Nonlinear Fluctuating Hydrodynamics in One Dimension: The Case of Two Conserved Fields

Abstract

We study the BS model, which is a one-dimensional lattice field theory taking real values. Its dynamics is governed by coupled differential equations plus random nearest neighbor exchanges. The BS model has two locally conserved fields. The peak structure of their steady state space–time correlations is determined through numerical simulations and compared with nonlinear fluctuating hydrodynamics, which predicts a traveling peak with KPZ scaling function and a standing peak with a scaling function given by the maximally asymmetric Lévy distribution with parameter $\alpha = 5/3$. As a by-product, we completely classify the universality classes for two coupled stochastic Burgers equations with arbitrary coupling coefficients.

1 Introduction

As recognized for some time [1–6], one-dimensional systems generically have anomalous transport properties. They can be observed through the super-diffusive spreading of small perturbations in a homogeneous steady state. An alternative, equally popular route is to consider a system of finite length, $L$, and to impose a fixed difference in the value of conserved fields at the two boundary points. For regular transport the resulting steady state current behaves as $L^{-1}$, while anomaly means an enhanced current of order $L^{-1 + \tilde{\alpha }}$ with some $\tilde{\alpha }> 0$.

Recently it has been proposed that such anomalous transport could be understood through a nonlinear extension of fluctuating hydrodynamics [7]. This method presupposes the availability of locally conserved fields, say $n$ in total number, where $n = 1,2,3$ mostly. The dynamics can be quite general, classical, quantum, stochastic, under the restriction of being translation invariant and having sufficiently local interactions. The basic construction is easily explained: One first has to identify all locally conserved fields, $n$ of them. Integrable systems are thereby ruled out because their number of conserved fields is proportional to system size. The dynamics admits then an $n$-parameter family of translation invariant steady states. For them one has to compute the steady state average currents, which thus are functions of the steady state average of the conserved fields. In order to have anomalous behavior these macroscopic current functions have to be nonlinear. There are models in which the currents are identically zero (or linear), which is then a strong indication for regular, diffusive transport.

Even if the current functions are nonlinear, there are still several distinct universality classes. To systematically explore their structure is one goal of our contribution. While one could consider the general case of $n$ conserved fields, it seems to us more instructive to stick to the simplest case of $n=2$, which already exhibits the main mechanisms at work. The one-component case has been studied in great detail under the heading of one-dimensional Kardar–Parisi–Zhang (KPZ) equation [8], see also the recent reviews [9–12]. We will make use of these results, but our focus is on the novel features arising for $n =2$.

In the following we will consider only the spreading of small perturbations, which is identical to investigating the steady state space–time correlations of the conserved fields. (The issue of steady states with open boundaries remains as a challenge for the future.) We will work out their scaling behavior on the basis of nonlinear fluctuating hydrodynamics, partially exact, partially approximate, and provide a complete classification of the universality classes for $n =2$.

Such results are of interest only when compared with microscopic models which are accessible through numerical simulations. This still leaves a very wide choice, but there are constraints. Firstly it is convenient to have a lattice type model. In addition, the steady states should be explicit. In the most favorable cases the steady states are of product form, for which static averages are then easily obtained. As an aside, thereby one can also compute explicitly the non-universal coefficients, making our predictions more pointed. Of course, we also would like the microscopic model to have features which have not been observed before. Our choice here is a one-dimensional lattice field theory taking real values. The dynamics is given by a system of coupled differential equations and admits two locally conserved quantities. To have good space–time mixing, and to avoid spurious conservation laws, we add a simple stochastic exchange term. Our model has been first introduced in [13]. Its novel feature is the two peak structure for the steady state correlation functions, one of them travels with a strictly negative velocity and has KPZ scaling, while the other one is standing still and has a scaling function given by the Lévy distribution with parameter $\alpha = 5/3$ and maximal asymmetry $b = 1$. Numerical simulations are performed for two different potentials, the exponential potential and an asymmetric FPU potential.

To provide a brief outline, in Sect. 2 we discuss the universality classes for a generic two-component system in the framework of nonlinear fluctuating hydrodynamics. The microscopic model is introduced in Sect. 3, while the results of the respective numerical solutions are reported in Sect. 4. Extra material, requiring more lengthy computations, is shifted to Appendices.

2 Two-Component Stochastic Burgers Equation

Stochastic Burgers equations are a convenient way to formulate systems of hyperbolic conservation laws including noise. The nonlinearity of the systematic currents are kept to quadratic order. A linear dissipative term is also included. All other degrees of freedom are subsumed as fluctuating currents, for simplicity modeled as space–time white noise. The resulting system of stochastic conservation laws is somewhat singular [14], but extremely useful in classifying the various universality classes. Applications concern suitably discrete versions. In our contribution we restrict ourselves to the case of two components, which already illustrates well the main features of systems with an arbitrary number of components.

2.1 One-Component Systems

Let us first briefly recall the case of a single component, $u_1(x,t)$, which by assumption is governed by the stochastic Burgers equation

$$\begin{aligned} \partial _tu_1+\partial _x\Big (c u_1 + G^1_{11}u^2_1 -D\partial _x u_1 +\sqrt{2D}\xi _1\Big )=0, \end{aligned}$$

(2.1)

where $D> 0$ is the viscosity, $c \in \mathbb {R}$ the velocity of propagation, $G_{11}^1 \in \mathbb {R}$ the strength of the nonlinearity, and $\xi _1$ a space–time white noise of unit strength. (We use redundant notation, as $u_1$, $G^1_{11}$ to be in accord with the case of two components). We are interested in the stationary process governed by (2.1). As proved in [15], spatial white noise with mean zero and unit variance is an invariant measure for (2.1). The most basic object of real interest is then the stationary space–time covariance $\langle u_1(x,t)u_1(0,0)\rangle$, where $\langle \cdot \rangle$ refers to the expectation with respect to the stationary process. Recently an exact solution has been accomplished [16], which turns out to validate prior non-rigorous replica computations [17]. For large $x,t$ the exact solution behaves as

$$\begin{aligned} \big \langle u_1(x,t)u_1(0,0)\big \rangle \simeq (\lambda _\mathrm {B} t)^{-2/3} f_{\mathrm {KPZ}} \left( (\lambda _\mathrm {B} t)^{-2/3}(x -ct)\right) , \end{aligned}$$

(2.2)

where $\lambda _{\mathrm {B}}= 2\sqrt{2}|G^1_{11}|$. The universal scaling function $f_{\mathrm {KPZ}}$ is tabulated in [18], there denoted by $f$, and has the following properties: $f_{\mathrm {KPZ}}\geqslant 0$, $f_{\mathrm {KPZ}}(x)=f_{\mathrm {KPZ}}(-x)$,

$$\begin{aligned} \int _\mathbb {R} f_{\mathrm {KPZ}}(x) \, dx = 1, \qquad \int _\mathbb {R} f_{\mathrm {KPZ}}(x) \, x^2 \, dx =0.510523\ldots . \end{aligned}$$

In fact, $f_{\mathrm {KPZ}}$ looks roughly like a Gaussian distribution but with faster decaying tails as $\exp (-0.295|x|^{3})$, see [19]. The $f_{\mathrm {KPZ}}$ scaling behavior for the stationary two-point function has been proved also for the PNG model [19], the TASEP [20], and the semi-discrete directed polymer model both at zero and positive temperature [16] and is expected to be valid for the entire KPZ universality class.

2.2 Classification of Two-Component Systems

Let us turn to the case of two components $\vec {u} = (u_1,u_2)$. The coupling constants become matrices and it is of advantage to stick to the most general form which reads

$$\begin{aligned} \partial _tu_\alpha +\partial _x\big (c_\alpha u_\alpha + \vec {u} \cdot G^\alpha \vec {u} -\partial _x (D\vec {u})_\alpha + \big (\sqrt{2D}\vec {\xi }\big )_\alpha \big )=0, \qquad \alpha = 1,2, \end{aligned}$$

(2.3)

where $c_\alpha$ is the propagation velocity of the $\alpha$-th component, the symmetric matrices $G^\alpha \in \mathbb {R}^{2 \times 2}$ determine the strength of the nonlinearity, the diffusion matrix $D \in \mathbb {R}^{2 \times 2}$ is symmetric positive, and $\vec {\xi }$ is a vector of two independent mean zero Gaussian white noises with covariance $\langle \xi _\alpha (x,t) \xi _{\alpha '}(x',t')\rangle = \delta _{\alpha \alpha '} \delta (x-x')\delta (t- t')$. Note that (2.3) is written already in normal coordinates, which are defined by the linear drift part of the current being diagonal, see [7] for more precision as well as (3.10). In (2.1) the term $cu_1$ can be removed by switching to a coordinate system moving with velocity $c$. Under the same transformation, for a two-component system the relative velocity necessarily persists, which is the origin for much richer properties.

As before, our interest is in the stationary process governed by (2.3), in particular its covariance matrix $\langle u_\alpha (x,t)u_{\alpha '}(0,0)\rangle$. No exact solutions are available and we have to work with approximations. The first issue is already the invariant measure of (2.3). Only if $G^1_{22} = G^2_{12}$ and $G^2_{11} = G^1_{12}$, the invariant measure is known to be white noise in $x$ with independent components. Since $G^1_{12}, \,G^2_{12}$ are subsub-leading couplings, one expects that the invariant measure has exponential mixing in space for general couplings. Our choice of the noise strength ensures unit strength for both components. The linear case, $G^1=0 = G^2$, is easily solved. If $c_1 \ne c_2$, then for large $x,t$, the covariance consists of two decoupled Gaussian peaks, respectively centered at $c_\alpha t$ and of width $\sqrt{D_{\alpha \alpha } t}$. Note that possible cross terms of $D$ do not show up, since the peaks move with distinct velocities. To a certain extent, this feature will still be valid, once the nonlinearity is included. Hence we will assume $c_1 \ne c_2$ throughout. The case $c_1 = c_2$ has to be studied separately, see [21] for an early discussion.

