Nonlinear Fluctuating Hydrodynamics for Anharmonic Chains

Abstract

With focus on anharmonic chains, we develop a nonlinear version of fluctuating hydrodynamics, in which the Euler currents are kept to second order in the deviations from equilibrium and dissipation plus noise are added. The required model-dependent parameters are written in such a way that they can be computed numerically within seconds, once the interaction potential, pressure, and temperature are given. In principle the theory is applicable to any one-dimensional system with local conservation laws. The resulting nonlinear stochastic field theory is handled in the one-loop approximation. Some of the large scale predictions can still be worked out analytically. For more details one has to rely on numerical simulations of the corresponding mode-coupling equations. In this way we arrive at detailed predictions for the equilibrium time correlations of the locally conserved fields of an anharmonic chain.

Appendices

Appendix 1: Coupling Constants

1.1 (a) Equilibrium Susceptibilities, Currents

Averages with respect to $Z^{-1}\exp [-\beta (V(y)+py)]dy$ are denoted in this appendix by $\langle \cdot \rangle$ with the dependence on $p,\beta$ being suppressed. $\langle X_1;X_2\rangle = \langle X_1 X_2\rangle - \langle X_1\rangle \langle X_2\rangle$ is the second cumulant and

$$\begin{aligned} \langle X_1;X_2;X_3\rangle&= \langle X_1 X_2 X_3 \rangle \nonumber \\&\quad -\, \langle X_1 X_2\rangle \langle X_3\rangle - \langle X_1 X_3\rangle \langle X_2\rangle - \langle X_2 X_3\rangle \langle X_1\rangle + 2 \langle X_1\rangle \langle X_2\rangle \langle X_3\rangle \qquad \end{aligned}$$

(8.1)

the third cumulant. It holds

$$\begin{aligned}&\quad \partial _p \langle X_1\rangle =-\beta \langle X_1;y\rangle ,\quad \partial _\beta \langle X_1\rangle =- \langle X_1;V+py\rangle , \nonumber \\&\quad \partial _p \langle X_1;X_2\rangle =-\beta \langle X_1;X_2;y\rangle ,\quad \partial _\beta \langle X_1;X_2\rangle =- \langle X_1;X_2;V+py\rangle . \end{aligned}$$

(8.2)

In equilibrium, $\{r_j,p_j, j\in \mathbb {Z}\}$ are independent random variables. $p_j$ has a Gaussian density with mean zero and variance $\beta ^{-1}$ and $r_j$ has the density $Z^{-1}\exp [-\beta (V(y)+py)]$. Hence $S(j,0) = \delta _{0j} C$ with

$$\begin{aligned} C=\left( \begin{array}{l@{\quad }l@{\quad }l} \langle y;y\rangle &{} 0 &{} \langle y;V\rangle \\ 0 &{} \beta ^{-1} &{} 0 \\ \langle y;V\rangle &{} 0 &{} \tfrac{1}{2}\beta ^{-2}+\langle V;V\rangle \end{array}\,\right) . \end{aligned}$$

(8.3)

The macroscopic conserved fields are $(\ell ,\mathsf {u},\mathfrak {e})$. Their currents are

$$\begin{aligned} \big ({-}\mathsf {u},p(\ell ,\mathfrak {e}-\tfrac{1}{2} \mathsf {u}^2), \mathsf {u}p(\ell ,\mathfrak {e}-\tfrac{1}{2}\mathsf {u}^2)\big ). \end{aligned}$$

(8.4)

We linearize the Euler equations at $(\ell , \mathsf {u}=0,\mathsf {e})$, $\mathsf {e}$ the internal energy, where

$$\begin{aligned} \ell =\langle y\rangle ,\quad \mathsf {e}= \tfrac{1}{2}\beta ^{-1} + \langle V\rangle , \end{aligned}$$

(8.5)

which defines $(p,\beta ) \mapsto (\ell (p,\beta ),\mathsf {e}(p,\beta ))$. Inverting this map yields

$$\begin{aligned} p=p(\ell ,\mathsf {e}),\quad \beta =\beta (\ell ,\mathsf {e}). \end{aligned}$$

(8.6)

1.2 (b) Linearization, Normal Modes

The linearized currents are

$$\begin{aligned} A=\left( \begin{array}{l@{\quad }l@{\quad }l} 0 &{} -1 &{} 0 \\ \partial _\ell p &{} 0 &{} \partial _\mathsf {e}p \\ 0 &{} p &{} 0 \end{array}\right) \end{aligned}$$

(8.7)

with the property that

$$\begin{aligned} AC=CA^{\mathrm {T}}. \end{aligned}$$

(8.8)

$A$ has the eigenvalues $c_\sigma =\sigma c$, $c_0=0$, $\sigma =\pm 1$, $c$ the sound speed,

$$\begin{aligned} c^2= -\partial _{\ell } p+p \partial _\mathsf {e}p>0. \end{aligned}$$

(8.9)

$A$ has right eigenvectors defined by $A|\psi _\alpha \rangle = c_\alpha |\psi _\alpha \rangle$ and given by

$$\begin{aligned} \psi _0=Z^{-1}_0\left( \begin{array}{l} \partial _\mathsf {e} p\\ 0\\ -\partial _\ell p \\ \end{array}\right) , \quad \psi _\sigma =Z^{-1}_\sigma \left( \begin{array}{l} -1\\ \sigma c\\ p \\ \end{array}\right) \end{aligned}$$

(8.10)

and left eigenvectors defined by $\langle \tilde{\psi }_\alpha |A = c_\alpha \langle \tilde{\psi }_\alpha |$ and given by

$$\begin{aligned} \tilde{\psi }_0=\tilde{Z}^{-1}_0 \left( \begin{array}{l} p\\ 0\\ 1 \\ \end{array}\right) , \quad \tilde{\psi }_\sigma =\tilde{Z}^{-1}_\sigma \left( \begin{array}{l} \partial _\ell p \\ \sigma c\\ \partial _\mathsf {e} p \\ \end{array}\right) , \end{aligned}$$

(8.11)

which satisfy $\langle \tilde{\psi }_\alpha |\psi _\beta \rangle =0$ for $\alpha \ne \beta$. The linear transformation to normal modes, $\vec {\phi }=R\vec {u}$, is defined through

$$\begin{aligned} RAR^{-1}= \mathrm {diag}(-c,0,c),\quad RCR^\mathrm {T}=1. \end{aligned}$$

(8.12)

The first identity can be achieved by setting

$$\begin{aligned}&\quad R= \left( \begin{array}{l} \langle \tilde{\psi }_-|\\ \langle \tilde{\psi }_0|\\ \langle \tilde{\psi }_+| \\ \end{array}\right) = (\tilde{Z}_{1})^{-1} \left( \begin{array}{l@{\quad }l@{\quad }l} \partial _\ell p&{}- c&{}\partial _ \mathsf {e}p\\ \tilde{\kappa }p&{}0&{} \tilde{\kappa }\\ \partial _\ell p&{} c &{}\partial _ \mathsf {e}p\\ \end{array}\right) ,\end{aligned}$$

(8.13)

$$\begin{aligned}&\quad R^{-1}=\big (|\psi _-\rangle |\psi _0\rangle |\psi _+\rangle \big ) = (Z_{1})^{-1} \left( \begin{array}{l@{\quad }l@{\quad }l} -1&{}\kappa \partial _ \mathsf {e}p&{}- 1\\ - c&{}0&{}c\\ p&{}-\kappa \partial _\ell p&{}p \end{array}\right) . \end{aligned}$$

(8.14)

The normalization factors are still free, but up to an overall factor of $-1$ determined by the second identity of (8.12),

$$\begin{aligned}&\quad Z_0\tilde{Z}_0=c^2,\quad \tilde{Z}^2_0=\tfrac{1}{2}\beta ^{-2}+\langle V+py;V+py\rangle = \Gamma c^2, \nonumber \\&\quad Z_\sigma \tilde{Z}_\sigma =2c^2,\quad \tilde{Z}^2_\sigma =2 \beta ^{-1}c^2,\quad \kappa = Z_1/Z_0, \quad \tilde{\kappa } = \tilde{Z}_1/ \tilde{Z}_0. \end{aligned}$$

(8.15)

The Euler part of the equations of motion reads

$$\begin{aligned} \partial _t u_\alpha + \partial _x\big ((A\vec {u})_\alpha + \tfrac{1}{2} \langle \vec {u}, H^{\alpha }\vec {u}\rangle \big )= 0. \end{aligned}$$

