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the latter. The idea of RG is to average equations of motion for low-momentum (slow) components over the fast oscillations. The averaging is trivially done in the last term of Eq. (A9). There it is sufficient to use zeroth order in θ $^{>}$. Using that sin $^{2}$( ω$_{Λ}$t ) , cos $^{2}$( ω$_{Λ}$t ) = 1 / 2 we find that averaging of the last term simply renormalizes the coupling g : 

where E$_{k}$ is the average energy of the mode k over the period (we used the fact that λk $^{2}$| θ$_{k}$ | 2 = E$_{k}$ ). We note that in a Boltzmann ensemble, we would have E$_{k}$ = 1, because the energies here are in units of the tempera- ture T . With this assumption we would recover the flow equation of the equilibrium case. 

Instead of this assumption, we proceed by noting that under RG transformations coupling constants slowly change in time. This implies that the adiabatic invariants per each mode are approximately conserved, as discussed in Ref. 34. For an oscillator the adiabatic invariant is I$_{k}$ = E$_{k}$/ω$_{k}$ . Thus we see that the energy of the mode is proportional to the frequency. Noting that at initial time E$_{k}$ ( t = 0) = 1 (in non-rescaled units this would be E$_{k}$ ( t = 0) = T$_{0}$ , where T$_{0}$ is the initial non-equilibrium temperature), one can rewrite Eq. (A10) as follows: 

where we introduced the analogue of the Luttinger- Liquid parameter K = √ λ/µ . v is the velocity which is now given by v = √$_{λµ}$ (note that in the original, not rescaled units, v = $_{T}$√$_{λµ}$$_{).}$ 

Next let us consider the second term in Eq. (A9). This term is more subtle since if we use θ > 0 the average over fast fluctuations will give zero. So we need to use the first correction θ > 1 , which would be a correction at sec- ond order in g . We note that the linear term in Eq. (A9) 

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can also be expanded to second in g , generating simi- lar contributions. However, when written as Eq. (A1), such a term a cancelled by a corresponding term from expanding d $^{2}$θ/dt$^{2}$. Alternatively we can view Eq. (A9) as written for the θ < component, this automatically en- sures that only the nonlinear term is responsible for the renormalization. 

Let us look closer into the Eq. (A7). We are dealing with the integral over the fast oscillating function of τ : sin( ω$_{Λ}$ ( t − τ )) and the slow oscillating function F . This integral can be evaluated by integrating by parts: 

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Note that the second integral contains a large denomina- tor 1 /ω$_{Λ}$ . In the first term only the limit τ = t gives a nonoscillating contribution to the integral. We can con- tinue the expansion in powers of 1 /ω$_{Λ}$ . Note that the next term proportional to ˙ F$_{1}$ will contain only highly os- cillatory part and can be neglected. So up to the third order in 1 /ω$_{Λ}$ we find: 

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Combining Eqs. (A7), (A8), and (A13) we find 

where we used Λ a = 1 again. Here we neglected by the term proportional to ( ˙ θ $^{<}$( r , t )) $^{2}$, because it leads to a subdominant (in the RG sense) contribution. From this we find that 

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We now consider the term ( g/a $^{2}$) cos( θ $^{<}$( r , t )) θ > 1 ( r , t ): 

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where we neglected terms such as sin(2 θ $^{<}$( r , t )). Because we are integrating over high momentum shell this integral will be suppressed unless x is close to r . This suggests change of variables x = r + ξ and Taylor expanding θ $^{<}$( x , t ) 

