On shell renormalization scheme

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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

  1. M. Kardar, G. Parisi, and Y.-C. Zhang, Dynamic Scaling of Growing Interfaces, Physical Review Letters, Vol. 56, 889 - 892 (1986). APS
  2. L. Bertini and G. Giacomin, Stochastic Burgers and KPZ equations from particle systems, Comm. Math. Phys., Vol. 183, 571-607 (1997) [1].
  3. 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].