(8.16)

Using $\vec {\phi } = R \vec {u}$, one arrives at

$$\begin{aligned} \partial _t \phi _\alpha + \partial _x\big (c_\alpha \phi _\alpha + \tfrac{1}{2}\sum _{\alpha ' =1}^3 R_{\alpha \alpha '} \langle R^{-1}\vec {\phi }, H^{\alpha '} R^{-1}\vec {\phi }\rangle \big )= 0, \end{aligned}$$

(8.17)

which implies

$$\begin{aligned} \partial _t \phi _\alpha + \partial _x\big ( c_\alpha \phi _\alpha + \langle \vec {\phi }, G^{\alpha }\vec {\phi }\rangle \big )= 0 \end{aligned}$$

(8.18)

with the normal mode coupling constants

$$\begin{aligned} G^\alpha _{\beta \gamma }=\tfrac{1}{2}\sum _{\alpha '=1}^3R_{\alpha \alpha '} \langle \psi _\beta , H^{\alpha '} \psi _\gamma \rangle . \end{aligned}$$

(8.19)

1.3 (c) Hessians and $G$ Couplings

The three Hessians are

$$\begin{aligned} H^\ell =0,\quad H^\mathsf {u}= \left( \begin{array}{l@{\quad }l@{\quad }l} \partial ^2_\ell p &{} 0 &{} \partial _\ell \partial _\mathsf {e}p \\ 0 &{} -\partial _\mathsf {e}p &{} 0 \\ \partial _\ell \partial _\mathsf {e}p &{} 0 &{} \partial ^2_\mathsf {e}p \end{array}\right) , \quad H^\mathsf {e}= \left( \begin{array}{l@{\quad }l@{\quad }l} 0 &{} \partial _\ell p &{} 0 \\ \partial _\ell p &{} 0 &{} \partial _\mathsf {e}p \\ 0 &{} \partial _\mathsf {e}p &{} 0 \end{array}\,\right) . \end{aligned}$$

(8.20)

The $H^\mathsf {u}$ matrix elements are given by

$$\begin{aligned}&\qquad \langle \psi _0,H^\mathsf {u} \psi _0\rangle =\frac{1}{Z^2_0} \big (\partial ^2_\ell p\partial _\mathsf {e}p \partial _\mathsf {e}p-2\partial _\ell \partial _\mathsf {e}p \partial _\ell p \partial _\mathsf {e}p +\partial ^2_\mathsf {e}p \partial _\ell p\partial _\ell p\big ),\end{aligned}$$

(8.21)

$$\begin{aligned}&\qquad \langle \psi _0,H^\mathsf {u} \psi _\sigma \rangle =\frac{1}{Z_0 Z_\sigma } \big (-\partial ^2_\ell p\partial _\mathsf {e}p-p \partial ^2_\mathsf {e}p\partial _\ell p +p \partial _\ell \partial _\mathsf {e}p \partial _\mathsf {e}p +\partial _\ell \partial _\mathsf {e} p \partial _\ell p\big ),\qquad \quad \end{aligned}$$

(8.22)

$$\begin{aligned}&\qquad \langle \psi _{\sigma },H^\mathsf {u} \psi _{\sigma '}\rangle =\frac{1}{Z_\sigma Z_{\sigma '}} \big (\partial ^2_\ell p- 2p\partial _\ell \partial _\mathsf {e}p +p^2 \partial ^2_\mathsf {e}p-\sigma \sigma ' c^2 \partial _\mathsf {e}p \big ) \end{aligned}$$

(8.23)

and the $H^\mathsf {e}$ matrix elements are

$$\begin{aligned} \langle \psi _0,H^\mathsf {e} \psi _0\rangle = 0,\quad \langle \psi _0,H^\mathsf {e}\psi _\sigma \rangle = 0,\quad \langle \psi _{\sigma },H^\mathsf {e} \psi _{\sigma '}\rangle = c(2\beta )^{-1}(\sigma + \sigma '). \end{aligned}$$

(8.24)

Denoting the standard basis vectors by $\mathrm {e}_1,\mathrm {e}_2,\mathrm {e}_3$, one has

$$\begin{aligned} R\mathrm {e}_2 = (\tfrac{1}{2}\beta )^{1/2}(- \mathrm {e}_1 + \mathrm {e}_3),\quad R\mathrm {e}_3 = c^{-1}(\tfrac{1}{2}\beta )^{1/2}\partial _\mathsf {e}p (\mathrm {e}_1 + \mathrm {e}_3) + (\tilde{Z}_0)^{-1} \mathrm {e}_2 \end{aligned}$$

(8.25)

and arrives at the coupling constants

$$\begin{aligned}&\qquad G^\sigma _{\beta \gamma }=\tfrac{1}{2}\sigma (\tfrac{1}{2}\beta )^{1/2} \langle \psi _\beta ,H^\mathsf {u}\psi _\gamma \rangle + \tfrac{1}{2}\partial _\mathsf {e}p (\tfrac{1}{2}\beta )^{1/2}{c} \langle \psi _\beta ,H^\mathsf {e}\psi _\gamma \rangle , \end{aligned}$$

(8.26)

$$\begin{aligned}&\qquad G^0_{\beta \gamma }=\frac{1}{2\tilde{Z}_0} \langle \psi _\beta ,H^\mathsf {e}\psi _\gamma \rangle . \end{aligned}$$

(8.27)

They have the symmetries

$$\begin{aligned}&\qquad G^\alpha _{\beta \gamma } = G^\alpha _{\gamma \beta }, \quad G^\sigma _{\alpha \beta } = - G^{-\sigma }_{-\alpha -\beta },\quad G^\sigma _{-10} = G^\sigma _{01}, \nonumber \\&\qquad G^0_{\sigma \sigma } = G^0_{-\sigma -\sigma },\quad G^0_{\alpha \beta } = 0\,\, \mathrm {otherwise}. \end{aligned}$$

(8.28)

1.4 (d) Transformation to Canonical Variables

We transform the couplings $G$ from microcanonical to canonical variables. In canonical variables, all $G$ coefficients are given by up to third order cumulants in $y,V,V+py$, which can be completely expressed in terms of one-dimensional integrals. A Mathematica program computes all $G$ coefficients for specified $V,p,\beta$ and by the same procedure also the $R,R^{-1}$ matrices.

Differentiating the identities $p(\ell (p,\beta ),\mathsf {e}(p,\beta ))= p$ and $\beta (\ell (p,\beta ),\mathsf {e}(p,\beta ))= \beta$ with respect to $p$ and $\beta$ yields

$$\begin{aligned} \left( \begin{array}{l} 1 \\ 0 \\ \end{array}\right) = \left( \begin{array}{l@{\quad }l} \partial _p\ell &{} \partial _p\mathsf {e} \\ \partial _\beta \ell &{} \partial _\beta \mathsf {e} \\ \end{array}\right) \left( \begin{array}{l} \partial _\ell p \\ \partial _\mathsf {e} p \\ \end{array}\,\right) , \end{aligned}$$

(8.29)

$$\begin{aligned} \left( \begin{array}{l} 0 \\ 1\\ \end{array}\right) = \left( \begin{array}{l@{\quad }l} \partial _p\ell &{} \partial _p\mathsf {e} \\ \partial _\beta \ell &{} \partial _\beta \mathsf {e} \\ \end{array}\right) \left( \begin{array}{l} \partial _\ell \beta \\ \partial _\mathsf {e} \beta \\ \end{array}\,\right) . \end{aligned}$$

(8.30)

Hence

$$\begin{aligned}&\partial _\ell p = \Gamma ^{-1}\partial _\beta \mathsf {e} =-\Gamma ^{-1} \big ( \tfrac{1}{2}\beta ^{-2} +\langle V;V+py\rangle \big ), \nonumber \\&\partial _\mathsf {e} p = -\Gamma ^{-1}\partial _\beta \ell =\Gamma ^{-1}\langle y;V+py\rangle ,\nonumber \\&\Gamma = \beta \big (\langle y;y\rangle \langle V;V\rangle - \langle y;V\rangle ^2\big ) +\tfrac{1}{2}\beta ^{-1}\langle y;y\rangle , \end{aligned}$$

(8.31)

and for the velocity of sound

$$\begin{aligned} c^2= \frac{1}{\Gamma }\big ( \tfrac{1}{2}\beta ^{-2}+\langle V+py;V+py\rangle \big ). \end{aligned}$$

(8.32)

