Shocks, Rarefaction Waves, and Current Fluctuations for Anharmonic Chains

Abstract

The nonequilibrium dynamics of anharmonic chains is studied by imposing an initial domain-wall state, in which the two half lattices are prepared in equilibrium with distinct parameters. We analyse the Riemann problem for the corresponding Euler equations and, in specific cases, compare with molecular dynamics. Additionally, the fluctuations of time-integrated currents are investigated. In analogy with the KPZ equation, their typical fluctuations should be of size $t^{1/3}$ and have a Tracy–Widom GUE distributed amplitude. The proper extension to anharmonic chains is explained and tested through molecular dynamics. Our results are calibrated against the stochastic LeRoux lattice gas.

Appendix: Coupling Matrices

We compute the G coupling matrices for the LeRoux model and anharmonic chains, following the derivation in [9].

1.1 LeRoux Model

The linearization matrix A and its left and right eigenvectors are stated in Eqs. (2.7) and (2.9). The transformation to normal modes is accomplished through the matrix R defined by

$$\begin{aligned} R = \begin{pmatrix} \langle \tilde{\psi }_{-1} |\\ \langle \tilde{\psi }_{1} |\end{pmatrix}, \quad R^{-1} = \Big ( |\psi _{-1} \rangle \ |\psi _{1} \rangle \Big ). \end{aligned}$$

(8.1)

By construction one has

$$\begin{aligned} R A R^{-1} = \mathrm {diag}(c_{-1}, c_1). \end{aligned}$$

(8.2)

As usual, the static susceptibility matrix, C, is given by

$$\begin{aligned} C = \begin{pmatrix} \langle 1 - |\eta _j|; 1 - |\eta _j |\rangle &{} \langle 1 - |\eta _j |; \eta _j\rangle \\ \langle 1 - |\eta _j |; \eta _j \rangle &{} \langle \eta _j ; \eta _j\rangle \end{pmatrix} = \begin{pmatrix} \rho (1 - \rho ) &{} -\rho v \\ -\rho v &{} 1 - \rho - v^2 \\ \end{pmatrix}. \end{aligned}$$

(8.3)

In addition we then require

$$\begin{aligned} R C R^{\mathrm {T}} = \mathbbm {1}, \end{aligned}$$

(8.4)

thereby fixing the normalizations $\tilde{Z}_{\sigma }$, $Z_{\sigma }$ in (2.9) to

$$\begin{aligned} \tilde{Z}_{\sigma } = \sqrt{2} \left( 4 \rho (1 - \rho ) + v^2 \big (1 - 5 \rho -v^2\big ) + \sigma \,v \big (3 \rho + v^2 - 1\big )\sqrt{4 \rho + v^2}\right) ^{1/2}, \end{aligned}$$

(8.5)

$$\begin{aligned} Z_{\sigma } = 2\,\tilde{Z}_{\sigma }^{-1} \left( 4 \rho - v \big (\sigma \sqrt{4 \rho + v^2} - v\big )\right) . \end{aligned}$$

(8.6)

To obtain the nonlinear couplings G, in particular $G^1_{11}$, we first compute the Hessians of the current as second derivatives of $\vec {\mathsf {j}}(\vec {u})$,

$$\begin{aligned} H^{\rho } = - \begin{pmatrix} 0 &{}\quad 1 \\ 1 &{}\quad 0 \end{pmatrix}, \qquad H^{v} = - \begin{pmatrix} 0 &{}\quad 0 \\ 0 &{}\quad 2 \end{pmatrix}. \end{aligned}$$

(8.7)

In normal coordinates

$$\begin{aligned}&\langle \psi _{\sigma } \vert H^{\rho } \vert \psi _{\tau } \rangle = \rho \,v \begin{pmatrix} 1 &{}\quad 0 \\ 0 &{}\quad 1 \end{pmatrix} + \frac{2 \rho }{\sqrt{4 \rho + v^2}} \big (1 - \rho - \tfrac{1}{2} v^2\big ) \begin{pmatrix} -1 &{}\quad 0 \\ 0 &{}\quad 1 \end{pmatrix} \nonumber \\&\qquad \qquad + \frac{v}{\sqrt{4 \rho + v^2}} \sqrt{\rho \big ((1 - \rho )^2 - v^2\big )} \begin{pmatrix} 0 &{}\quad 1 \\ 1 &{}\quad 0 \end{pmatrix} \end{aligned}$$

