On shell renormalization scheme: Difference between revisions
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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 | |||
: <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> | |||
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. | |||
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. | |||
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 /> | |||
<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 | |||
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> | |||
W(L,t) \approx L^{\alpha} f(t/L^z) , | |||
</math> | |||
<br /> | |||
with a scaling function <math alt>f(u)</math> satisfying <br /> | |||
<math alt> | |||
f(u) \propto \left \{ \begin{array}{lr} u^{\beta} & \ u\ll 1 \\ | |||
1 & \ u\gg1\end{array} \right. | |||
</math> | |||
== Sources == | |||
<references/> | |||
<!--- 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. ---> | |||
* A.-L. Barabási and H.E. Stanley, ''Fractal concepts in surface growth'' (Cambridge University Press, 1995) | |||
* Lecture Notes by Jeremy Quastel http://math.arizona.edu/~mathphys/school_2012/IntroKPZ-Arizona.pdf | |||
* Lecture Notes by Ivan Corwin http://arxiv.org/abs/1106.1596 | |||
<noinclude> | |||
[[Category:Statistical mechanics]] | |||
</noinclude> | |||
Revision as of 07:46, 2 November 2013
The KPZ-equation[1] (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 at place and time . It is formally 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 has been conjectured that the KPZ equation is the field theory of many surface growth models, such as the Eden model, ballistic deposition, and the SOS model. A rigorous proof has been given by Bertini and Giacomin[2] in the case of the SOS model.
Many models in the field of interacting particle systems, 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, , defined as
where 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 can be characterized by the
Family-Vicsek scaling relation[3] of the roughness, where we have
with a scaling function satisfying
Sources
- ↑ M. Kardar, G. Parisi, and Y.-C. Zhang, Dynamic Scaling of Growing Interfaces, Physical Review Letters, Vol. 56, 889 - 892 (1986). APS
- ↑ L. Bertini and G. Giacomin, Stochastic Burgers and KPZ equations from particle systems, Comm. Math. Phys., Vol. 183, 571-607 (1997) [1].
- ↑ 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) [2].
- A.-L. Barabási and H.E. Stanley, Fractal concepts in surface growth (Cambridge University Press, 1995)
- Lecture Notes by Jeremy Quastel http://math.arizona.edu/~mathphys/school_2012/IntroKPZ-Arizona.pdf
- Lecture Notes by Ivan Corwin http://arxiv.org/abs/1106.1596