We collect first and second derivatives of $\ell ,\mathsf {e}$ and first derivatives of $\Gamma$,

$$\begin{aligned}&\partial _p \ell = - \beta \langle y;y\rangle ,\quad \partial _\beta \ell = - \langle y;V + py\rangle ,\end{aligned}$$

(8.33)

$$\begin{aligned}&\partial _p \mathsf {e} = -\beta \langle y;V\rangle ,\quad \partial _\beta \mathsf {e}= - \tfrac{1}{2} \beta ^{-2}- \langle V;V +py \rangle ,\end{aligned}$$

(8.34)

$$\begin{aligned}&\partial _p\partial _\beta \ell = - \langle y;y\rangle +\beta \langle y;y;V+ py\rangle ,\quad \partial _\beta ^2 \ell = \langle y;V+py;V +py\rangle ,\end{aligned}$$

(8.35)

$$\begin{aligned}&\partial _p\partial _\beta \mathsf {e} = - \langle y;V\rangle + \beta \langle y;V; V +py\rangle , \quad \partial _\beta ^2\mathsf {e}= \beta ^{-3} + \langle V;V+py;V+py\rangle ,\end{aligned}$$

(8.36)

$$\begin{aligned}&\partial _p \Gamma \!=\! \beta ^2\big ( -\langle y;y;y\rangle \langle V;V\rangle \!-\! \langle y;y\rangle \langle y;V;V\rangle \!+\! 2 \langle y;V\rangle \langle y;y;V\rangle \big ) -\tfrac{1}{2}\langle y;y;y\rangle ,\qquad \end{aligned}$$

(8.37)

$$\begin{aligned}&\partial _\beta \Gamma = \langle y;y\rangle \langle V;V\rangle - \langle y;V\rangle ^2 - \tfrac{1}{2}\beta ^{-2}\langle y;y\rangle +\beta \big ( - \langle y;y;V+py\rangle \langle V;V\rangle \nonumber \\&\qquad \quad \,\,\, - \langle y;y\rangle \langle V;V;V\!+\!py\rangle +2 \langle y;V\rangle \langle y;V;V\!+\!py\rangle ) \!-\! \tfrac{1}{2} \beta ^{-1} \langle y;y;V+py\rangle \big ).\qquad \quad \end{aligned}$$

(8.38)

To complete our task we still need the second derivatives $\partial _\ell \partial _\ell p$, $\partial _\ell \partial _\mathsf {e} p$, $\partial _\mathrm {e}\partial _\mathsf {e} p$, where we start from

$$\begin{aligned} \left( \begin{array}{l} \partial _p(\Gamma ^{-1} \partial _\beta \mathsf {e}) \\ \partial _\beta (\Gamma ^{-1} \partial _\beta \mathsf {e} )\\ \end{array}\right) = \left( \begin{array}{l@{\quad }l} \partial _p\ell &{} \partial _p\mathsf {e} \\ \partial _\beta \ell &{} \partial _\beta \mathsf {e} \\ \end{array}\right) \left( \begin{array}{l} \partial _\ell \partial _\ell p \\ \partial _\mathsf {e} \partial _\ell p \\ \end{array}\right) , \end{aligned}$$

(8.39)

$$\begin{aligned} -\left( \begin{array}{l} \partial _p(\Gamma ^{-1} \partial _\beta \ell )\\ \partial _\beta ( \Gamma ^{-1} \partial _\beta \ell )\\ \end{array}\right) = \left( \begin{array}{l@{\quad }l} \partial _p\ell &{} \partial _p\mathsf {e} \\ \partial _\beta \ell &{} \partial _\beta \mathsf {e} \\ \end{array}\right) \left( \begin{array}{l} \partial _\ell \partial _\mathsf {e} p \\ \partial _\mathsf {e} \partial _\mathsf {e} p \\ \end{array}\right) . \end{aligned}$$

(8.40)

Inverting (8.39), (8.40) we arrive at

$$\begin{aligned}&\partial ^2_\ell p= - \Gamma ^{-2}(\partial _p\mathsf {e}\partial _\beta ^2\mathsf {e} -\partial _\beta \mathsf {e} \partial _p\partial _\beta \mathsf {e} )+ \Gamma ^{-3}(\partial _p\mathsf {e}\partial _\beta \mathsf {e}\partial _\beta \Gamma -\partial _\beta \mathsf {e}\partial _\beta \mathsf {e}\partial _p\Gamma ),\end{aligned}$$

(8.41)

$$\begin{aligned}&\partial _\ell \partial _\mathsf {e} p = -\Gamma ^{-2}(- \partial _p\mathsf {e} \partial _\beta ^2 \mathsf {e}+ \partial _\beta \mathsf {e} \partial _p\partial _\beta \ell ) + \Gamma ^{-3}(-\partial _p\mathsf {e} \partial _\beta \ell \partial _\beta \Gamma + \partial _\beta \mathsf {e} \partial _\beta \ell \partial _p\Gamma ),\qquad \quad \end{aligned}$$

(8.42)

$$\begin{aligned}&\partial _\mathsf {e}^2p = \Gamma ^{-2}(- \partial _p\ell \partial _\beta ^2\ell + \partial _\beta \ell \partial _p\partial _\beta \ell ) - \Gamma ^{-3}(-\partial _\beta \ell \partial _p \ell \partial _\beta \Gamma + \partial _\beta \ell \partial _\beta \ell \partial _p\Gamma ). \end{aligned}$$

(8.43)

By successive substitutions, the coupling constants in (8.26), (8.26) are expressed in terms of cumulants in $y, V, V+py$ at most of order three.

1.5 (e) Even Potential, Zero Pressure

For an even potential and $p=0$ by symmetry $\langle y;V\rangle =0$, $\langle y;V;V\rangle =0$, $\langle y;y;y\rangle =0$. This simplifies the expressions for the couplings and $c^2$. It holds

$$\begin{aligned} c^2 = (\beta \langle y;y\rangle )^{-1}. \end{aligned}$$

(8.44)

The only non-zero couplings are

$$\begin{aligned}&G^\sigma _{0 \sigma '} = G^\sigma _{\sigma '0}= \tfrac{1}{2}\sigma ' c^3 \big ((2\beta ^2)^{-1}+ \langle V;V\rangle \big )^{-1/2},\quad \sigma ,\sigma '=\pm 1,\nonumber \\&G^0_{\sigma \sigma } = \tfrac{1}{2}\sigma c \beta ^{-1} \big ((2\beta ^2)^{-1}+ \langle V;V\rangle \big )^{-1/2}. \end{aligned}$$

(8.45)

1.6 (f) Second Sum Rule

As claimed in Eq. (3.9), the linearized Euler currents, $A$, and the susceptibility matrix, $C$, satisfy

$$\begin{aligned} AC = CA^\mathrm {T}. \end{aligned}$$

(8.46)

This relation is well known for classical fluids, see e.g. [70]. In fact, (8.46) is very general and relies only on space–time stationarity. Let us denote, independently of any particular model, the conserved fields by $\eta _\alpha (j,t)$, $j\in \mathbb {Z}$, $t \in \mathbb {R}$, $\alpha = 1,\ldots ,n$, which are assumed to be space–time stationary with zero mean. The corresponding currents are denoted by $\mathcal {J}_\alpha (j,t)$. By stationarity

$$\begin{aligned} S_{\alpha \beta }(j,t) = \langle \eta _\alpha (j,t) \eta _\beta (0,0)\rangle = S_{\beta \alpha }(-j,-t). \end{aligned}$$

(8.47)

Using the conservation law,

$$\begin{aligned} \begin{aligned} \frac{d}{dt}\sum _{j \in \mathbb {Z}} jS_{\alpha \beta }(j,t)&= \sum _{j \in \mathbb {Z}} j \langle (\mathcal {J}_\alpha (j-1,t) - \mathcal {J}_\alpha (j,t))\eta _\beta (0,0)\rangle \\&= \sum _{j \in \mathbb {Z}} \langle \mathcal {J}_\alpha (j,t)\eta _\beta (0,0)\rangle = \sum _{j \in \mathbb {Z}} \langle \mathcal {J}_\alpha (0,0)\eta _\beta (-j,-t)\rangle \\&= \sum _{j \in \mathbb {Z}} \langle \mathcal {J}_\alpha (0,0)\eta _\beta (j,0)\rangle . \end{aligned} \end{aligned}$$

(8.48)