(8.8)

and

$$\begin{aligned}&\langle \psi _{\sigma } \vert H^{v} \vert \psi _{\tau } \rangle = -\big (1 - \rho - v^2\big ) \begin{pmatrix} 1 &{}\quad 0 \\ 0 &{}\quad 1 \end{pmatrix} + \frac{v}{\sqrt{4 \rho + v^2}} \big (1 - 3 \rho - v^2\big ) \begin{pmatrix} -1 &{}\quad 0 \\ 0 &{}\quad 1 \end{pmatrix} \nonumber \\&\qquad \qquad - \frac{2}{\sqrt{4 \rho + v^2}} \sqrt{\rho \big ((1 - \rho )^2 - v^2\big )} \begin{pmatrix} 0 &{}\quad 1 \\ 1 &{}\quad 0 \end{pmatrix}. \end{aligned}$$

(8.9)

The coupling matrices are thus obtained as

$$\begin{aligned} \begin{array}{ll} G^{\sigma } &{}= \tfrac{1}{2} \sum \limits _{i=\{\rho ,v\}} R_{\sigma i} \, R^{-\mathrm {T}} H^{i} R^{-1} \\ &{}= Z_{\sigma }^{-1} \big (\sqrt{4 \rho +v^2} - \sigma v\big ) \begin{pmatrix} \tfrac{1}{2}(1 - \sigma ) &{}\quad 0 \\ 0 &{}\quad \tfrac{1}{2}(1 + \sigma ) \end{pmatrix}\\ &{}\quad + \tilde{Z}_{\sigma }^{-1} \sqrt{\rho \big ((1 - \rho )^2 - v^2\big )} \begin{pmatrix} 0 &{}\quad 1 \\ 1 &{}\quad 0\end{pmatrix}. \end{array} \end{aligned}$$

(8.10)

In particular, comparison with (2.10) shows that

$$\begin{aligned} G^{\sigma }_{\sigma \sigma } = \tfrac{1}{2} \psi _\sigma \cdot D c_{\sigma }. \end{aligned}$$

(8.11)

1.2 General Anharmonic Chain

To set the scale for the Tracy–Widom distribution, one has to compute $G_{11}^1$. For a general anharmonic chain, in the special case $v = 0$, the coupling matrices are derived in [9], a result which should be extended to $v \ne 0$. In fact, it turns out that the coupling matrices do not depend on v.

Following the notation of [9], the static susceptibility matrix is given by

$$\begin{aligned} C = \begin{pmatrix} \langle r_j; r_j \rangle &{} \langle r_j; v_j \rangle &{} \langle r_j; e_j \rangle \\ \langle r_j; v_j \rangle &{} \langle v_j; v_j \rangle &{} \langle v_j; e_j \rangle \\ \langle r_j; e_j \rangle &{} \langle v_j; e_j \rangle &{} \langle e_j; e_j \rangle \end{pmatrix} = \begin{pmatrix} \langle y; y \rangle &{} 0 &{} \langle y; V \rangle \\ 0 &{} 1/(m \beta ) &{} v/\beta \\ \langle y; V \rangle &{} v/\beta &{} \frac{1}{2}\beta ^{-2} + m v^2 \beta ^{-1} + \langle V; V\rangle \end{pmatrix}.\nonumber \\ \end{aligned}$$

(8.12)

The linearization matrix A in (3.14) and its right and left eigenvectors in (3.16) and (3.17), respectively, define the transformation to normal modes via

$$\begin{aligned} R = \begin{pmatrix} \langle \tilde{\psi }_{-1} |\\ \langle \tilde{\psi }_{0} |\\ \langle \tilde{\psi }_{1} |\end{pmatrix}, \quad R^{-1} = \Big ( |\psi _{-1} \rangle \ |\psi _{0} \rangle \ |\psi _{1} \rangle \Big ) \end{aligned}$$

(8.13)

such that

$$\begin{aligned} R A R^{-1} = \mathrm {diag}(-c, 0, c), \quad R C R^{\mathrm {T}} = \mathbbm {1}. \end{aligned}$$

(8.14)

To have $R R^{-1} = \mathbbm {1}$, the normalization constants of the eigenvectors must satisfy

$$\begin{aligned} Z_{0} \tilde{Z}_{0} = m c^2, \qquad Z_{\sigma } \tilde{Z}_{\sigma } = 2 m c^2 \ \ \text {for}\ \ \sigma = \pm 1. \end{aligned}$$

(8.15)

An explicit computation of the diagonal entries of $R C R^{\mathrm {T}}$ shows that the velocity terms cancel. Hence the relations

$$\begin{aligned} \tilde{Z}_0 = \sqrt{m \Upsilon }\,c, \qquad \tilde{Z}_{\sigma } = \sqrt{2 m/\beta }\,c \ \ \text {for}\ \ \sigma = \pm 1 \end{aligned}$$

(8.16)

from [9] remain valid in general, where $\Upsilon = \beta \left( \langle y; y \rangle \langle V; V \rangle - \langle y; V \rangle ^2\right) + \frac{1}{2} \beta ^{-1} \langle y; y \rangle$.