Since the mode velocities differ, the linear drift term is dominant and one expects that in general the two equations in (2.3) decouple for large $x,t$. However, if one of the leading non-linear couplings, $G^\alpha _{\alpha \alpha }$, vanishes, the argument becomes more subtle. To gain some insight we turn directly to the mode-coupling approximation for (2.3). It is based on a suitable Gaussian approximation together with the observation that the off-diagonal terms of the covariance are very small, see [7, Appendix C]. More precisely,

$$\begin{aligned} \big \langle u_\alpha (x,t)u_{\alpha '}(0,0)\big \rangle \simeq \delta _{\alpha \alpha '}f_\alpha (x,t), \end{aligned}$$

(2.4)

where initially $f_\alpha (x,0) = \delta (x)$, and the functions $f_\alpha$ satisfy the memory equation

$$\begin{aligned} \partial _t f_\alpha (x,t)= \left( -c_\alpha \partial _x+D_\alpha \partial ^2_x\right) f_\alpha (x,t) + \int ^t_0 \int _{\mathbb {R}} f_\alpha (x-y,t-s) \partial ^2_y M_{\alpha \alpha }(y,s)\,dy \, ds, \end{aligned}$$

(2.5)

$\alpha = 1,2$, where we have introduced $D_{\alpha \alpha }=D_\alpha$ and the memory kernel

$$\begin{aligned} M_{\alpha \alpha }(x,t)=2 \sum _{\alpha ',\alpha ''=1,2} \left( G^\alpha _{\alpha '\alpha ''}\right) ^2 f_{\alpha '}(x,t) f_{\alpha ''}(x,t). \end{aligned}$$

If $\alpha ' \ne \alpha ''$, the product $f_{\alpha '}(x,t) f_{\alpha ''}(x,t)$ is very small everywhere and hence can safely be neglected. Thereby the memory kernel simplifies to

$$\begin{aligned} M_{\alpha \alpha }(x,t)=2 \sum _{\alpha '=1,2} \left( G^\alpha _{\alpha '\alpha '}\right) ^2 f_{\alpha '}(x,t)^2. \end{aligned}$$

(2.6)

To obtain the asymptotic behavior, one makes an educated scaling ansatz for $f_\alpha$, the precise computation being shifted to Appendix 1. Particular cases were already presented in [7]. The universality classes are labeled according to whether the leading coefficient $G^\alpha _{\alpha \alpha }$ vanishes or not. Each class still subdivides according to the sub-leading terms $G^{\alpha }_{\alpha '\alpha '}$. In our tables “1” indicates any value different from $0$, “KPZ” labels the scaling reported in (2.2), “diff” means a Gaussian peak with width proportional to $\sqrt{t}$, and “$\alpha$-Lévy” indicates a scaling function given by a maximally asymmetric $\alpha$-stable law with exponent $\alpha$, see (3.12) and Appendix 1.3 for more details.

$G^1_{11}= 1$, $G^2_{22} = 1$	$G^1_{22}$	$G^2_{11}$	Peak 1	Peak 2
	0,1	0,1	KPZ	KPZ

In fact, the KPZ scaling function is not a solution of the fixed point equation derived from the mode-coupling equations (2.5)–(2.6), but it turns out to be very close to this solution, see the discussion in [22].

$G^1_{11}= 1$, $G^2_{22} = 0$	$G^1_{22}$	$G^2_{11}$	Peak 1	Peak 2
	0,1	1	KPZ	$\tfrac{5}{3}$-Lévy
	1	0	mod. KPZ	Diff
	0	0	KPZ	Diff

As explained in more detail in Appendices 1 and 2, if $G_{11}^2 \ne 0$, then the KPZ peak 1 feeds into mode 2 to generate a peak 2 with $\tfrac{5}{3}$-Lévy asymptotics, while the reverse process yields only a subdominant contribution to peak 1. On the other hand, if $G_{11}^2 =0$ but $G_{22}^1 \ne 0$, then on the level of mode-coupling the diffusive peak 2 generates a feed back on mode 1 which has also a dynamical exponent $z = 3/2$. In principle this should lead to a modified KPZ scaling function for peak 1. If actually correct, the decoupling hypothesis would have to be slightly modified.

$G^1_{11}= 0$, $G^2_{22} = 0$	$G^1_{22}$	$G^2_{11}$	Peak 1	Peak 2
	1	1	Gold-Lévy	Gold-Lévy
	1	0	$\tfrac{3}{2}$-Lévy	Diff
	0	1	Diff	$\tfrac{3}{2}$-Lévy
	0	0	Diff	Diff

The case gold-Lévy is discussed in Appendix 1. If one peak is diffusive, it feeds back to the other peak, which then becomes $\tfrac{3}{2}$-Lévy.

The maximal asymmetry of the Lévy distributions follows from the mode-coupling equations. But there is also a more qualitative argument. Physically one expects to have exponentially small correlations away from the sound cone $[c_1t,c_2t]$. If the Lévy distribution would not be maximally asymmetric, then it would exhibit both-sided power law tails which necessarily have slow decay outside the sound cone. Only for the maximal asymmetric distribution there is rapid decay to the outside and slow decay to the inside of the sound cone (see the discussion in Appendix 1.3). In fact, in accordance with the general principle, in numerical simulations one always observes the Lévy tail to be cut off at the other peak.

While we explained the asymptotic behavior of two coupled Burgers equation, one still has to relate them to a microscopic type model. In principle the theory should be applicable to any system with local interactions, either classical or quantum Hamiltonian, or classical with stochastic dynamics. Of course the model must have exactly two conservation laws and the dynamics should be sufficiently chaotic so to have good space–time mixing properties. In all examples investigated in more detail the steady state can be written in product form. This has the advantage that the Euler currents and $c_\alpha$ are known explicitly. After transformation to normal modes, the universality class for the model under consideration can be easily determined. In fact, beyond the specific predictions, one strength of the theory is to capture exceptional classes which would be hard to guess from a mere inspection of the equations of motion.

Below we report on numerical solutions of a one-dimensional lattice theory, for which the field takes real values and is governed by a deterministic differential equation plus random exchanges. We will present two examples for KPZ plus $\tfrac{5}{3}$-Lévy peak, corresponding to second table, row 1. The same model with a harmonic interaction belongs to the class diffusive plus $\tfrac{3}{2}$-Lévy peak (see Appendix 3.3). In this case a complete mathematical proof is available [23], which validates the prediction from mode-coupling. Also stochastic lattice gas models with two species of particles have been investigated. In [24] both peaks are KPZ. The two-lane model [25] has more parameters. Generically the two peaks are KPZ, but also the universality class studied here can be realized. In the very recent contribution [26] even more classes, including gold-Lévy, are obtained. Finally we should mention the discrete non-linear Schrödinger equation on a one-dimensional lattice with repulsive on-site interactions [27–29]. At low temperatures the model has the usual three conservation laws to a very good approximation. However the heat mode has a very small amplitude and one is reduced to an effective two-component system governing superfluid density and momentum. In this case both peaks are predicted to be KPZ, which is well confirmed through numerical simulations.

3 The BS Model with Random Exchanges

We consider the model as proposed and studied in [13], called ‘BS’ for short. Originally the model was motivated by anharmonic chains, for which stochastic collisions are added so to improve space–time mixing properties. One considers a real-valued field, denoted by $\eta _i\in \mathbb {R}$, $i \in \mathbb {Z}$. To define the model we first take a finite volume with $0 \leqslant i \leqslant N-1$. We call $\varvec{\eta } = (\eta _0,\dots ,\eta _{N-1})$ the displacement field, also ‘volume’ and ‘height’ have been proposed. The dynamics of the BS model consists of a deterministic part, which describes forces exerted by neighboring displacements and a stochastic part in which neighboring displacements are exchanged at random. The deterministic part is governed by the first order differential equations

$$\begin{aligned} \frac{d}{dt} \eta _i= V'(\eta _{i+1})-V'(\eta _{i-1}) \end{aligned}$$

(3.1)

and has the corresponding generator

$$\begin{aligned} \mathcal {A}_N = \sum _{i=0}^{N-1} \big ( V'(\eta _{i+1})-V'(\eta _{i-1}) \big ) \partial _{\eta _i}. \end{aligned}$$

Periodic boundary conditions are imposed as $\eta _{i+N} = \eta _i$. Note that the vector field on the left of (3.1) is divergence free and hence the Lebesgue measure is preserved under the time evolution. The potential $V$ is bounded from below with at least a one-sided growth to infinity as $|\eta _i |\rightarrow \infty$. In addition, at independent random times distributed according to an exponential law with parameter $\gamma$, neighboring displacements are exchanged. The generator for the random part is $\gamma \mathcal {S}_N$ with

$$\begin{aligned} \mathcal {S}_Nf(\varvec{\eta }) \!=\! \sum _{i=0}^{N-1}\left( f\left( \varvec{\eta }^{i,i+1}\right) \!-\! f(\varvec{\eta })\right) , \qquad \varvec{\eta }^{i,i+1} \!=\! \left( \eta _0,\dots ,\eta _{i-1},\eta _{i+1},\eta _i,\eta _{i+2},\dots ,\eta _{N-1}\right) . \end{aligned}$$

Clearly the displacement field is locally conserved. Note that under the deterministic part

$$\begin{aligned} \frac{d}{dt} V(\eta _i) = V'(\eta _{i+1})V'(\eta _{i})-V'(\eta _{i})V'(\eta _{i-1}). \end{aligned}$$

(3.2)

Thus, including random exchanges, also $V(\eta _i)$ is locally conserved. This field is called the (potential) energy field. As a consequence, the BS model has a two-parameter family of invariant measures. The parameter dual to $V(\eta _i)$ is called inverse temperature, denoted by $\beta >0$, and the parameter dual to $\eta _i$ is called tension, denoted by $\tau \in \mathbb {R}$. Hence the invariant measures are written as

$$\begin{aligned} \mu _{\tau ,\beta }\left( d\eta _0\dots d\eta _{N-1}\right) = \prod _{i=0}^{N-1} Z_{\tau ,\beta }^{-1} \, \mathrm {e}^{-\beta (V(\eta _i)+\tau \eta _i)} \, d\eta _i. \end{aligned}$$

(3.3)

If the potential increases too slowly as $\eta _i \rightarrow \pm \infty$, the range of admissible values of $\tau$ may have to be restricted in order for the density $\mathrm {e}^{-\beta (V(\eta )+\tau \eta )}$ to be integrable. (We use $\eta \in \mathbb {R}$ as standing for one of the $\eta _i$’s.) Averages with respect to $\mu _{\tau ,\beta }$ are denoted by $\langle \cdot \rangle _{\tau ,\beta }$.