As standard for mechanical systems in thermal equilibrium, but also valid for stochastic lattice gases with conservation laws [71, 72], the infinite volume average can be obtained from a system on a ring by introducing a chemical potential, $\mu _\alpha$, for the density $\rho _\alpha$. Hence

$$\begin{aligned} \sum _{j \in \mathbb {Z}} \langle \mathcal {J}_\alpha (0,0)\eta _\beta (j,0)\rangle&= \frac{\partial }{\partial \mu _{\beta }} \langle \mathcal {J}_\alpha (0,0)\rangle _{\vec {\mu }}\end{aligned}$$

(8.49)

$$\begin{aligned}&= \sum _{\gamma =1}^{n} \frac{\partial }{\partial \rho _{\gamma }} \langle \mathcal {J}_\alpha (0,0)\rangle _{\vec {\rho }} \,\frac{\partial \rho _{\gamma }}{\partial \mu _{\beta }} = (AC)_{\alpha \beta } \end{aligned}$$

(8.50)

and

$$\begin{aligned} \sum _{j \in \mathbb {Z}} jS_{\alpha \beta }(j,t) = (AC)_{\alpha \beta }t +\sum _{j \in \mathbb {Z}} jS_{\alpha \beta }(j,0). \end{aligned}$$

(8.51)

Summing in Eq. (8.47) over $j$ yields

$$\begin{aligned} (AC)_{\alpha \beta }t = (AC)_{\beta \alpha }t, \end{aligned}$$

(8.52)

which is the desired identity.

For many-component lattice gases Eq. (3.9) was noted by Tóth and Valkó [73] in a special case and proved in generality by Grisi and Schütz [72].

Appendix 2: The Hard-Point Gas with Alternating Masses

As the name suggests, the hard-point gas consists of point particles which collide elastically. This amounts to merely exchanging the labels and thus ideal gas dynamics. To introduce more chaotic elements one modifies the model to have alternating masses, say $m_0$ and $m_1$. Numerically the choice $m_1/m_0 = 3$ seems to have good time mixing. The simulation runs from collision to collision which is much faster than solving differential equations. Regarded as a chain, the hard-point gas has maximal simplicity and appears to be a favorable candidate the check our predictions. At the same time it serves as a nice illustration of the method.

As a novel feature, the unit cell consists of two particles. Let us first consider the case of hard-points with a general potential $V$ and let us reintroduce the mass $m$ of a particle. Then the hydrodynamic currents from (2.22) are modified as

$$\begin{aligned} -\tfrac{1}{m}\mathsf {u}, p, \tfrac{1}{m}\mathsf {u}p, \quad p = p(\ell ,\mathfrak {e} - \tfrac{1}{2m}\mathsf {u}^2), \end{aligned}$$

(9.1)

$\mathsf {u}$ the momentum density. Alternating masses modify the currents. We claim that in (9.1) one merely has to substitute for $m$ the average mass

$$\begin{aligned} \bar{m} = \tfrac{1}{2}(m_0 +m _1). \end{aligned}$$

(9.2)

To verify the claim we use the relation $AC = CA$, which holds in generality. The static correlator $C$ is modified to

$$\begin{aligned} C= \left( \begin{array}{l@{\quad }l@{\quad }l} \langle y;y\rangle &{} 0 &{} \langle y;V\rangle \\ 0 &{} \beta ^{-1}\bar{m} &{} 0 \\ \langle y;V\rangle &{} 0 &{} \tfrac{1}{2}\beta ^{-2}+\langle V;V\rangle \end{array}\right) , \end{aligned}$$

(9.3)

since

$$\begin{aligned} \tfrac{1}{2}\langle (p_0 + p_1)^2\rangle = \beta ^{-1}\bar{m}. \end{aligned}$$

(9.4)

Hence the linearization $A$ is modified to

$$\begin{aligned} A= \left( \begin{array}{l@{\quad }l@{\quad }l} 0 &{} -\tfrac{1}{\bar{m}} &{} 0 \\ \partial _\ell p &{} 0 &{} \partial _\mathsf {e}p \\ 0 &{} \tfrac{1}{\bar{m}}p &{} 0 \end{array}\right) \end{aligned}$$

(9.5)

as claimed.

The hard-point gas has ideal gas thermodynamics, which means

$$\begin{aligned} p = \frac{\mathsf {2e}}{\ell },\quad \beta = \frac{1}{2\mathsf {e}}. \end{aligned}$$

(9.6)

Thus $\partial _\ell p= -\beta p^2$, $\partial _\mathsf {e}p= 2\beta p$ and the sound speed reads

$$\begin{aligned} c_{\bar{m}} = (3\beta /\bar{m})^{1/2}p. \end{aligned}$$

(9.7)

The transformation matrix is obtained to

$$\begin{aligned} R = \frac{1}{\sqrt{6}} \left( \begin{array}{l@{\quad }l@{\quad }l} -\beta p&{}- \sqrt{3\beta /\bar{m}}&{}2\beta \\ 2\beta p&{}0&{} 2\beta \\ -\beta p&{} \sqrt{3\beta /\bar{m}}&{}2\beta \\ \end{array}\right) , \end{aligned}$$

(9.8)

$$\begin{aligned} R^{-1}= \frac{1}{\sqrt{6}\beta p} \left( \begin{array}{l@{\quad }l@{\quad }l} -1&{}2&{}-1\\ -\sqrt{3\beta \bar{m}} \,p&{}0&{} \sqrt{3\beta \bar{m}} \,p\\ p&{}p&{}p\\ \end{array}\right) . \end{aligned}$$

(9.9)

The correlations of the physical fields are then given through $S= R^{-1}S^\sharp R^{-1\mathrm {T}}$, where $S^\sharp$ is assumed to be approximately diagonal,

$$\begin{aligned} S^\sharp _{\alpha \beta } = \delta _{\alpha \beta }f_\alpha . \end{aligned}$$

(9.10)

Using (9.9) one obtains

$$\begin{aligned}&\ell {-}\ell \;\;\mathrm {correlations}{:}\quad \frac{1}{6\beta ^2 p^2}(f_{-1} + 4 f_0 +f_1),\nonumber \\&\mathsf {u}{-}\mathsf {u} \;\;\mathrm {correlations}{:}\quad \frac{\bar{m}}{2\beta }(f_{-1} + f_1),\nonumber \\&\mathsf {e}{-}\mathsf {e} \;\;\mathrm {correlations}{:}\quad \frac{1}{6\beta ^2}(f_{-1} + f_0 +f_1). \end{aligned}$$

(9.11)

Next we compute the $G$ matrices. Firstly, by direct differentiation of $p$,

$$\begin{aligned} H^\ell =0,\quad H^\mathsf {u}= \frac{2}{\ell ^3} \left( \begin{array}{l@{\quad }l@{\quad }l} 2 \mathsf {e} &{} 0 &{} -\ell \\ 0 &{} -\bar{m}^{-1}\ell ^2 &{} 0 \\ -\ell &{} 0 &{} 0 \end{array}\right) ,\quad H^\mathsf {e}= \frac{2}{\bar{m} \ell ^2} \left( \begin{array}{l@{\quad }l@{\quad }l} 0 &{} - \mathsf {e}&{} 0 \\ - \mathsf {e} &{} 0 &{} \ell \\ 0 &{} \ell &{} 0 \end{array}\right) \end{aligned}$$

(9.12)

and transformed as

$$\begin{aligned} (R^{-1})^{\mathrm {T}}H^\mathsf {u} R^{-1}= p D_0, \quad (R^{-1})^{\mathrm {T}}H^\mathsf {e} R^{-1} = \frac{c_{\bar{m}}}{\beta } D_1 \end{aligned}$$

(9.13)

with the matrices

$$\begin{aligned} D_0= \left( \begin{array}{l@{\quad }l@{\quad }l} 0 &{} -1 &{} 2 \\ -1 &{} 0 &{} -1 \\ 2 &{} -1 &{} 0 \end{array}\right) ,\quad D_1= \left( \begin{array}{l@{\quad }l@{\quad }l} -1 &{} 0&{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \end{array}\right) . \end{aligned}$$

(9.14)

We conclude that

$$\begin{aligned} G^{\pm 1} = \frac{c_{\bar{m}}}{2 \sqrt{6}}(\pm D_0 + 2D_1), \quad G^0 = \frac{c_{\bar{m}}}{ \sqrt{6}} D_1 \end{aligned}$$

(9.15)

and more explicitly

$$\begin{aligned} G^{\pm 1} = \pm \frac{c_{\bar{m}}}{ 2\sqrt{6}} \left( \begin{array}{lll} \mp 2 &{} -1 &{} 2 \\ -1 &{} 0 &{} -1 \\ 2 &{} -1 &{} \pm 2 \end{array}\right) , \quad G^0= \frac{c_{\bar{m}}}{ \sqrt{6}} \left( \begin{array}{lll} -1 &{} 0&{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \end{array}\right) . \end{aligned}$$

(9.16)

As a special feature of the hard-point gas, all non-universal parameters are accounted for by the sound speed $c_{\bar{m}}$. Most importantly, it should be noted that $G^1_{00} = 0$, which implies that to leading order there is no coupling of the heat mode to the sound mode. This is an indication that for the sound mode peaks the finite time corrections are less severe than in asymmetric FPU chains.

