TY - JOUR
T1 - Nonlocal growth equations - A test case for dynamic renormalization group analysis
AU - Schwartz, Moshe
AU - Katzav, Eytan
PY - 2003/12/1
Y1 - 2003/12/1
N2 - In this paper we discuss nonlocal growth equations such as the generalization of the Kardar-Parisi-Zhang (KPZ) equation that includes long-range interactions, also known as the Nonlocal-Kardar-Parisi-Zhang (NKPZ) equation, and the nonlocal version of the molecular-beam-epitaxy (NMBE) equation. We show that the steady-state strong coupling solution for nonlocal models such as NKPZ and NMBE can be obtained exactly in one dimension for some special cases, using the Fokker-Planck form of these equations. The exact results we derive do not agree with previous results obtained by Dynamic Renormalization Group (DRG) analysis. This discrepancy is important because DRG is a common method used extensively to deal with nonlinear field equations. While difficulties with this method for d > 1 has been realized in the past, it has been believed so far that DRG is still safe in one dimension. Our result shows differently. The reasons for the failure of DRG to recover the exact one-dimensional results are also discussed.
AB - In this paper we discuss nonlocal growth equations such as the generalization of the Kardar-Parisi-Zhang (KPZ) equation that includes long-range interactions, also known as the Nonlocal-Kardar-Parisi-Zhang (NKPZ) equation, and the nonlocal version of the molecular-beam-epitaxy (NMBE) equation. We show that the steady-state strong coupling solution for nonlocal models such as NKPZ and NMBE can be obtained exactly in one dimension for some special cases, using the Fokker-Planck form of these equations. The exact results we derive do not agree with previous results obtained by Dynamic Renormalization Group (DRG) analysis. This discrepancy is important because DRG is a common method used extensively to deal with nonlinear field equations. While difficulties with this method for d > 1 has been realized in the past, it has been believed so far that DRG is still safe in one dimension. Our result shows differently. The reasons for the failure of DRG to recover the exact one-dimensional results are also discussed.
KW - Exact result
KW - KPZ equation
KW - Nonlocal models
UR - http://www.scopus.com/inward/record.url?scp=0344118206&partnerID=8YFLogxK
U2 - 10.1016/j.physa.2003.08.013
DO - 10.1016/j.physa.2003.08.013
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AN - SCOPUS:0344118206
SN - 0378-4371
VL - 330
SP - 91
EP - 98
JO - Physica A: Statistical Mechanics and its Applications
JF - Physica A: Statistical Mechanics and its Applications
IS - 1-2
T2 - Randomes and Complexity
Y2 - 5 January 2003 through 9 January 2003
ER -