At finite volume the micro-canonical measures are time-invariant, but they could be not ergodic. Such possible pathology disappears in the infinite volume limit. In [13] it is established that the infinite volume dynamics is ergodic: A probability measure for the infinite system, which (i) is stationary in time, (ii) is translation-invariant, and (iii) has a finite relative entropy per unit length with respect to the infinite dimensional analogue of $\mu _{0,1}$, is necessarily a convex combinations of canonical measures. Hence, in the infinite volume limit, displacement and energy are the only conserved fields.

The local conservation of displacement and energy implies the existence of local displacement and energy currents. They have a deterministic and random part with the former given by

$$\begin{aligned} \frac{d}{dt}\begin{pmatrix} \eta _i \\ V(\eta _i) \end{pmatrix} = J^{i-1,i} - J^{i,i+1}, \qquad J^{i,i+1} = \begin{pmatrix} j_h^{i,i+1} \\ j_e^{i,i+1} \end{pmatrix} = - \begin{pmatrix} V'(\eta _i) + V'(\eta _{i+1}) \\ V'(\eta _i)V'(\eta _{i+1}) \end{pmatrix}. \end{aligned}$$

(3.4)

To apply the theory from Sect. 2, we first have to obtain the macroscopic Euler equations. In the continuum limit, studied in [13], the displacement field becomes $h(x,t)$ and the energy field $e(x,t)$. The currents of the Euler equations are determined by averaging the currents in a local equilibrium state. On that scale random exchanges make no contribution yet and it suffices to compute the average of the currents in (3.4) with respect to $\mu _{\tau ,\beta }$. Since

$$\begin{aligned} \big \langle V'(\eta _i) \big \rangle _{\tau ,\beta } = -\tau , \qquad \big \langle V'(\eta _i) V'(\eta _{i+1}) \big \rangle _{\tau ,\beta } = \tau ^2, \end{aligned}$$

the Euler currents for the conserved fields $h,e$ are respectively $j_h = 2 \tau$ and $j_e = -\tau ^2$, where the tension $\tau$ is considered as a function of the average displacement and energy as defined through the implicit relation

$$\begin{aligned} h_{\tau ,\beta } = \langle \eta _i \rangle _{\tau ,\beta }, \qquad e_{\tau ,\beta } = \langle V(\eta _i) \rangle _{\tau ,\beta }. \end{aligned}$$

(3.5)

In the hydrodynamic limit the system of conservation laws then reads

$$\begin{aligned} \partial _t \begin{pmatrix} h(x,t) \\ e(x,t) \end{pmatrix} + \partial _x \begin{pmatrix} 2 \tau (h(x,t),e(x,t)) \\ -\tau (h(x,t),e(x,t))^2 \end{pmatrix} = 0. \end{aligned}$$

(3.6)

The linearization of this system around a uniform background profile $(h_0,e_0)$, obtained by writing $h(x,t) = h_0 + \tilde{h}(x,t)$ and $e(x,t) = e_0 + \tilde{e}(x,t)$, yields

$$\begin{aligned} \partial _t \!\begin{pmatrix} \tilde{h}(x,t) \\ \tilde{e}(x,t) \end{pmatrix} + A(h_0,e_0) \partial _x \!\begin{pmatrix} \tilde{h}(x,t) \\ \tilde{e}(x,t) \end{pmatrix} = 0, \end{aligned}$$

(3.7)

where

$$\begin{aligned} A = 2 \begin{pmatrix} \partial _h \tau &{} \partial _e \tau \\ -\tau \partial _h \tau &{} -\tau \partial _e \tau \end{pmatrix}. \end{aligned}$$

(3.8)

In (3.8) the dependence of $A$ on $h_0,e_0$ has already been suppressed. In the sequel we will regard $\tau ,\beta$ as given and thereby via (3.5) also the value of the background fields $h_0,e_0$.

We now follow the strategy in [7] in order to study the space–time correlation matrix $S(i,t) \in \mathbb {R}^{2 \times 2}$ of the conserved fields whose entries read

$$\begin{aligned} S_{\alpha \alpha '}(i,t) = \left\langle g_\alpha (\eta _{i,t}) g_{\alpha '}(\eta _{0,0})\right\rangle _{\tau ,\beta } - \left\langle g_\alpha (\eta _{i,t}) \right\rangle _{\tau ,\beta } \left\langle g_{\alpha '}(\eta _{0,0})\right\rangle _{\tau ,\beta }. \end{aligned}$$

Here $g_1(\eta ) = \eta$ and $g_2(\eta ) = V(\eta )$ and, in slight abuse, $\langle \cdot \rangle _{\tau ,\beta }$ refers to average in the stationary process with starting measure $\mu _{\tau ,\beta }$. For the said purpose we expand the Euler equations (3.6) to second order in the currents and add dissipation plus noise. The resulting Langevin equations have a structure similar to (2.3), but with the linear drift term not yet diagonal. The latter feature is accomplished through the transformation matrix $R$ defined by the properties

$$\begin{aligned} RAR^{-1} = \mathrm {diag}(c,0),\qquad RS(0,0)R^\mathrm {T} = 1, \end{aligned}$$

where it is already anticipated that $A$ has the eigenvalues $0$ and

$$\begin{aligned} c = 2 (\partial _h -\tau \partial _e) \tau < 0, \end{aligned}$$

(3.9)

see (8.1). We use the convention that the left moving mode has label $1$, while the standing mode has label $2$. In analogy to anharmonic chains, mode $1$ is called sound mode and mode $2$ heat mode. After this transformation the equations of nonlinear fluctuating hydrodynamics are exactly of the form of two coupled Burgers equations as in (2.3). The transformation matrix $R$ and the nonlinear coupling matrices $G^{\alpha }$ are tabulated in (8.2) and Appendix 3.2 respectively. Because of the particular form of the Euler currents, one has $G^2_{22} = 0$, $G^2_{12} = G^2_{21} =0$ always, while $G^2_{11} < 0$. Thus the heat peak is non-KPZ, but coupled to the sound peak. According to our classification, this leaves only the two cases: (i) $G^1_{11} \ne 0$ implying KPZ for mode $1$ and $\tfrac{5}{3}$-Lévy for mode $2$, (ii) $G^1_{11} = 0$ implying diffusive for mode $1$ and $\tfrac{3}{2}$-Lévy for mode $2$.

The case (ii) is exceptional, a more explicit condition being

$$\begin{aligned} (\partial _h - \tau \partial _e)^2\tau = 0, \end{aligned}$$

see (8.4). One example is the harmonic potential $V(\eta ) = \eta ^2$ discussed in [23], for which $G^1_{11} = 0$ identically. In general, there could be special values of $\tau ,\beta$ at which $G^1_{11} = 0$.

The matrix $S(i,t)$ is transformed to normal modes as $S^\sharp (i,t) = R S(i,t) R^\mathrm {T}$. On sufficiently large scales $S^\sharp (i,t)$ should be determined through the stationary covariance of the coupled Burgers equations (2.3) and we can use directly the results from Sect. 2 and Appendix 1. They assert that $R S(i,t) R^\mathrm {T}$ is approximately diagonal,

$$\begin{aligned} S^\sharp (i,t) = R S(i,t) R^\mathrm {T} \simeq \delta _{\alpha \alpha '} \, f_\alpha (N^{-1}i,t), \quad \mathrm {mod}\,\,N \, . \end{aligned}$$

(3.10)

The sound peak scales asymptotically as

$$\begin{aligned} f_1(x,t) \simeq (\lambda _1 t)^{-2/3} f_\mathrm{KPZ}\Big ((\lambda _1 t)^{-2/3}(x-ct)\Big ), \qquad \lambda _1 = 2\sqrt{2} \left| G_{11}^1 \right| , \end{aligned}$$

(3.11)

and the heat peak as

(3.12)

with the constant¹

$$\begin{aligned} a_\mathrm{h} = \sqrt{3} \, \Gamma \big (\tfrac{1}{3}\big ) \int _\mathbb {R} \left( f_\mathrm{KPZ}\right) ^2 \simeq 1.81. \end{aligned}$$

Here $f_{\mathrm {L\acute{e}vy},5/3,1}$ denotes the maximally asymmetric Lévy distribution with parameters $\alpha = 5/3$ and asymmetry $b= 1$, see Appendix 1.3 for the definition.

4 Numerical Simulations for the BS Model

To assess the validity of (3.11)–(3.12), we perform numerical simulations obtained by integrating the dynamics (3.1) and adding random exchanges when starting from initial conditions distributed according to the canonical measure (3.3). This is done for two potentials: the FPU-$\alpha$ potential (simply abbreviated as FPU in the sequel)

$$\begin{aligned} V(\eta ) = \frac{1}{2} \eta ^2 + \frac{a}{3}\eta ^3 + \frac{1}{4} \eta ^4, \end{aligned}$$

(4.1)

with $a = 2$ (as in [22]), and the Kac-Van Moerbeke potential (abbreviated as KvM in the sequel)

$$\begin{aligned} V(\eta ) = \frac{\mathrm {e}^{-\kappa \eta }+ \kappa \eta -1}{\kappa ^2}, \end{aligned}$$

(4.2)

with $\kappa =1$. Note that $\tau > -1/\kappa$ is necessary to ensure the normalization of the canonical measures in this case. The KvM potential is special since it makes the system integrable in the absence of stochastic exchanges, i.e. when $\gamma = 0$, see [30]. In fact, the corresponding system is related by a simple transformation to the famous Toda lattice [31], which is a chain of oscillators coupled through the potential (4.2) to nearest neighbors and evolving according to Hamiltonian dynamics.

In Sect. 4.1 we explain how the correlators are computed and fitted to the respective scaling functions. In Sect. 4.2 we quantify the agreement between the numerically computed correlators and the theoretical predictions.