Appendix 3: Discretized Multi-Component KPZ, Mode-Coupling

1.1 (a) Discretized KPZ

The Langevin equation (3.21) is somewhat formal. To have a well-defined evolution, we lattice discretize space by a lattice of $N$ sites. The field $\phi (x,t)$ then becomes $\phi _j(t)$ with components $\phi _{j,\alpha }(t)$, $j=1,\ldots ,N$, $\alpha =1,\ldots ,n$. The spatial finite difference operator is denoted by $\tau$, $\tau f_j=f_j-f_{j-1}$, with transpose $\tau ^\mathrm {T} f_j=f_j-f_{j+1}$. Then the discretized KPZ equation reads

$$\begin{aligned} \partial _t \phi _{j,\alpha }+\tau \big (c_\alpha \phi _{j,\alpha } +\mathcal {N}_{j,\alpha } + \tau ^\mathrm {T} D\phi _{j,\alpha } + \sqrt{2D}\xi _{j,\alpha }\big )=0 \end{aligned}$$

(10.1)

with $\phi _{0}=\phi _N$, $\phi _{N+1}=\phi _1$, $\xi _0=\xi _N$, where $\xi _{j,\alpha }$ are independent Gaussian white noises with covariance

$$\begin{aligned} \langle \xi _{j,\alpha }(t) \xi _{j',\alpha '} (t')\rangle =\delta _{jj'} \delta _{\alpha \alpha '} \delta (t-t'). \end{aligned}$$

(10.2)

The diffusion matrix $D$ acts on components, while the difference $\tau$ acts on the lattice site variable $j$.

$\mathcal {N}_{j,\alpha }$ is quadratic in $\phi$. But let us first consider the case $\mathcal {N}_{j,\alpha } =0$. Then, since according to  (10.1) the drift is linear in $\phi$, $\phi _{j,\alpha }(t)$ a Gaussian process. The noise strength has been chosen such that one invariant measure is the Gaussian

$$\begin{aligned} \prod ^N_{j=1} \prod ^n_{\alpha =1} \big (\exp [-\tfrac{1}{2}\phi ^2_{j,\alpha }] (2\pi )^{-1/2} d \phi _{j,\alpha }\big )= \rho _\mathrm {G} (\phi ) \prod ^N_{j=1} \prod ^n_{\alpha =1} d \phi _{j,\alpha }. \end{aligned}$$

(10.3)

The set of all extremal invariant measures are obtained by conditioning (10.3) on the hyperplanes

$$\begin{aligned} \sum ^N_{j=1} \phi _{j,\alpha }=N\rho _\alpha , \end{aligned}$$

(10.4)

which for large $N$ would become independent Gaussians with mean $\rho _\alpha$. In the following we fix $\rho _\alpha =0$ and denote the average with respect to $\rho _\mathrm {G}$ by $\langle \cdot \rangle _{\mathrm {eq}}$.

The generator corresponding to (10.1) with $\mathcal {N}_{j,\alpha }=0$ is given by

$$\begin{aligned} L_0=\sum ^N_{j=1} \Big (-\sum ^n_{\alpha =1} \tau \big (c_\alpha \phi _{j,\alpha }+\tau ^\mathrm {T} D\phi _{j,\alpha }\big ) \partial _{\phi _{j,\alpha }}+\sum ^n_{\alpha ,\beta =1} 2D_{\alpha \beta } (\tau \partial _{\phi _{j,\alpha }} )(\tau \partial _{\phi _{j,\beta }})\Big ). \end{aligned}$$

(10.5)

The invariance of $\rho _\mathrm {G}(\phi )$ can be checked through

$$\begin{aligned} L^*_0 \rho _\mathrm {G}(\phi )=0, \end{aligned}$$

(10.6)

where $^*$ is the adjoint with respect to the flat volume measure. Furthermore linear functions evolve to linear functions according to

$$\begin{aligned} \mathrm {e}^{L_0 t}\phi _{j,\alpha }= \sum ^N_{j'=1} \sum ^n_{\alpha '=1} (\mathrm {e}^{tB})_{j\alpha ,j'\alpha '} \phi _{j',\alpha '}, \end{aligned}$$

(10.7)

where the matrix $B=-\tau \otimes \mathrm {diag} (c_1,\ldots , c_n) -\tau \tau ^\mathrm {T} \otimes D$.

We now add the nonlinearity $\mathcal {N}_{j,\alpha }$. In general, this will modify the invariant measure and we have little control how. Therefore we propose to choose $\mathcal {N}_{j,\alpha }$ such that $\rho _\mathrm {G}$ is left invariant under the deterministic flow generated the evolution equation $\tfrac{d}{dt}\phi = -\tau \mathcal {N}$, i.e. under the generator

$$\begin{aligned} L_1= - \sum ^N_{j=1} \sum ^n_{\alpha =1} \tau \mathcal {N}_{j,\alpha } \partial _{\phi _{j,\alpha }}. \end{aligned}$$

(10.8)

The invariance of $\rho _\mathrm {G}$ under $L_1$ is equivalent to the condition

$$\begin{aligned} \sum ^N_{j=1} \sum ^n_{\alpha =1} \phi _{j,\alpha } \tau \mathcal {N}_{j,\alpha }=0. \end{aligned}$$

(10.9)

If $\mathcal {N}_{j,\alpha }$ depends only on the field at sites $j$ and $j+1$, then the most general solution to (10.9) reads

$$\begin{aligned} \mathcal {N}_{j,\alpha }=\tfrac{1}{3}\sum ^n_{\beta ,\gamma =1} G^\alpha _{\beta \gamma }\big (\phi _{j,\beta }\phi _{j,\gamma }+\phi _{j,\beta }\phi _{j+1,\gamma }+\phi _{j+1,\beta }\phi _{j+1,\gamma }\big ) \end{aligned}$$

(10.10)

under the constraint that

$$\begin{aligned} G^\alpha _{\beta \gamma }=G^\alpha _{\gamma \beta }=G^\beta _{\alpha \gamma } \end{aligned}$$

(10.11)

for all $\alpha ,\beta ,\gamma =1,\ldots ,n$. Denoting the generator of the Langevin equation (10.1) by

$$\begin{aligned} L=L_0+L_1, \end{aligned}$$

(10.12)

one concludes $L^*\rho _\mathrm {G}=0$, i.e. the invariance of $\rho _\mathrm {G}$.

In the continuum limit the condition (10.9) reads

$$\begin{aligned} \sum ^n_{\alpha ,\beta ,\gamma =1} G^\alpha _{\beta \gamma } \int dx\phi _\alpha (x) \partial _x \big (\phi _\beta (x) \phi _\gamma (x)\big )=0, \end{aligned}$$

(10.13)

where $G^\alpha _{\beta \gamma }=G^\alpha _{\gamma \beta }$. By partial integration

$$\begin{aligned} 2 \sum ^n_{\alpha ,\beta ,\gamma =1} G^\alpha _{\beta \gamma } \int dx \phi _\alpha (x) \phi _\beta (x) \partial _x \phi _\gamma (x) =- \sum ^n_{\alpha ,\beta ,\gamma } G^\alpha _{\beta \gamma } \int dx \phi _\beta (x) \phi _\gamma (x) \partial _x \phi _\alpha (x)\nonumber \\ \end{aligned}$$

(10.14)

and (10.13) is satisfied only if $G^\gamma _{\beta \alpha }=G^\alpha _{\beta \gamma }$, which is the condition (10.11) obtained already in the discrete setting.

