that, as is well known [2], is identically zero. A numerical analysis of Eq. (53) indicates that this expression is several orders of magnitude smaller than the value of the pdf’s exponent [Eq. (52)], and typically behaves as O (1 /N ), where N is the number of spatial points used in the discretization. Moreover, using expressions with higher accuracy for the differential operators one gets an even faster approach to zero. This indicates that the problem with the fluctuation–dissipation theorem in 1+1, discussed in [11, 15] can be just circumvented using more accurate expressions. It is also worth commenting that, if a consistent discrete scheme is built from the discrete scheme in [11], it would also violate the fluctuation–dissipation relation. In Ref. [26], a variational formulation was introduced for the KPZ equation. There it was shown that Eq. (1) can be written as where (for F = 0) and the function Γ( h ) is given by The way in which the functionals F [ φ ] and G [ h ] are related is also shown in Ref. [26]. It is also easy to prove that the functional G [ h ] fulfills the Lyapunov property ∂$_{t}$ G [ h ] ≤ 0. According to the previous results, we can write the discrete version of Eq. (55) as Now, introducing this expression into the discrete version of Eq. (54), and through a simple algebra, we reobtain Eq. (7). This reinforces our result, and clearly indicates the need to be consistent when considering a discrete version of the KPZ equation. 16 G [ h ] = λ 2 8 ν ∑ j e λ $_{$_{ν}$h$_{j}$}$1 2 [ ( H j +1 j ) 2 + ( H j − 1 j ) 2 ] . Γ( h ) = ( 2 ν λ ) 2 e − λ $^{$_{ν}$h}$. G [ h ] = ∫ $_{Ω}$e λ $_{ν}$h ( x,t $_{)}$λ 2 8 ν [ ∂$_{x}$h ( x, t )] $^{2}$d x, (55) ∂$_{t}$h ( x, t ) = − Γ( h ) δ G [ h ] δh ( x, t ) + ε ξ ( x, t ); (54) VI. ON THE VARIATIONAL FORMULATION OF KPZ