4.1 Numerical Computation of the Correlation Functions

4.1.1 Generation of Initial Conditions

Initial conditions are sampled according to the canonical measure (3.3). Since this measure is of product form, one can sample independently the initial values $(\eta _{i,0})_{i=0,\dots ,N-1}$ for each site according to the measure $Z_{\tau ,\beta }^{-1} \, \mathrm {e}^{-\beta (V(\eta )+\tau \eta )} \, d\eta$. One way to do so is to start from $\eta ^\mathrm{init}_i = 0$ and to evolve according to the SDE

$$\begin{aligned} d\eta _{i,t}^\mathrm{init} = - \big (V'(\eta _{i,t}^\mathrm{init}) + \tau \big )dt + \sqrt{\frac{2}{\beta }} \, dW_{i,t}^\mathrm{init}. \end{aligned}$$

In practice this is done by a discretization through an Euler-Maruyama scheme, using a time step $\Delta t_\mathrm{thm}$,

$$\begin{aligned} \eta _{i,n+1}^{\mathrm{init}} = \eta _{i,n}^{\mathrm{init}} - \Delta t_\mathrm{thm} \Big ( V'(\eta _{i,n}^{\mathrm{init}}) + \tau \Big ) + \sqrt{\frac{2 \Delta t_\mathrm{thm}}{\beta }} \, G_{i,n}^{\mathrm{init}}, \end{aligned}$$

where $G_{i,n}^{\mathrm{init}}$ are independent and identically distributed (i.i.d.) standard Gaussian random variables. We use $\Delta t_\mathrm{thm} = 0.005$, and integrate over $N_\mathrm{thm} = 1000$ steps. We finally set $\eta _{i,0} = \eta _{i,N_\mathrm{thm}}^{\mathrm{init}}$. To check the sampling of the initial conditions, we compared the reference distribution $Z_{\tau ,\beta }^{-1} \, \mathrm {e}^{-\beta (V(\eta )+\tau \eta )}$ and the histogram of the displacements $(\eta _{i,N_\mathrm{thm}}^{\mathrm{init}})_{0 \leqslant i \leqslant N-1}$.

4.1.2 Numerical Integration of the Dynamics

The dynamics (3.1) is integrated with a timestep $\Delta t > 0$ using the algorithm presented in [13, Sect. 6.3.1], adapted here to the periodic setting. In a nutshell, the numerical method first integrates the deterministic part of the dynamics over a time increment $\Delta t$ with a splitting strategy where even and odd sites are evolved separately over time increments $\Delta t$, in accordance with the hidden Hamiltonian structure of the deterministic part of the dynamics. Next one has to include the random exchanges: independent exponential clocks are attached to each pair $(\eta _i,\eta _{i+1})$, and the current clock times are decreased by $\Delta t$ at each time step. When a clock time becomes negative, the corresponding neighboring displacements are exchanged, and a new exponential time of mean $1/\gamma$ is sampled.

We produce $K$ samples of initial conditions of the system (starting from independent initial conditions $\eta _{i,0}^k$), and denote by $(\eta _{i,n}^{k})_{0 \leqslant i \leqslant N-1}$ an approximation of the state of the $k$th sample at time $n\Delta t$. The time step is set to $\Delta t = 0.005$, a value sufficiently small to ensure relative energy variations of order $10^{-3}$ or less over very long times for stochastic rates in the range $0 \leqslant \gamma \leqslant 1$ and for the system sizes up to $N = 8000$. Note that the splitting algorithm respects the underlying symplectic structure of the differential equation part, while the exchange does not. As a consequence the near energy conservation observed for the deterministic dynamics is degraded by the exchange noise, although no systematic drift is observed.

4.1.3 Computation of the Correlators

The correlation matrices are computed at times $n\Delta t$. To this end, we first evaluate the empirical average over the replicas of the displacements and energies at each site $i$,

$$\begin{aligned} \overline{h}_{i,n} = \frac{1}{K} \sum _{k=1}^K \eta _{i,n}^{k}, \qquad \overline{e}_{i,n} = \frac{1}{K} \sum _{k=1}^K V\left( \eta _{i,n}^{k}\right) , \end{aligned}$$

and then compute the entries of the correlation matrix by the following space- and sample-average,

$$\begin{aligned} \left[ C_{N,K}(i,n) \right] _{\alpha ,\alpha '} = \frac{1}{NK} \sum _{k=1}^K \sum _{i=0}^{N-1} u_{\alpha ,i+j,n}^{k} u_{\alpha ',j,0}^{k}, \end{aligned}$$

with

$$\begin{aligned} u_{1,i,m}^{k} = \eta _{i,m}^{k} - \overline{h}_{i,m}, \qquad u_{2,i,m}^{k} = V\left( \eta _{i,m}^{k}\right) - \overline{e}_{i,m}. \end{aligned}$$

The numerical results reported below are obtained using $K = 10^5$ samples and chains of lengths $N=~$2000–8000. We checked that

$$\begin{aligned} C_{N,K}(i,0) \simeq \delta _{i0} \, \mathcal {C}, \qquad \mathcal {C} = \begin{pmatrix} \langle \eta ;\eta \rangle _{\tau ,\beta } &{} \langle \eta ;V(\eta )\rangle _{\tau ,\beta } \\ \langle \eta ;V(\eta )\rangle _{\tau ,\beta } &{} \langle V(\eta );V(\eta )\rangle _{\tau ,\beta } \end{pmatrix}, \end{aligned}$$

where $\left\langle A;B\right\rangle _{\tau ,\beta } = \left\langle A(\eta _0) B(\eta _0) \right\rangle _{\tau ,\beta } - \left\langle A(\eta _0) \right\rangle _{\tau ,\beta }\left\langle B(\eta _0) \right\rangle _{\tau ,\beta }$. We also checked that the sum rules hold, up to very small errors related to the only approximate conservation of the energies and to the finiteness of the number of samples $K$,

$$\begin{aligned} \sum _{i=0}^{N-1} C_{N,K}(i,n) = \sum _{i=0}^{N-1} C_{N,K}(i,0). \end{aligned}$$

After a normal mode transformation as in (3.10), based on the matrix $R$ defined in (8.2), the correlation matrix is almost diagonal,

$$\begin{aligned} C^\sharp _{N,K}(i,n) = R C_{N,K}(i,n) R^\mathrm {T} \simeq \begin{pmatrix} f_1^\mathrm{num}(i,n) &{} 0 \\ 0 &{} f_2^\mathrm{num}(i,n) \end{pmatrix}. \end{aligned}$$

4.1.4 Computation of the Scaling Factors

In order to check quantitatively the agreement between the numerically computed correlation functions $f_\alpha ^\mathrm{num}$ and the theoretically predicted values (3.11)–(3.12), following [32], we optimize the parameters in the ansatz (3.11)–(3.12) such as to minimize the $L^1$ distance,

$$\begin{aligned} \inf _{\begin{array}{c} x_n \in \mathbb {R} \\ \Lambda _n > 0 \end{array}} \left\{ \sum _{i=0}^{N-1} \left| f_\alpha ^\mathrm{num}(i,n) - (\Lambda _n)^{-1} f_\alpha ^\mathrm{mc}\big ((\Lambda _n)^{-1} (i-x_n)\big )\right| \right\} . \end{aligned}$$

(4.3)

Here, $f_\alpha ^\mathrm{mc}$ denotes the theoretical scaling function, namely KPZ for mode 1 and maximally asymmetric $\tfrac{5}{3}$-Lévy for peak 2. In fact, in order to have a more stable minimization procedure, we use the prior knowledge on how $x_n,\Lambda _n$ should scale and write

$$\begin{aligned} x_n = c_\mathrm{theor} n\Delta t + \widetilde{x}_n, \qquad \Lambda _n = \widetilde{\Lambda }_n \, \left( n \Delta t\right) ^{\delta _\mathrm{theor}}. \end{aligned}$$

(4.4)

The value $c_\mathrm{theor}$ is the theoretical peak velocity, to say, 0 for the heat peak and (3.9) for the sound peak, and $\delta _\mathrm{theor}$ the theoretically predicted scaling exponent, 3/5 for the heat mode and 2/3 for the sound peak.

The optimization in (4.3) is now performed over $\widetilde{x}_n$ and $\widetilde{\Lambda }_n$ at the various times $n\Delta t$ at which the correlation matrix is computed. In practice, the sum in (4.3) is not performed over all indices $i$ but restricted to the indices $i$ which are close to the center of the peak under investigation, since far away from the peak center the correlation is almost zero and the dominance of statistical noise makes those values irrelevant. The center of the peak at time index $n$ is defined as the index $i^n_\mathrm{center}$ for which $f_\alpha ^\mathrm{num}(i,n)$, as a function of $i$, is maximal. A cut-off range $R_\mathrm{cut} > 0$ is then introduced to limit the sum in (4.3) to indices $i^n_\mathrm{center} - R_\mathrm{cut} t^{\delta _\mathrm{theor}} \leqslant i \leqslant i^n_\mathrm{center} + R_\mathrm{cut} t^{\delta _\mathrm{theor}}$.

The values $\widetilde{x}_n,\widetilde{\Lambda }_n$ may be drifting in time when the expected scaling (4.4) is not completely exact. It may happen for instance that, due to errors related to the use of finite stepsizes $\Delta t$, the actual velocity is not exactly equal to $c_\mathrm{theor}$. We therefore fit $\widetilde{x}_n,\widetilde{\Lambda }_n$ as

$$\begin{aligned} \widetilde{x}_n = c_\mathrm{crt} \, n\Delta t + \hbox {x}_0, \qquad \widetilde{\Lambda }_n = \widetilde{\Lambda }_0 \, (n\Delta t)^{\delta _\mathrm{crt}}, \end{aligned}$$

(4.5)

these fits being performed using a standard least-square minimization for $\widetilde{x}_n$ and a least-square minimization based on $\log \widetilde{\Lambda }_n$ to find the correction to the scaling exponent. The actual velocity observed in the numerical experiments is then $c_\mathrm{num} = c_\mathrm{theor}+c_\mathrm{crt}$ and the actual scaling exponent is $\delta _\mathrm{num} = \delta _\mathrm{theor}+\delta _\mathrm{crt}$. Once the corrected scaling exponent is determined, the scaling factor is obtained as

$$\begin{aligned} \lambda _\mathrm{num} = \widetilde{\Lambda }_0^{1/\delta _\mathrm{num}}. \end{aligned}$$

(4.6)

We have checked that the final outputs, in particular the actual scaling exponent $\delta _\mathrm{num}$ and the associated scaling factor $\lambda _\mathrm{num}$ are insensitive to the choice of the surrogate scaling exponent $\delta _\mathrm{theor}$. This procedure also allows to check how fast the “instantaneous” estimates of the scaling factor, defined as

$$\begin{aligned} \lambda _n = \left( \widetilde{\Lambda }_n (n\Delta t)^{-\delta _\mathrm{crt}} \right) ^{1/\delta _\mathrm{num}}, \end{aligned}$$

(4.7)

stabilize around the average value $\lambda _\mathrm{num}$ given by (4.6), see Figs. 3, 4, 6, and 7, left.