We henceforth assume (10.11) although it will not hold for anharmonic chains, in general. The leading coefficients $G^\alpha _{\alpha \alpha }$’s are not constrained and one can still freely choose the sub-leading $G^\alpha _{\beta \beta }$. The symmetry thus restricts the coefficients corresponding to sub-sub-leading terms. Appealing to universality we expect that the true invariant measure for general $G$ will have short range correlations and nonlinear fluctuating hydrodynamics remains a valid approximation to the microscopic dynamics.

1.2 (b) Mode-Coupling

We consider the stationary $\phi _{j,\alpha }(t)$ process, average denoted by $\langle \cdot \rangle$, governed by (10.1) with $\rho _\mathrm {G}$ as $t=0$ measure. For $t\ge 0$ the stationary covariance reads

$$\begin{aligned} S_{\alpha \beta }(j,t)=\langle \phi _{j,\alpha }(t)\phi _{0,\beta }(0)\rangle =\langle \phi _{0,\beta }\mathrm {e}^{Lt}\phi _{j,\alpha }\rangle _{\mathrm {eq}}, \end{aligned}$$

(10.15)

where for easier reading we leave out the superscript $^{\sharp \phi }$. By construction

$$\begin{aligned} S_{\alpha \beta }(j,0)=\delta _{\alpha \beta } \delta _{j0}. \end{aligned}$$

(10.16)

The time derivative reads

$$\begin{aligned} \frac{d}{dt} S_{\alpha \beta }(j,t)=\langle \phi _{0,\beta }(\mathrm {e}^{Lt}L_0 \phi _{j,\alpha })\rangle _{\mathrm {eq}}+ \langle \phi _{0,\beta }(\mathrm {e}^{Lt}L_1 \phi _{j,\alpha })\rangle _{\mathrm {eq}}. \end{aligned}$$

(10.17)

We insert

$$\begin{aligned} \mathrm {e}^{Lt}=\mathrm {e}^{L_0 t}+\int ^t_0 ds\, \mathrm {e}^{L_0(t-s)} L_1 \mathrm {e}^{Ls} \end{aligned}$$

(10.18)

in the second summand of (10.17). The term containing $\mathrm {e}^{L_0 t}$ does not show, since it is cubic in the time zero fields and the average $\langle \cdot \rangle _\mathrm {eq}$ vanishes. Therefore one arrives at

$$\begin{aligned} \frac{d}{dt} S_{\alpha \beta }(j,t)= \sum _{j'\in \mathbb {Z}} \sum ^n_{\alpha '=1} \big (B_{\alpha j,\alpha ' j'} S_{\alpha '\beta }(j',t) + \int ^t_0 ds \langle \phi _{0,\beta } \mathrm {e}^{L_0(t-s)}L_1(\mathrm {e}^{Ls} L_1 \phi _{j,\alpha }) \rangle _\mathrm {eq}\big ). \end{aligned}$$

(10.19)

For the adjoint of $\mathrm {e}^{L_0(t-s)}$ we use (10.7) and for the adjoint of $L_1$ we use

$$\begin{aligned} \langle \phi _{j,\alpha } L_1 F(\phi )\rangle _\mathrm {eq} = - \langle (L_1 \phi _{j,\alpha }) F(\phi )\rangle _\mathrm {eq}. \end{aligned}$$

(10.20)

Furthermore

$$\begin{aligned} L_1 \phi _{j,\alpha } = - \tau \mathcal {N}_{j,\alpha }. \end{aligned}$$

(10.21)

Inserting in (10.19) one arrives at the identity

$$\begin{aligned} \frac{d}{dt} S_{\alpha \beta }(j,t)&= \sum _{j'\in \mathbb {Z}} \sum ^n_{\alpha '=1} \big (B_{\alpha j,\alpha ' j'} S_{\alpha '\beta }(j',t)\nonumber \\&\quad - \int ^t_0 ds (\mathrm {e}^{B^\mathrm {T}(t-s)})_{0\beta ,j'\alpha '} \langle \tau \mathcal {N}_{j',\alpha '}(\mathrm {e}^{Ls} \tau \mathcal {N}_{j,\alpha }) \rangle _\mathrm {eq}\big ). \end{aligned}$$

(10.22)

To obtain a closed equation for $S$ we note that the average $\langle \tau \mathcal {N}_{j,\alpha }(s)\tau \mathcal {N}_{j',\alpha '}(0)\rangle$ appearing in (10.22) is a four-point correlation, for which we invoke a Gaussian factorization as

$$\begin{aligned} \langle \phi (s)\phi (s)\phi (0)\phi (0)\rangle \cong \langle \phi (s)\phi (s)\rangle \langle \phi (0)\phi (0)\rangle +2\langle \phi (s)\phi (0)\rangle \langle \phi (s)\phi (0)\rangle . \end{aligned}$$

(10.23)

The first summand vanishes because of the difference operator $\tau$. Secondly we replace the bare propagator $\mathrm {e}^{B(t-s)}$ by the interacting propagator $S(t-s)$. Finally we take a limit of zero lattice spacing. This step could be avoided, and is done so in numerical schemes. We could also maintain the ring geometry. As one example of interest, thereby one could investigate collisions between the moving peaks. Universality is only expected for large $j,t$, hence in the limit of zero lattice spacing. The continuum limit of $S(j,t)$ is denoted by $S(x,t)$, $x\in \mathbb {R}$. With these steps we arrive at the mode-coupling equation

$$\begin{aligned} \partial _t S_{\alpha \beta }(x,t)&= \sum ^n_{\alpha '=1} \big ({-}c_\alpha \delta _{\alpha \alpha '}\partial _x +D_{\alpha \alpha '}\partial ^2_x\big ) S_{\alpha '\beta }(x,t)\nonumber \\&+ \int ^t_0 ds \int _{\mathbb {R}} dy \partial ^2_y M_{\alpha \alpha '}(y,s) S_{\alpha '\beta }(x-y,t-s) \end{aligned}$$

(10.24)

with the memory kernel

$$\begin{aligned} M_{\alpha \alpha '}(x,t)= 2\sum ^n_{\beta ',\beta '',\gamma ',\gamma ''=1} G^\alpha _{\beta '\gamma '} G^{\alpha '}_{\beta ''\gamma ''} S_{\beta '\beta ''}(x,t) S_{\gamma '\gamma ''}(x,t). \end{aligned}$$

(10.25)

1.3 (c) Fourier Transform Conventions

Our equations of motion are written mostly in terms of the index $j$ and its continuum approximation $x$. For numerical simulations, and also for the asymptotic analysis, it is convenient to use Fourier space. We list here the conventions used throughout.

We naively discretize (10.24) in replacing $x\in \mathbb {R}$ by $j\in \mathbb {Z}$, $\partial _x f(x)$ by $\frac{1}{2}( f(j+1)-f(j-1))$, and $\partial ^2_x f(x)$ by $f(j+1)-2f(j)+f(j-1)$. Then (10.24) becomes

$$\begin{aligned} \partial _t S_{\alpha \beta } (j,t)&= -c_\alpha \tfrac{1}{2}\big (S_{\alpha \beta }(j+1,t)-S_{\alpha \beta }(j-1,t)\big )\nonumber \\&\quad +\sum ^n_{\alpha '=1}\Big (D_{\alpha \alpha '} \big (S_{\alpha '\beta }(j+1,t) -2S_{\alpha '\beta }(j,t)+S_{\alpha '\beta }(j-1,t)\big ) \nonumber \\&\quad +\int ^t_0 ds \sum _{j'\in \mathbb {Z}} \big (M_{\alpha \alpha '}(j' +1,s) {-}2 M_{\alpha \alpha '}(j',s)\nonumber \\&\quad +\, M_{\alpha \alpha '}(j'-1,s)\big ) S_{\alpha '\beta }(j-j',t-s)\Big ). \end{aligned}$$

(10.26)

For finite $N$, $j=1,\ldots ,N$, periodic boundary conditions are understood.

