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.

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)