4.2 Comparison with Theoretical Predictions

The numerical results reported here have been obtained at the fairly low temperature of $\beta ^{-1}= \tfrac{1}{2}$, with a tension $\tau = 1$, using a noise intensity $\gamma = 1$, and the value $R_\mathrm{cut} = 9$ for the sound peak and $R_\mathrm{cut} = 11$ for the heat peak to compute the $L^1$ error in the minimization procedure (4.3). A plot summarizing the evolution of the sound and heat peaks is presented in Fig. 1. In all cases, the value $\hbox {x}_0$ in (4.5) is very small and is henceforth set to 0.

Fig. 1

Evolution of the heat peak (centered at $x=0$) and the sound peak, traveling to the left, for the KvM potential (4.2). Note that the heat peak is not symmetric, the rapid decay being away from the sound peak

In the pictures, we call “rescaled peak” the plots for which the renormalized numerical correlation functions $(\lambda _\mathrm{num}n\Delta t)^{\delta _\mathrm{num}} f_\alpha ^\mathrm{num}(i,n)$ are plotted, at a given time index $n$, as a function of the renormalized spatial variable $(i-c_\mathrm{num}n\Delta t)/(\lambda _\mathrm{num}n\Delta t)^{\delta _\mathrm{num}}$.

4.2.1 FPU Potential

The rescaled sound and heat peaks for the FPU potential are presented in Fig. 2. The agreement with the predicted scaling functions is qualitatively excellent. On a quantitative level, the numerical parameters obtained by the minimization procedure are:

• for sound peaks, exponent $\delta _\mathrm{num} = 2/3$, fixed to its theoretical value since $\delta _\mathrm{num}$ turns out to be extremely close to 2/3, velocity $c_\mathrm{num} = -5.24$, compared to the theoretical value $c_\mathrm{theor} = -5.28$, and scaling factor $\lambda _{1} \simeq 6.36$. The scaling factor is in excellent agreement with the theoretical value $\lambda _{1} = 2\sqrt{2} |G^1_{11}| = 6.32$ predicted by (3.11).

• for heat peaks, the reference being the maximally asymmetric Lévy distribution with $\alpha = 5/3$: velocity $c_\mathrm{num} = 0$, exponent $\delta _\mathrm{num} = 0.605$, very close to the theoretical value $3/5$, scaling factor $\lambda _2 \simeq 3.70$. The scaling factor is in very good agreement with the theoretical value $3.46$ predicted by (3.12). A slightly better agreement between the scaling functions could be obtained by decreasing a little bit the parameter of the Lévy distribution from $5/3$ to values around to 1.64 in order to have a sharper decrease on the right.

The evolution of the non-universal scaling factors as a function of the time index is reported in Figs. 3 and 4, together with the $L^1$ error. Note that the error very quickly decreases at the beginning of the simulation but, after reaching an absolute minimum, slowly increases again due to the increase of the statistical noise. The initial decrease is faster for the sound peak, which attains its asymptotic shape more rapidly. Also, the scaling factor settles down slightly faster for the sound peak.

Fig. 2

(FPU potential) Comparison of rescaled sound and heat peaks. The first line corresponds to the sound peak, the second one to the heat peak. The reference for the heat peak is the Lévy stable distribution with parameter $\alpha =5/3$ and maximal asymmetry. Logarithmic plots are provided in the right column

Fig. 3

(FPU potential, sound peak) Left Optimal value of the scaling parameter for a given time, as given by (4.7). Right $L^1$ error (4.3) for the optimal value of the parameters

Fig. 4

(FPU potential, heat peak) Left Optimal value of the scaling parameter for a given time, as given by (4.7). Right $L^1$ error (4.3) for the optimal value of the parameters

4.2.2 KVM Potential

The rescaled sound and heat peaks for the KvM potential are presented in Fig. 5. The agreement with the predicted scaling function is again qualitatively excellent. On a quantitative level, the numerical parameters obtained by the minimization procedure are:

• for sound peaks, exponent $\delta _\mathrm{num} = 2/3$, fixed to its theoretical value since $\delta _\mathrm{num}$ turns out to be extremely close to 2/3, velocity $c_\mathrm{num} = -3.996$, compared to the theoretical value $c_\mathrm{theor} = -4$, and scaling factor $\lambda _\mathrm{s} \simeq 2.81$. The scaling factor is in excellent agreement with the theoretical value $\lambda _1 = 2\sqrt{2} |G^1_{11}| = 2.83$ predicted by (3.11).

• for heat peaks, we consider as a reference the maximally asymmetric Lévy distribution with $\alpha = 1.57$ instead of $5/3$ since this value of the $\alpha$ parameter allows to further decrease the error (4.3). We find a velocity $c_\mathrm{num} \simeq 0$, an exponent $\delta _\mathrm{num} = 0.633$, somewhat away from the theoretically predicted value $3/5$, and a scaling factor $\lambda _2 \simeq 2.51$. The latter value is quite off the theoretical value $4.21$ predicted by (3.12).

The evolution of the scaling factors as a function of the time index is reported in Figs. 6 and 7, together with the $L^1$ error. The behavior and orders of magnitude of the error are similar to what is observed with the FPU potential. We again see in this example that the convergence to the limiting regime for the sound peak is slightly faster than for the heat peak.

Fig. 5

(KvM potential) Comparison of rescaled sound and heat peaks. The first line corresponds to the sound peak, the second one to the heat peak. The reference for the heat peak is a maximally asymmetric Lévy distribution with parameter $\alpha = 1.57$ instead of $1.67$. Logarithmic plots are provided in the right column

Fig. 6

(KvM potential, sound peak) Left Optimal value of the scaling parameter for a given time, as given by (4.7). Right $L^1$ error (4.3) for the optimal value of the parameters

Fig. 7

(KvM potential, heat peak) Left Optimal value of the scaling parameter for a given time, as defined by (4.7). Right $L^1$ error (4.3) for the optimal value of the parameters

5 Conclusions

Comparable simulations have been carried out for a two-component stochastic lattice gas [24] and for Hamiltonian anharmonic chains with hard core collisions [32], resp. with an asymmetric FPU potential [33, 34]. The latter two models have three conserved fields. The lattice gas has two KPZ peaks with distinct velocities. The agreement with KPZ is of a similar quality as obtained here, including the values for the non-universal coefficients. Possibly the $G$-matrices of these two models favor small finite time corrections. For the anharmonic chains the agreement is less perfect. For hard-core collisions the predicted shape of the peaks is achieved with an $L^1$ error of the order of $3~\%$, but the non-universal coefficients deviate considerably from their predicted values. Such deviations are even more pronounced for the FPU chains. For instance, at the largest time and size available, the sound peaks still show a slight asymmetry.

It is remarkable that the space–time correlation functions obtained from numerical simulations of the BS model are in such a good agreement with the ones of nonlinear fluctuating hydrodynamics.

KPZ scaling is based on decoupling and is expected to be exact for sufficiently long times. On the other hand, the Lévy distribution is based on mode-coupling, which is an approximation. As also observed in other models, for our simulations the fit to the maximally asymmetric $\tfrac{5}{3}$-Lévy distribution is so precise that one is tempted to conjecture it to be the true long time scaling function.

Appendices

Appendix 1: Scaling Functions for Two Cross-Coupled Modes

We study the asymptotic behavior of two cross-coupled Burgers equations of the form

$$\begin{aligned} \partial _t u_{\sigma }+\partial _x\big (\sigma c u_{\sigma } + \lambda (u_{-\sigma })^2 -D\partial _x u_{\sigma } +\sqrt{2D}\xi _{\sigma }\big )=0,\quad \sigma = \pm 1, \end{aligned}$$

(6.1)

for velocity $c >0$, diffusion constant $D>0$, and strength of nonlinearity $\lambda > 0$. This is the case “gold-Lévy” from third table, row 1, in Sect. 2 (with the simplification that the strength of the nonlinearity is assumed to be the same for both modes). Note that, compared to (2.3) the index of the modes is $\pm 1$ instead of $1,2$ and the frame of reference is such that the modes have opposite velocities.

In the diagonal approximation, compare with (2.5)–(2.6), the respective mode-coupling equations read

$$\begin{aligned} \partial _t f_\sigma (x,t) = \left( -\sigma c \partial _x+D \partial ^2_x\right) f_\sigma (x,t)+ 2\lambda ^2 \int ^t_0 \int _{\mathbb {R}} f_\sigma (x-y,t-s) \partial ^2_y\big ( f_{-\sigma }(y,s)^2\big )dy \, ds . \end{aligned}$$

(6.2)

Initially $f_\sigma (x,0) = \delta (x)$ and the normalization is preserved,

$$\begin{aligned} \int _\mathbb {R} f_{\sigma }(x,t) \, dx = 1. \end{aligned}$$

(6.3)

Furthermore, by symmetry of the equations,

$$\begin{aligned} f_{\sigma }(x,t)= f_{-\sigma }(-x,t). \end{aligned}$$

(6.4)

Our goal is to find the long-time self-similar solution to (6.2). We will establish that the appropriate space–time scaling is $x/t^{1/\gamma }$ with $\gamma$ the golden mean,

$$\begin{aligned} \gamma = \frac{1 + \sqrt{5}}{2} \simeq 1.618. \end{aligned}$$

(6.5)

The scaling function turns out to be the maximally asymmetric $\gamma$-Lévy distribution, see (6.14) and Appendix 1.3 for a discussion of its tail properties.