We adopt the standard discrete Fourier transform,

$$\begin{aligned} \hat{f}(k)= \sum _{j\in \mathbb {Z}} f(j) \mathrm {e}^{-\mathrm {i} 2\pi jk},\quad f(j)=\int ^{1/2}_{-1/2} dk \hat{f}(k) \mathrm {e}^{\mathrm {i}2\pi jk}. \end{aligned}$$

(10.27)

$\hat{f}$ is one-periodic and the standard Brillouin zone is $k\in [-\frac{1}{2},\frac{1}{2}]$. For finite $N$,

$$\begin{aligned} \hat{f}(k)= \sum ^N_{j=1} f(j) \mathrm {e}^{-\mathrm {i}2\pi jk},\quad f(j)=\frac{1}{N}\sum _{k\in [0,1]_N} \hat{f}(k) \mathrm {e}^{\mathrm {i}2\pi jk} \end{aligned}$$

(10.28)

with $[0,1]_N = \{k|k=0,1,\ldots ,N-1\}$. In Fourier space (10.26) becomes

$$\begin{aligned} \partial _t \hat{S}_{\alpha \beta }(k,t)&= -\mathrm {i} c_\alpha \sin (2\pi k) \hat{S}_{\alpha \beta }(k,t) -2(1-\cos (2\pi k)) \nonumber \\&\sum ^n_{\alpha '=1} \Big (D_{\alpha \alpha '}\hat{S}_{\alpha '\beta }(k,t)+\int ^t_0 ds \hat{M}_{\alpha \alpha '}(k,s) \hat{S}_{\alpha '\beta }(k,t-s)\Big ), \end{aligned}$$

(10.29)

where

$$\begin{aligned} \hat{M}_{\alpha \alpha '} (k,s)= 2 \sum ^n_{\beta ,\beta ',\gamma ,\gamma '=1} G^\alpha _{\beta \gamma } G^{\alpha '}_{\beta '\gamma '} \int ^{1/2}_{-1/2} dq \hat{S}_{\beta \beta '} (k-q,s) \hat{S}_{\gamma \gamma '} (q,s). \end{aligned}$$

(10.30)

Since $S$ refers here to the normal mode covariance, the initial conditions are

$$\begin{aligned} S_{\alpha \beta } (j,0)=\delta _{\alpha \beta } \delta _{j0},\quad \hat{S}_{\alpha \beta } (k,0)=\delta _{\alpha \beta }. \end{aligned}$$

(10.31)

Correspondingly we use for the continuum Fourier transform

$$\begin{aligned} \hat{f}(k)= \int dx f(x) \mathrm {e}^{-\mathrm {i}2\pi xk},\quad f(x)=\int dk \hat{f}(k) \mathrm {e}^{\mathrm {i}2\pi xk}. \end{aligned}$$

(10.32)

Convolutions come with no extra factor of $\pi$, while $\partial _x f(x)$ is transformed to $\mathrm {i}2\pi k \hat{f}(k)$. Also $\hat{f}(0)=\int dx f(x)$.

Appendix 4: Levy Asymptotics for the Heat Mode

The starting equation reads

$$\begin{aligned} \partial _t \hat{f}(k,t)=- (2\pi k\lambda _0)^2 \int ^t_0 ds \hat{f}(k,t-s) \int _\mathbb {R} dq \hat{g}(k-q,s)\hat{g}(q,s), \end{aligned}$$

(11.1)

$\hat{f}(k,0)=1$. The function $\hat{g}$ is assumed to have the scaling form

$$\begin{aligned} \hat{g}(k,t)= \mathrm {e}^{\mathrm {i}2\pi kct} \hat{g}_0\big ((\lambda _\mathrm {s} t)^\beta k\big ),\quad \hat{g}_0(0)=1, \end{aligned}$$

(11.2)

with $0<\beta <1$, $\lambda _\mathrm {s} >0$. We claim that with $\gamma =1+\beta$

$$\begin{aligned} \lim _{k\rightarrow 0} \hat{f}(k,|k|^{-\gamma }t)= \mathrm {e}^{-\lambda _{\beta } t}, \end{aligned}$$

(11.3)

where the non-universal coefficient, $\lambda _\beta$, reads

$$\begin{aligned} \lambda _\beta = (2\pi \lambda _0)^2(\lambda _\mathrm {s})^{-\beta } \int ^\infty _0 dt t^{-\beta } \mathrm {e}^{\mathrm {i}2\pi \mathrm {sgn}(k)ct} \int _\mathbb {R} dk |\hat{g}_0(k)|^2. \end{aligned}$$

(11.4)

To prove (11.3), (11.4) we look for a scaling function $\hat{h}$ such that, for $t$ of order $|k|^\gamma$,

$$\begin{aligned} \hat{f}(k,t )= \hat{h}(|k|^\gamma t). \end{aligned}$$

(11.5)

We first set $k> 0$. Inserting in (11.1) one obtains

$$\begin{aligned} \hat{h}'(|k|^\gamma t) |k|^\gamma = - (2\pi k \lambda _0)^2 \int ^t_0 ds \hat{h}(|k|^\gamma (t -s)) (\lambda _\mathrm {s}s)^{-\beta } \mathrm {e}^{\mathrm {i}2\pi kcs} \int _\mathbb {R} dq |\hat{g}_0(q)|^2. \end{aligned}$$

(11.6)

We substitute $w = |k|^\gamma t$ and $s$ by $s/|k|$. Then

$$\begin{aligned}&\hat{h} '(w) |k|^\gamma \nonumber \\&\!=\! - (2\pi \lambda _0)^2|k|^{(1+\beta )} \int ^{|k|^{(-\gamma +1)}w}_0 ds \hat{h}(w\! -\! |k|^{\gamma - 1}s) s^{-\beta } \mathrm {e}^{\mathrm {i}2\pi cs} (\lambda _\mathrm {s})^{-\beta }\int _\mathbb {R} dq |\hat{g}_0(q)|^2.\qquad \quad \end{aligned}$$

(11.7)

Taking the limit $k \rightarrow 0$, $\hat{h}$ can be taken out of the $s$-integral with the result $\hat{h}' = -\lambda \hat{h}$, which yields (11.3). In repeating the same computation for $k< 0$, only the sign in the oscillating factor changes.

Working out the integration in (11.4), the scaling function at $t=1$ becomes

$$\begin{aligned} \hat{h}(|k|^\gamma )= \exp \big [-\lambda _\beta |k|^\gamma \big (1 - \mathrm {i}\,\mathrm {sgn}(k)\tan (\tfrac{1}{2}\pi \gamma ) \big )\big ], \end{aligned}$$

(11.8)

where

$$\begin{aligned} \lambda _\beta = (2 \pi \lambda _0)^2(\lambda _\mathrm {s})^{-\beta } (2\pi c)^{-1+\beta } \tfrac{1}{2}\pi \frac{1}{\Gamma (\beta )} \frac{1}{\cos (\tfrac{1}{2}\pi \beta )} \int _\mathbb {R} dk |\hat{g}_0(k)|^2. \end{aligned}$$

(11.9)

One recognizes (11.9) as the Fourier transform of asymmetric $\alpha$-stable law with $\alpha = \gamma$. The asymmetry parameter is at its limiting allowed value. This implies a decay in position space as $|x|^{-\gamma -1}$ to the left and as $\exp [-|x|^{\gamma /(\gamma -1)}]$ to the right, see [74] for the asymptotics. In the context of fluctuating hydrodynamics this feature implies that outside the sound cone the correlations are suppressed faster than exponential.

Appendix 5: Anharmonic Chains and One-Dimensional Fluids

One-dimensional fluids are governed by the hamiltonian

$$\begin{aligned} H_\mathrm {f}=\sum ^N_{j=1} \tfrac{1}{2}p^2_j +\tfrac{1}{2}\sum ^N_{i\ne j=1} V(q_i-q_j) \end{aligned}$$

(12.1)

with an even interaction potential, $V(x)=V(-x)$. $V$ must be chosen such that the system is thermodynamically stable, to say $V$ may have a hard core and $V$ decays sufficiently fast at infinity, minimal requirements being $V(x)\rightarrow 0$ as $|x|\rightarrow \infty$, faster than $- |x|^{-2}$ to avoid phase transitions, $V$ is bounded from below, and not “too negative”. Particles move on the interval $[0,L]$ with periodic boundary conditions. Particle number, momentum, and energy are locally conserved. Their corresponding microscopic fields are written as

$$\begin{aligned} \sum ^N_{j=1}\delta (q_j-x),\quad \sum ^N_{j=1}\delta (q_j-x)p_j,\quad \sum ^N_{j=1}\delta (q_j-x) \Big (\tfrac{1}{2}p^2_j +\tfrac{1}{2}\sum ^N_{i=1,i\ne j} V(q_i-q_j)\Big ). \end{aligned}$$

(12.2)

From the evolution equations one finds the corresponding microscopic currents as

$$\begin{aligned} J_1(x)&= \sum ^N_{j=1}\delta (q_j-x)p_j,\nonumber \\ J_2(x)&= \sum ^N_{j=1}\delta (q_j-x)p^2_j -\tfrac{1}{2}\sum ^N_{i=1,i\ne j} V'(q_i-q_j)(q_i-q_j) \int ^1_0 dx \delta (\lambda q_i +(1-\lambda ) q_j-x),\nonumber \\ J_3(x)&= \sum ^N_{j=1}\delta (q_j-x)p_j\Big (\tfrac{1}{2}p^2_j +\tfrac{1}{2}\sum ^N_{i=1,i\ne j} V(q_i-q_j)\Big ),\nonumber \\&+\,\tfrac{1}{2}\sum ^N_{i,j=1}\tfrac{1}{2}(p_i+p_j) V'(q_i-q_j)(q_i-q_j) \int ^1_0 d\lambda \delta (\lambda q_i +(1-\lambda ) q_j-x). \end{aligned}$$

(12.3)

Even without averaging, the densities from (12.2) and the currents from (12.3) satisfy a system of conservation laws.

