran Kardar]], [[Giorgio Parisi]], and Yi-Cheng Zhang) is a non-linear [[stochastic partial differential equation]]. It describes the temporal change of the height <math>h(\vec x,t)</math> at place <math>\vec x</math> and time <math>t</math>. It is formally given by
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| : <math>\frac{\partial h(\vec x,t)}{\partial t} = \nu \nabla^2 h + \frac{\lambda}{2} \left(\nabla h\right)^2 + \eta(\vec x,t) \; ,</math>
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| where <math>\eta(\vec x,t)</math> is [[White noise|white]] [[Gaussian noise]] with average <math>\langle \eta(\vec x,t) \rangle = 0</math> and second moment <math>\langle \eta(\vec x,t) \eta(\vec x',t') \rangle = 2D\delta^d(\vec x-\vec x')\delta(t-t')</math>. <math>\nu</math>, <math>\lambda</math>, and <math>D</math> are parameters of the model and <math>d</math> is the dimension.
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| By use of [[renormalization group]] techniques it has been conjectured that the KPZ equation is the field theory of many [[surface growth]] models, such as the [[Eden growth model|Eden model]], ballistic deposition, and the SOS model. A rigorous proof has been given by Bertini and Giacomin<ref>L. Bertini and G. Giacomin, ''Stochastic Burgers and KPZ equations from particle systems'', Comm. Math. Phys., Vol. '''183''', 571-607 (1997) [http://link.springer.com/article/10.1007%2Fs002200050044].</ref> in the case of the SOS model.
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| Many models in the field of [[interacting particle system]]s, such as the totally [[asymmetric simple exclusion process]], also lie in the KPZ universality class. This class is characterised by models which, in one spatial dimension (1+1 dimension) have a roughness exponent α=1/2, growth exponent β=1/3 and dynamic exponent z=3/2. In order to check if a growth model is within the KPZ class, one can calculate the width of the surface, <math>W(L,t)</math>, defined as <br />
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| <math alt>W(L,t)=\Big\langle\frac1L\int_0^L \big( h(x,t)-\bar{h}(t)\big)^2 dx\Big\rangle^{1/2},</math> <br /> where <math alt=> \bar{h}(t) </math> is the mean surface height at time t and L is the size of the system. For models within the KPZ class, the main properties of the surface <math alt> h(x,t) </math> can be characterized by the
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| Family-Vicsek scaling relation<ref>F. Family and T. Vicsek, ''Scaling of the active zone in the Eden process on percolation networks and the ballistic deposition model'', J. Phys. A: Math. Gen., Vol. '''18''', L75-L81 (1985) [http://iopscience.iop.org/0305-4470/18/2/005].</ref> of the roughness, where we have <br /> <math alt>
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| W(L,t) \approx L^{\alpha} f(t/L^z) ,
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| </math>
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| <br />
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| with a scaling function <math alt>f(u)</math> satisfying <br />
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| <math alt>
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| f(u) \propto \left \{ \begin{array}{lr} u^{\beta} & \ u\ll 1 \\
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| 1 & \ u\gg1\end{array} \right.
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| </math>
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| == Sources == | |
| <references/>
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| <!--- After listing your sources please cite them using inline citations and place them after the information they cite. Please see http://en.wikipedia.org/wiki/Wikipedia:REFB for instructions on how to add citations. --->
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| * A.-L. Barabási and H.E. Stanley, ''Fractal concepts in surface growth'' (Cambridge University Press, 1995)
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| * Lecture Notes by Jeremy Quastel http://math.arizona.edu/~mathphys/school_2012/IntroKPZ-Arizona.pdf
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| * Lecture Notes by Ivan Corwin http://arxiv.org/abs/1106.1596
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| <noinclude>
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| [[Category:Statistical mechanics]]
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| </noinclude>
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