1.1 Appendix 1.1: Equation for the Scaling Functions

We use the same Fourier transform conventions as in [7],

$$\begin{aligned} \hat{g}(k) = \int _\mathbb {R} g(x) \, \mathrm {e}^{-2\mathrm {i}\pi kx} \, dx. \end{aligned}$$

Taking the spatial Fourier transform of (6.2) leads to

$$\begin{aligned} \begin{array}{ll} \partial _t \hat{f}_\sigma (k,t) &{}\displaystyle = -\left( 2\mathrm {i}\pi \sigma c k + (2\pi k)^2 D \right) \hat{f}_\sigma (k,t) \\ &{}\displaystyle \quad - 2(2\pi k)^2 \lambda ^2 \int ^t_0 \hat{f}_\sigma (k,t-s) \Big ( \int _{\mathbb {R}} \hat{f}_{-\sigma }(k-q,s)\, \hat{f}_{-\sigma }(q,s) \, dq \Big ) ds . \end{array} \end{aligned}$$

(6.6)

We assume that, relative to $\sigma ct$, $f_\sigma$ is a self-similar solution with still to be determined space–time scale. Recall that if a function $f$ is self-similar,

$$\begin{aligned} f(x,t) = t^{-a}F(t^{-a}(x\mp ct)), \end{aligned}$$

then $\hat{f}(k,t) = \mathrm {e}^{\mp 2\mathrm {i}\pi kct} \hat{F}(k t^a)$. We therefore make the following scaling ansatz

$$\begin{aligned} \hat{f}_1(k,t) = \mathrm {e}^{-2\mathrm {i}\pi kct} h(k^\gamma t), \qquad \hat{f}_{-1}(k,t) = \mathrm {e}^{2\mathrm {i}\pi kct} g(k t^\beta ), \end{aligned}$$

(6.7)

which is expected to be valid asymptotically only, as made precise in (6.10).

We consider the forcing exerted by $f_{-1}$ on $f_{1}$, which amounts to regarding the function $g$ as the input and $h$ as the output. Let us first state some properties of the functions $g,h$. Since $f_\sigma$ is real-valued, $h(-w) = \overline{h(w)}$ and $g(-w) = \overline{g(w)}$. We therefore restrict ourselves to $k >0$ in the sequel. Plugging the ansatz (6.7) into (6.6), the equation for $\hat{f}_1$ turns into

$$\begin{aligned} k^\gamma h'(k^\gamma t) =&- (2\pi k)^2 D h(k^\gamma t) \\&- 2(2\pi k)^2 \lambda ^2 \int ^t_0 h\big (k^\gamma (t-s)\big )\int _\mathbb {R} g\big ((k-q)s^\beta \big ) g\big (q s^\beta \big )\mathrm {e}^{4\mathrm {i}\pi cks} \, dq \, ds , \end{aligned}$$

so that, introducing the new variables $w = k^\gamma t$ and $u = q s^\beta$,

$$\begin{aligned} \begin{array}{ll} h'(w) =&{}\displaystyle -4\pi ^2D\, k^{2-\gamma } h(w) \\ &{}\displaystyle - 8\pi ^2 \lambda ^2 \, k^{2-\gamma } \int ^{k^{-\gamma } w}_0 \mathrm {e}^{4\mathrm {i}\pi cks}s^{-\beta } \, h\left( w-k^\gamma s\right) \Big (\int _\mathbb {R} g\left( ks^\beta -u\right) g\left( u\right) \, du\Big ) ds . \end{array} \end{aligned}$$

Here $t$ has been eliminated and we study the limit $k \rightarrow 0$. We rescale the time integration variable as $s = k^{-a} \theta$ and obtain

$$\begin{aligned} h'(w)&= -4\pi ^2D\, k^{2-\gamma } h(w)- 8\pi ^2 \lambda ^2 \, k^{2-\gamma +a(\beta -1)} \\&\int ^{k^{a-\gamma }w}_0 \mathrm {e}^{4\mathrm {i}\pi ck^{1-a}\theta }\theta ^{-\beta } \, h\left( w-k^{\gamma -a} \theta \right) \Big (\int _\mathbb {R} g\left( k^{1-a\beta }\theta ^\beta -u\right) g\left( u\right) \, du\Big ) d\theta . \end{aligned}$$

The choice $a=1$ is the only one leading to a non-trivial limit in the integral over $\theta$ as $k\rightarrow 0$. Indeed, for $a < 1$, the exponential factor converges to 1 and the integrand is proportional to $\theta ^{-\beta }$ which is not integrable over $\mathbb {R}_+$, while for $a > 1$ the integral converges to 0, since the exponential factor oscillates more and more. Setting

$$\begin{aligned} \gamma = 1 + \beta , \qquad 0< \beta < 1, \end{aligned}$$

(6.8)

one arrives at

$$\begin{aligned} \begin{array}{ll} h'(w) &{}\displaystyle = -4\pi ^2 D\, k^{2-\gamma } h(w) \\ &{}\displaystyle \quad - 8\pi ^2 \lambda ^2 \, \int ^{k^{1-\gamma }w}_0 \mathrm {e}^{4\mathrm {i}\pi c \theta }\theta ^{-\beta } \, h\left( w-k^{\beta } \theta \right) \Big (\int _\mathbb {R} g\left( k^{1-\beta }\theta ^\beta -u\right) g\left( u\right) \, du\Big ) d\theta . \end{array} \end{aligned}$$

In the limit $k\rightarrow 0$,

$$\begin{aligned} h'(w) = - h(w) (4\pi \lambda )^2 \Big (\int _0^\infty |g(u)|^2 du\Big ) \Big (\int _0^\infty \mathrm {e}^{4\mathrm {i}\pi c \theta }\theta ^{-\beta }\,d\theta \Big ), \end{aligned}$$

(6.9)

which determines $h$ once the values of the integrals on the right-hand side are known. We can now give a precise meaning to the limiting procedure, namely

$$\begin{aligned} \lim _{k \rightarrow 0} \mathrm {e}^{2\mathrm {i}\pi c k^{1-\gamma } w } \hat{f}_1\left( k,k^{-\gamma }w\right) = h(w), \end{aligned}$$

(6.10)

where $h$ is the solution of (6.9).

1.2 Appendix 1.2: Cross-Coupled Scaling Functions

The time integral in (6.9) can be computed analytically as (see [35, Sect. 6.33])

$$\begin{aligned} \int _0^\infty \mathrm {e}^{4\mathrm {i}\pi c \theta }\theta ^{-\beta }\,d\theta = (4\pi c)^{-1+\beta } \int _0^\infty \mathrm {e}^{\mathrm {i} s}s^{-\beta }\,ds = a \Big (1 + \frac{\mathrm {i}}{\tan (\pi \beta /2)}\Big ) \end{aligned}$$

(6.11)

with

$$\begin{aligned} a = (4\pi c)^{-1+\beta } \, \frac{\pi }{2\Gamma (\beta ) \cos (\pi \beta /2)}. \end{aligned}$$

We repeat now the derivation in Appendix 1.1 considering $h$ as input and $g$ as the output. By symmetry (6.4), one concludes that

$$\begin{aligned} h(k^\gamma t) = \overline{g\big ((k^{1/\beta } t)^\beta \big )}, \end{aligned}$$

which forces $h(w) = \overline{g(w^\beta )}$ and

$$\begin{aligned} \gamma = \frac{1}{\beta }, \end{aligned}$$

which, combined with (6.8), implies that $\gamma$ equals the golden mean (6.5). The normalization condition (6.3) implies $h(0) = 1$. Hence, noting that $\tan (\pi \beta /2) = -1/\tan (\pi \gamma /2)$,

$$\begin{aligned} h(w) = \exp \left( -(4\pi \lambda )^2 a\big (1-\mathrm {i}\tan (\pi \gamma /2)\big ) A w\right) , \end{aligned}$$

(6.12)

with

$$\begin{aligned} A = \int _0^\infty |g(u)|^2 du = \int _0^\infty |h(w^\gamma )|^2 dw. \end{aligned}$$

Inserting (6.12) in the latter expression, it follows that

$$\begin{aligned} A = (a (4\pi \lambda )^2)^{-1/\gamma ^2}\widetilde{A}^{1/\gamma }, \qquad \widetilde{A} = \int _0^\infty \mathrm {e}^{-2 w ^{\gamma }}\,dw = \frac{2^{-1/\gamma }}{\gamma } \Gamma \left( \frac{1}{\gamma }\right) . \end{aligned}$$

(6.13)

Thereby we finally obtain the scaling function

$$\begin{aligned} f_\sigma (k,t) = \exp \left( -2\mathrm {i}\pi \sigma k c t - C\,|2\pi k|^\gamma \big ( 1 -\mathrm {i}\sigma \mathrm {sgn}(k)\tan (\pi \gamma /2) \big ) t \right) , \end{aligned}$$

(6.14)

with

$$\begin{aligned} C = 2^{-1/\gamma } \lambda ^{2/\gamma } \left( \frac{1}{\gamma \sin (\pi \gamma /2)}\right) ^{1/\gamma } c^{1-2/\gamma }, \end{aligned}$$

which one recognizes as the Fourier transform of an $\alpha$-stable law with $\alpha = \gamma$ and maximal asymmetry $b = \sigma$, see Appendix 1.3. This expression reduces to the more general expression derived in [26] in the case when the strengths of the nonlinearities for the cross-coupled modes are different.

For two components the Lévy distribution is necessarily maximally asymmetric. For three modes, there is the possibility to sandwich the Lévy peak inbetween two sound peaks with a rapid fall off, as KPZ or Gaussian. Then the Lévy distribution could be partially asymmetric with the tails cut off at the location of the sound peaks. Such a situation is realized in all anharmonic chains. For them the two sound peaks are mirror images relative to $0$, and the heat peak reflection symmetric, which implies that the Lévy distribution has to be symmetric, $b = 0$. On a mathematical level, the only result available is the harmonic chain with random velocity exchanges. In this case the sound peaks are Gaussian and the heat peak is Lévy with parameters $\alpha = \tfrac{3}{2}$ and $b = 0$ [36].