If the fluid starts and approximately remains in local equilibrium, one can equilibrium average the microscopic conservation laws and arrives at the Euler equations of a one-dimensional fluid. The hydrodynamic fields are $\rho$, $\rho v$, $\rho \mathfrak {e}_\mathrm {f}$, which depend on $x,t$. $\rho$ is the local particle density, $v$ the local velocity per particle and $\mathfrak {e}_\mathrm {f}$ the local energy per particle. Then the Euler equations read

$$\begin{aligned}&\partial _t \rho +\partial _x (\rho v) =0,\nonumber \\&\partial _t (\rho v)+\partial _x (\rho v^2 +p_\mathrm {f}) =0,\nonumber \\&\partial _t (\rho \mathfrak {e}_\mathrm {f})+ \partial _x (\rho \mathfrak {e}_\mathrm {f} v +p_\mathrm {f}v)=0. \end{aligned}$$

(12.4)

Here $p_\mathrm {f}$ is the local thermodynamic pressure, which depends on $\rho$ and the internal energy. We use the subscript “f” to distinguish from the corresponding quantity for the anharmonic chain. To construct nonlinear fluctuating hydrodynamics we may proceed as in the main text. In particular, $G^0_{00}=0$ always and generically $G^1_{11}= -G^{-1}_{-1-1}\ne 0$. Thus up to model-dependent coefficients the overall structure remains unaltered. There is one technical difference. To determine the pressure $p_\mathrm {f}$ requires a full many-body computation, in general. One has then to resort to series expansions and Monte Carlo techniques. In particular, to obtain with sufficient accuracy the coupling constants $G$ will be more costly than for the chain.

If one introduces a hard core, the ordering of particles is preserved and by a suitable choice of parameters one can achieve that only the nearest neighbor contribution to the potential term remains. An explicit example is

$$\begin{aligned} V_{\mathrm {hc}}(x)= \left\{ {\begin{array}{ll} \infty , &{} |x|\le a/2, \\ \text {``arbitrary''}, &{} a/2\le |x|\le a, \\ 0, &{} a\le |x|. \\ \end{array} } \right. \end{aligned}$$

(12.5)

Then, on the restricted configuration space,

$$\begin{aligned} \tfrac{1}{2}\sum ^N_{i=1,i\ne j} V_{\mathrm {hc}}(q_i-q_j) = \sum ^N_{j=1} V_{\mathrm {hc}}(q_{j+1}-q_j) \end{aligned}$$

(12.6)

Thus for $V_{\mathrm {hc}}$ one may adopt the field theory point of view with coupling only between nearest neighbors.

At second thoughts one is puzzled, since the hydrodynamic equations for a fluid look different from the ones of the field theory. One reason is that $\rho =\ell ^{-1}$. More importantly $\partial _x$ above refers to per unit length while the $\partial _x$ in (2.21), (2.22) refers to the particle label. Still, because it is the same physical system, there must be a transformation which converts (12.4) to (2.21), (2.22). I did not find a discussion in the literature and hence explain how the transformation is done.

First one has to properly distinguish between “$x$” in (2.21) and in (12.5). To do so, only in this section, we replace $x$ in (2.21) by $y$. Let us define

$$\begin{aligned} \int ^x_{-\infty } dx' \rho (x',t)= y_t(x),\quad \int ^y_{-\infty } dy' \ell (y',t)= x_t(y). \end{aligned}$$

(12.7)

$y_t$ is the inverse function of $x_t$. It follows that

$$\begin{aligned} \rho (x,t) \ell (y,t)=1, \end{aligned}$$

(12.8)

where we used

$$\begin{aligned} \partial _x y_t(x)=\rho (x,t),\quad \partial _y x_t(y)=\ell (y,t). \end{aligned}$$

(12.9)

If in a single equation, as (12.8), both $x$ and $y$ appear, they are understood to be related as in (12.9). Using the conservation of mass one obtains

$$\begin{aligned} \partial _t y_t(x)=-\rho (x,t) v(x,t). \end{aligned}$$

(12.10)

Other quantities flow along, i.e.

$$\begin{aligned} v(x,t)= u(y,t),\quad \mathfrak {e}_\mathrm {f}(x,t)=\mathfrak {e}(y,t),\quad p_\mathrm {f}(x,t)=p(y,t). \end{aligned}$$

(12.11)

We first discuss mass conservation and write

$$\begin{aligned} 0&= \partial _t \rho + \partial _x(\rho v)=\partial _t\big (\ell (y_t(x),t)^{-1}\big )+\rho \partial _x u(y_t(x),t)+v \partial _x\big (\ell (y_t(x),t)^{-1}\big )\nonumber \\&= -\ell ^{-2} (\partial _y\ell )(\partial _t y_t(x))-\ell ^{-2} \partial _t \ell +\rho \big (\partial _y u(y,t)\big )\big (\partial _x y_t(x)\big ) -v\ell ^{-2}\big (\partial _y \ell (y,t)\big )\big (\partial _x y_t(x)\big )\nonumber \\&= \ell ^{-3} v \partial _y\ell -\ell ^{-2} \partial _t\ell +\rho ^2 \partial _y u -v\ell ^{-3}\partial _y \ell \nonumber \\&= -\ell ^{-2} (\partial _t \ell - \partial _y u), \end{aligned}$$

(12.12)

as to be shown. We turn to momentum conservation

$$\begin{aligned} 0&= \partial _t (\rho v) + \partial _x(\rho v^2) + \partial _x p_\mathrm {f}= \rho \partial _t v + \rho v \partial _x v +\partial _x p_\mathrm {f} \nonumber \\&= \rho \partial _t u (y_t(x),t) + \rho v\partial _x u(y_t(x),t) +\partial _x p_\mathrm {f}\nonumber \\&= -\rho (\partial _y u)\rho v +\rho \partial _t u+\rho v (\partial _y u) \rho +\partial _x p_\mathrm {f}\nonumber \\&= \rho \partial _t u + \partial _x p_\mathrm {f}. \end{aligned}$$

(12.13)

Now for any pair of functions $g$, $g_\mathrm {f}$ such that $g_\mathrm {f}(x,t)=g(y,t)$, one has

$$\begin{aligned} \partial _x g_\mathrm {f}(x,t)= \partial _x g\big (y_t(x),t\big )=\big (\partial _y g(y,t)\big )\big (\partial _x y_t(x)\big )=\big (\partial _y g(y,t)\big ) \rho (x,t). \end{aligned}$$

(12.14)

From (12.13) one concludes that

$$\begin{aligned} \partial _t u +\partial _y p=0. \end{aligned}$$

(12.15)

Finally we turn to energy conservation, with similar steps as in (12.15)

$$\begin{aligned} 0&= \partial _t (\rho \mathfrak {e}_\mathrm {f}) + \partial _x(\rho \mathfrak {e}_\mathrm {f} v) + \partial _x (vp_\mathrm {f}) \nonumber \\&= \rho \partial _t \mathfrak {e}_\mathrm {f} + \rho v\partial _x \mathfrak {e}_\mathrm {f} +\partial _x (v p_\mathrm {f})\nonumber \\&= \rho \partial _t \mathfrak {e} +\partial _x (v p_\mathrm {f})= \rho \big (\partial _t \mathfrak {e}+\partial _y (p u)\big ), \end{aligned}$$

(12.16)

where we used (12.14) with $g_\mathrm {f}=v p_\mathrm {f}$, $g= up$. The pressure $p_\mathrm {f}$ depends on $x,t$ through $\rho$ and $\mathsf {e}_\mathrm {f}$ and the pressure $p$ depends on $y,t$ through $\ell$ and $\mathsf {e}$.