Discretization-related issues in the KPZ equation: Consistency, Galilean-invariance violation, and fluctuation–dissipation relation H. S. Wio, 1 J. A. Revelli, 1 R. R. Deza, 2 C. Escudero, 3 and M. S. de La Lama 1 $^{1}$Instituto de F´ısica de Cantabria (UC and CSIC), Avda. de los Castros, s/n, E-39005 Santander, Spain. $^{2}$Instituto de Investigaciones F´ısicas Mar del Plata (UNMdP and CONICET), De´ an Funes 3350, B7602AYL Mar del Plata, Argentina. $^{3}$ICMAT (CSIC-UAM-UC3M-UCM), Departamento de Matem´ aticas, Facultad de Ciencias, Universidad Aut´ onoma de Madrid, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain. (Dated: October 29, 2018) In order to perform numerical simulations of the KPZ equation, in any dimensionality, a spatial discretization scheme must be prescribed. The known fact that the KPZ equation can be obtained as a result of a Hopf–Cole transformation applied to a diffusion equation (with multiplicative noise) is shown here to strongly restrict the arbitrariness in the choice of spatial discretization schemes. On one hand, the discretization prescriptions for the Laplacian and the nonlinear (KPZ) term cannot be independently chosen. On the other hand, since the discretization is an operation performed on space and the Hopf–Cole transformation is local both in space and time, the former should be the same regardless of the field to which it is applied. It is shown that whereas some discretization schemes pass both consistency tests, known examples in the literature do not. The requirement of consistency for the discretization of Lyapunov functionals is argued to be a natural and safe starting point in choosing spatial discretization schemes. We also analyze the relation between real-space and pseudo-spectral discrete representations. In addition we discuss the relevance of the Galilean invariance violation in these consistent discretization schemes, and the alleged conflict of standard discretization with the fluctuation–dissipation theorem, peculiar of 1D. arXiv:1001.2351v1 [cond-mat.stat-mech] 14 Jan 2010 1 Abstract