1.3 Appendix 1.3: Lévy Distributions and Their Asymptotic Properties

The Lévy distributions are defined through their Fourier transform as

(6.15)

There are two parameters: $\alpha$ controls the steepness, $0 < \alpha < 2$, and $b$ controls the asymmetry, $|b| \leqslant 1$. For $|b| > 1$ the Fourier integral no longer defines a non-negative function. At the singular point $\alpha = 2$ only $b = 0$ is admitted and the distribution is Gaussian. If $|b|< 1$, the asymptotic decay of $f_{\mathrm{L\acute{e}vy},\alpha ,b}(x)$ is determined by $\alpha$ and is given by $|x|^{-\alpha -1}$ for $|x| \rightarrow \infty$. At $|b| = 1$ the two tails show different decay. The functions corresponding to $b = 1$ and $b = -1$ are mirror images, for $b=1$ the slow decay being for $x \rightarrow -\infty$ and still as $|x|^{-\alpha -1}$. For $0 < \alpha \leqslant 1$, $f_{\mathrm{L\acute{e}vy},\alpha ,1}(x) = 0$ for $x > 0$, while for $1 < \alpha < 2$ the decay becomes stretched exponential as $\exp (-c_0 x^{\alpha /(1 - \alpha )})$ with known constant $c_0$. We refer to [37] for more details. In our context only the maximal asymmetry $b = \pm 1$ with $1 <\alpha < 2$ is realized.

Appendix 2: Modified KPZ Scaling

In this section we study modified KPZ from second table, row 2 of Sect. 2, in which case $G^1_{11} \ne 0$ and mode 2 is diffusive, but has a non-trivial feed back to mode 1 since $G^1_{22} \ne 0$. More precisely, upon changing the frame of reference, we assume that

$$\begin{aligned} f_2(x,t) = \frac{1}{\sqrt{4\pi D t}} \, \mathrm {e}^{-(x + ct)^2/4D t}, \end{aligned}$$

$c > 0$, while $f_1$ evolves according to

$$\begin{aligned} \begin{array}{ll} \partial _t f_1 = D_1 \partial _x^2f_1 &{}\displaystyle + 2 \left( G^1_{11} \right) ^2 \int ^t_0 \int _{\mathbb {R}} \partial ^2_xf_1(x-y,t-s) f_1(y,s)^2 \, dy \, ds \\ &{}\displaystyle + 2 \left( G^1_{22} \right) ^2 \int ^t_0 \int _{\mathbb {R}} \partial ^2_xf_1(x-y,t-s) f_2 (y,s)^2 \, dy \, ds, \end{array} \end{aligned}$$

compare with (2.5)–(2.6). Through Fourier transform in space one obtains

$$\begin{aligned} \begin{array}{ll} \partial _t \hat{f}_1(k,t) &{}\displaystyle = - D_1 (2\pi k)^2 \hat{f}_1(k,t) \\ &{}\displaystyle \quad - 2(2\pi k)^2 \left( G^1_{11} \right) ^2 \int ^t_0 \hat{f}_1(k,t-s) \Big ( \int _{\mathbb {R}} \hat{f}_1(k-q,s)\, \hat{f}_1(q,s) \, dq \Big ) ds \, \\ &{}\displaystyle \quad - 2(2\pi k)^2 \left( G^1_{22} \right) ^2 \int ^t_0 \hat{f}_1(k,t-s) \Big ( \int _{\mathbb {R}} \hat{f}_2(k-q,s)\, \hat{f}_2(q,s) \, dq \Big ) ds . \end{array} \end{aligned}$$

As in Appendix 1.1, it suffices to consider $k>0$. Following the scheme in [7, Sect. 4], we make the ansatz

$$\begin{aligned} f_1(k,t) = F\big ((\lambda _\mathrm {s} t)^{2/3} k\big ) \end{aligned}$$

(7.1)

with $\lambda _\mathrm {s} = 2 \sqrt{2} |G^1_{11}|$. Setting momentarily $G^1_{22} = 0$, and substituting $u = (\lambda _\mathrm {s} t)^{2/3} k$, one arrives at

$$\begin{aligned} \tfrac{2}{3}u F'(u) = - \pi ^2 u^2 \int ^1_0 F\big ((1-\theta )^{2/3} u\big ) \Big ( \int _{\mathbb {R}} F\big (\theta ^{2/3} (u-v)\big )\,F\big (\theta ^{2/3} v\big ) \, dv \Big ) d\theta . \end{aligned}$$

(7.2)

Next we set momentarily $G^1_{11} = 0$. Then we are back to the problem discussed in Appendix 1.1 with $\beta =1/2$, $\gamma = 3/2$ and input function $\hat{f}_{2}(k,t) = \mathrm {e}^{2\mathrm {i}\pi kct} g(k t^{1/2})$ with $g(k) = \exp \big (-D(2\pi k)^2\big )$. In the scaling limit the output function is $h(k^{3/2}t)$, which satisfies

$$\begin{aligned} h'(w) = - h(w) (4\pi G^1_{22})^2 \Big (\int _0^\infty |g(u)|^2 du\Big ) \Big (\int _0^\infty \mathrm {e}^{2\mathrm {i}\pi c \theta }\theta ^{-1/2}\,d\theta \Big ). \end{aligned}$$

Working out the integrals yields

$$\begin{aligned} h'(w) = - h(w) (4\pi G^1_{22})^2 ( 4\sqrt{\pi D})^{-1} (1 + \mathrm {i})(2 \sqrt{c})^{-1}. \end{aligned}$$

Since $w = k^{3/2}t$, one concludes $h(w) = F\big ((\lambda _\mathrm {s}w)^{2/3}\big )$. The linear equation $h'(w) = ah(w)$ translates into

$$\begin{aligned} \tfrac{2}{3} F'(u) = a (\lambda _\mathrm {s})^{-1} \sqrt{u} F(u). \end{aligned}$$

(7.3)

Combining (7.2) and (7.3) one arrives at the fixed point equation for the scaling function $F$,

$$\begin{aligned} \begin{array}{ll} \tfrac{2}{3} \, F'(u) &{}\displaystyle = - \pi ^2 u \int ^1_0 F\big ((1-\theta )^{2/3} u\big ) \left( \int _{\mathbb {R}} F\big (\theta ^{2/3} (u-v)\big )\,F\big (\theta ^{2/3} v\big ) \, dv \right) d\theta \, \\ &{}\displaystyle \ \ \ - (4\pi G^1_{22})^2 ( 4 \sqrt{\pi D})^{-1} (1 + \mathrm {i})(2 \sqrt{c})^{-1} (2 \sqrt{2}|G_{11}^1|)^{-1} \, \sqrt{u} F(u) . \end{array} \end{aligned}$$

(7.4)

If $G^1_{22} =0$, then (7.4) reduces to the fixed point equation for $f_\mathrm{KPZ}$ in the mode-coupling approximation. Now a term linear in $F$ is added. Presumably this results in a one-parameter family of scaling functions, depending on the prefactor of the linear term. Most likely such a behavior persists for the true coupled Burgers equations.

Appendix 3: Expressions for the Coupling Constants

We follow here the strategy presented in [7, Appendix A] to compute the various coefficients appearing in the mode-coupling equations. Mode 1 corresponds to the sound mode, while mode 2 represents the heat mode.

For three random variables $\mathcal {A},\mathcal {B},\mathcal {C}$, we denote the third cumulant by

$$\begin{aligned} \left\langle \mathcal {A};\mathcal {B};\mathcal {C}\right\rangle _{\tau ,\beta }&= \left\langle \mathcal {A}(\eta _0) \mathcal {B}(\eta _0) \mathcal {C}(\eta _0) \right\rangle _{\tau ,\beta } \\&\quad - \left\langle \mathcal {A}(\eta _0)\mathcal {B}(\eta _0)\right\rangle _{\tau ,\beta }\left\langle \mathcal {C}(\eta _0)\right\rangle _{\tau ,\beta } - \left\langle \mathcal {A}(\eta _0)\mathcal {C}(\eta _0)\right\rangle _{\tau ,\beta }\left\langle \mathcal {B}(\eta _0)\right\rangle _{\tau ,\beta } \\&\quad - \left\langle \mathcal {B}(\eta _0)\mathcal {C}(\eta _0)\right\rangle _{\tau ,\beta }\left\langle \mathcal {A}(\eta _0)\right\rangle _{\tau ,\beta } \\&\quad +2 \left\langle \mathcal {A}(\eta _0) \right\rangle _{\tau ,\beta }\left\langle \mathcal {B}(\eta _0) \right\rangle _{\tau ,\beta }\left\langle \mathcal {C}(\eta _0) \right\rangle _{\tau ,\beta }. \end{aligned}$$

1.1 Appendix 3.1: Matrix $R$ and Sound Velocity

The right eigenvectors of the matrix $A$ are proportional to

$$\begin{aligned} \psi _1 = Z_1^{-1} \begin{pmatrix} 1 \\ -\tau \end{pmatrix}, \qquad \psi _2 = Z_2^{-1} \begin{pmatrix} \partial _e \tau \\ -\partial _h \tau \end{pmatrix}, \end{aligned}$$

with, respectively, associated eigenvalues 0 and

$$\begin{aligned} c = 2 (\partial _h -\tau \partial _e) \tau . \end{aligned}$$

The corresponding left eigenvectors are proportional to

$$\begin{aligned} \widetilde{\psi }_1 = \widetilde{Z}_1^{-1} \begin{pmatrix} \partial _h \tau \\ \partial _e \tau \end{pmatrix}. \qquad \widetilde{\psi }_2 = \widetilde{Z}_2^{-1} \begin{pmatrix} \tau \\ 1 \end{pmatrix}, \end{aligned}$$

The coefficients $\widetilde{Z}_1,\widetilde{Z}_2$ are obtained from the diagonal conditions $RCR^\mathrm {T} = 1$, the $R$ matrix being constructed from the left eigenvectors. The coefficients $Z_1,Z_2$ are determined by the condition $R R^{-1} = 1$, with the inverse $R^{-1}$ constructed from the right eigenvectors. By some computations one obtains

$$\begin{aligned} c = -2 \Gamma ^{-1}\langle V + \tau \eta ; V + \tau \eta \rangle _{\tau ,\beta } <0, \qquad \Gamma = \beta \big ( \left\langle \eta ; \eta \right\rangle _{\tau ,\beta }\left\langle V ; V\right\rangle _{\tau ,\beta } - \left\langle \eta ; V\right\rangle _{\tau ,\beta }^2 \big ), \end{aligned}$$

(8.1)

as well as

$$\begin{aligned} \widetilde{Z}_1 = \sqrt{-\frac{c}{2\beta }}, \qquad \widetilde{Z}_2 = \sqrt{-\frac{\Gamma c}{2}}. \end{aligned}$$

Moreover,

$$\begin{aligned} R = \begin{pmatrix} \partial _h \tau /\widetilde{Z}_1 &{} \partial _e \tau /\widetilde{Z}_1 \\ \tau /\widetilde{Z}_2 &{} 1/\widetilde{Z}_2 \\ \end{pmatrix}, \end{aligned}$$

(8.2)

with

$$\begin{aligned} \partial _h \tau = -\Gamma ^{-1}\left\langle V ; V + \tau \eta \right\rangle _{\tau ,\beta }, \qquad \partial _e \tau = \Gamma ^{-1}\left\langle \eta ; V + \tau \eta \right\rangle _{\tau ,\beta }. \end{aligned}$$

(8.3)

Finally,

$$\begin{aligned} Z_1 = \frac{c}{2\widetilde{Z}_1} = -\sqrt{-\frac{\beta c}{2}}, \qquad Z_2 = -\frac{c}{2\widetilde{Z}_2} = \sqrt{-\frac{c}{2\Gamma }}. \end{aligned}$$

1.2 Appendix 3.2: Hessians and Coupling Matrices $G$

The Hessians of the currents $j_h = 2\tau$ and $j_e = -\tau ^2$ are

$$\begin{aligned} H_h = \begin{pmatrix} \partial _h^2 j_h &{} \partial _h \partial _e j_h \\ \partial _h \partial _e j_h &{} \partial _e^2 j_h \\ \end{pmatrix} = 2 \begin{pmatrix} \partial _h^2 \tau &{} \partial _h \partial _e \tau \\ \partial _h \partial _e \tau &{} \partial _e^2 \tau \\ \end{pmatrix} , \end{aligned}$$

and

$$\begin{aligned} H_e = \begin{pmatrix} \partial _h^2 j_e &{} \partial _h \partial _e j_e \\ \partial _h \partial _e j_e &{} \partial _e^2 j_e \\ \end{pmatrix} = -\tau H_h - 2 \widehat{H}_e, \qquad \widehat{H}_e = \begin{pmatrix} (\partial _h \tau )^2 &{} \partial _h \tau \partial _e \tau \\ \partial _h \tau \partial _e \tau &{} (\partial _e \tau )^2 \\ \end{pmatrix}. \end{aligned}$$

The second derivatives of $\tau$ with respect to $h,e$, which are required in order to evaluate the Hessian matrices $H_h,H_e$, are obtained by inverting the following systems,

$$\begin{aligned} \begin{pmatrix} \partial _\tau h &{} \partial _\tau e \\ \partial _\beta h &{} \partial _\beta e \end{pmatrix} \begin{pmatrix} \partial _h^2 \tau \\ \partial _h \partial _e \tau \end{pmatrix} = \begin{pmatrix} \partial _\tau (\partial _h \tau ) \\ \partial _\beta (\partial _h \tau ) \end{pmatrix}, \qquad \begin{pmatrix} \partial _\tau h &{} \partial _\tau e \\ \partial _\beta h &{} \partial _\beta e \end{pmatrix} \begin{pmatrix} \partial _h\partial _e \tau \\ \partial ^2_e \tau \end{pmatrix} = \begin{pmatrix} \partial _\tau (\partial _e \tau ) \\ \partial _\beta (\partial _e \tau ) \end{pmatrix}, \end{aligned}$$

and using the expressions (8.3) for the partial derivatives $\partial _h \tau ,\partial _e \tau$, as well as the rules

$$\begin{aligned} \partial _\tau \left\langle \mathcal {A};\mathcal {B}\right\rangle _{\tau ,\beta } = -\beta \left\langle \mathcal {A};\mathcal {B};\eta \right\rangle _{\tau ,\beta }, \qquad \partial _\beta \left\langle \mathcal {A};\mathcal {B}\right\rangle _{\tau ,\beta } = -\left\langle \mathcal {A};\mathcal {B};V+\tau \eta \right\rangle _{\tau ,\beta }. \end{aligned}$$

The elements of the $G$ matrices are then computed as

$$\begin{aligned} \begin{array}{ll} G_{\alpha \alpha '}^1 &{} = \tfrac{1}{2} \Big ( R_{11}\big ( \psi _\alpha ^\mathrm {T} \cdot H_h \psi _{\alpha '}\big ) + R_{12}\big ( \psi _\alpha ^\mathrm {T}\cdot H_e\psi _{\alpha '}\big ) \Big ), \\ G_{\alpha \alpha '}^2 &{} = \tfrac{1}{2} \Big (R_{21} \big ( \psi _\alpha ^\mathrm {T}\cdot H_h \psi _{\alpha '}\big ) + R_{22} \big ( \psi _\alpha ^\mathrm {T}\cdot H_e \psi _{\alpha '}\big )\Big ). \end{array} \end{aligned}$$

Note that, since $R_{21} = \tau R_{22}$ and $\widehat{H}_e \psi _2 = 0$, the only non-zero coefficient of the heat mode coupling matrix $G^2$ is $G^2_{11}$, which can be written more concisely as

$$\begin{aligned} G^2_{11} = -R_{22} \big (\psi _1^\mathrm {T} \cdot \widehat{H}_e \psi _1\big ) = - R_{22}\frac{(\partial _h \tau -\tau \partial _e \tau )^2}{Z_1^2} = -\left( \frac{c}{2}\right) ^2 \frac{1}{\widetilde{Z}_2 Z_1^2} = - \frac{1}{\beta }\sqrt{-\frac{c}{2\Gamma }} <0. \end{aligned}$$

On the other hand, there seems to be no simplified expression for the sound mode coupling matrix  $G^1$ and, a priori, all entries $G^1_{\alpha \alpha '}$ are non-zero. A straightforward computation shows that

$$\begin{aligned} G^1_{11} = \frac{c}{2 Z_1^2 \widetilde{Z}_1} \left( \partial _h - \tau \partial _e\right) ^2\tau , \end{aligned}$$

(8.4)

which has no definite sign, in general.

1.3 Appendix 3.3: Specific Potentials

There are simplifications for the expression of the components of the matrix $G^1$ for specific potentials such as the Kac-van Moerbeke potential (4.2). In the latter case, a simple computation based on the identity $V'(\eta ) = -\kappa V(\eta ) + \eta$ shows that

$$\begin{aligned} \tau = \kappa e - h, \end{aligned}$$

so that $c = -2(1+\kappa \tau )$, $H_h = 0$,

$$\begin{aligned} H_e = -2 \begin{pmatrix} 1 &{} -\kappa \\ -\kappa &{} \kappa ^2 \end{pmatrix}, \qquad R = \frac{1}{\sqrt{1+\kappa \tau }} \begin{pmatrix} -\sqrt{\beta } &{} \kappa \sqrt{\beta } \\ \tau /\sqrt{\Gamma } &{} 1/\sqrt{\Gamma } \\ \end{pmatrix}. \end{aligned}$$

In addition,

$$\begin{aligned} \psi _1 = \sqrt{\frac{1}{\beta (1+\kappa \tau )}} \begin{pmatrix} 1 \\ -\tau \end{pmatrix}, \qquad \psi _2 = \sqrt{\frac{\Gamma }{1+\kappa \tau }} \begin{pmatrix} \kappa \\ 1 \end{pmatrix}, \end{aligned}$$

so that $H_e \psi _2 = 0$. The only non-zero coefficient of $G^1$ therefore is $G_{11}^1$, which reads

$$\begin{aligned} G^1_{11} = -\kappa \sqrt{\frac{1+\kappa \tau }{\beta }}. \end{aligned}$$

Note that the harmonic potential $V(\eta ) = \frac{\eta ^2}{2}$ is obtained from the KvM potential (4.2) in the limit $\kappa \rightarrow 0$. Hence also the coupling matrices are obtained in the same limit, implying that $G^1 = 0$ for the harmonic potential.

1.4 Appendix 3.4: Coupling Matrices for the Numerically Simulated Systems

Recall that we choose $\tau = 1$ and $\beta = 2$ in both cases. For the FPU potential (4.1) with $a=2$, we obtain $c = -5.28$,

$$\begin{aligned} R = \begin{pmatrix} -0.401 &{} 1.90 \\ 2.55 &{} 2.55 \\ \end{pmatrix}, \qquad R^{-1} = \begin{pmatrix} -0.435 &{} 0.323 \\ 0.435 &{} 0.0683 \\ \end{pmatrix} \end{aligned}$$

and

$$\begin{aligned} G^1 = \begin{pmatrix} -2.23 &{} 0.431 \\ 0.431 &{} 0.333 \\ \end{pmatrix}, \qquad G^2 = \begin{pmatrix} -3.37 &{} 0 \\ 0 &{} 0\\ \end{pmatrix}. \end{aligned}$$

(8.5)

For the KvM potential (4.2) with $\kappa =1$, we obtain $c=-4$,

$$\begin{aligned} R = \begin{pmatrix} -1 &{} 1 \\ 2.72 &{} 2.72 \\ \end{pmatrix}, \qquad R^{-1} = \begin{pmatrix} -1/2 &{} 0.184 \\ 1/2 &{} 0.184 \\ \end{pmatrix} \end{aligned}$$

and

$$\begin{aligned} G^{1} = \begin{pmatrix} -1 &{} 0 \\ 0 &{} 0 \\ \end{pmatrix}, \qquad G^{2} = \begin{pmatrix} -2.719 &{} 0 \\ 0 &{} 0\\ \end{pmatrix}. \end{aligned}$$

(8.6)