As in [9], we denote the Hessian matrices of the average current by

$$\begin{aligned} H^{i}_{\alpha \beta } = \partial _{u_\alpha } \partial _{u_\beta }\,\mathsf {j}_{i} \end{aligned}$$

(8.17)

with the conserved fields $\vec {u} = (r, v, \mathfrak {e})$ and the current vector defined in (3.10). The coupling matrices are then given by

$$\begin{aligned} G^{\sigma } = \tfrac{1}{2} \sum _{i=1}^3 R_{\sigma i} \, R^{-\mathrm {T}} H^{i} R^{-1} \end{aligned}$$

(8.18)

for $\sigma = -1, 0, 1$. While the Hessian matrices $H^{i}$ depend on v, the coupling matrices are actually independent of v. Thus using the formulas in [9] one arrives at

$$\begin{aligned} G^0 = \frac{1}{2 \beta \sqrt{m \Upsilon }} \begin{pmatrix} -1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 \end{pmatrix} \end{aligned}$$

(8.19)

and for $\sigma = \pm 1$

$$\begin{aligned} \begin{array}{lll} G^{\sigma } &{}= \frac{P\,\partial _e c - \partial _r c }{2 \sqrt{2 m \beta }\,c} \begin{pmatrix} 1 &{}\quad 0 &{}\quad -1 \\ 0 &{}\quad 0 &{}\quad 0 \\ -1 &{}\quad 0 &{}\quad 1 \end{pmatrix} - \frac{\partial _e P}{\sqrt{2 m \beta }} \begin{pmatrix} \frac{1}{2}(1+\sigma ) &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \frac{1}{2}(1-\sigma ) \end{pmatrix} \\ &{}\quad + \frac{\Upsilon }{2 \sqrt{2m/\beta }\,m c^2} \left[ (\partial _r P)^2 (\partial _e^2 P) - 2 (\partial _r P) (\partial _r \partial _e P) (\partial _e P) + (\partial _r^2 P) (\partial _e P)^2 \right] \begin{pmatrix} 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \end{pmatrix} \\ &{}\quad + \frac{\sqrt{\Upsilon }}{2 \sqrt{m}\,c} \left[ (\partial _r P) (\partial _e c) - (\partial _e P) (\partial _r c) \right] \begin{pmatrix} 0 &{}\quad 1 &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad -1 \\ 0 &{}\quad -1 &{}\quad 0 \end{pmatrix}. \end{array}\nonumber \\ \end{aligned}$$

(8.20)

The relation (6.12) follows by using on both sides the expressions provided above.

Note that the signs of some entries in $G^{\sigma }$ are flipped compared to [9], which is due to different sign conventions for the eigenvectors of A.

1.3 Hard-Point and Square-Well Potential

The coupling constants for these models have been discussed already in Appendix A of [28]. For completeness, here we adapt to the current sign convention for the eigenvectors and using the velocity (instead of momentum) as field variable. The linearization matrix A and its right eigenvectors are stated in Eqs. (5.7) and (5.9). The corresponding left eigenvectors of A are

$$\begin{aligned} \tilde{\psi }_{0,h} = \tilde{Z}_{0,h}^{-1} \begin{pmatrix} 2 e h \\ -m v \\ 1 \end{pmatrix}, \qquad \tilde{\psi }_{\sigma ,h} = \tilde{Z}_{\sigma ,h}^{-1} \begin{pmatrix} 2 e \sigma h' \\ m(c_h - 2 \sigma v h) \\ 2 \sigma h \end{pmatrix}. \end{aligned}$$

(8.21)

Since the interaction potential is either zero or infinite, $\Upsilon _h = \frac{1}{2} \beta ^{-1} \langle y; y \rangle = -e/h'$, and the normalization constants in (8.16) become

$$\begin{aligned} \tilde{Z}_{0,h} = \sqrt{m e}\,c_h/\sqrt{-h'}, \qquad \tilde{Z}_{\sigma ,h} = 2 \sqrt{m e}\,c_h \,. \end{aligned}$$

(8.22)

Specializing (8.19) and (8.20) to the square-well interaction potential leads to the coupling matrices

$$\begin{aligned} G_h^0 = \sqrt{- h'\,e/m} \begin{pmatrix} -1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 \end{pmatrix} \end{aligned}$$

(8.23)

and for $\sigma = \pm 1$

$$\begin{aligned} G_h^{\sigma } = \frac{1}{2} \sqrt{e/m} \left[ \frac{1}{2(2 h^2 - h')} \begin{pmatrix} a_3 &{}\quad a_1 &{}\quad -a_3 \\ a_1 &{}\quad a_2 &{}\quad -a_1 \\ -a_3 &{}\quad -a_1 &{}\quad a_3 \end{pmatrix} - 4 h \begin{pmatrix} \frac{1}{2}(1+\sigma ) &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \frac{1}{2}(1-\sigma ) \end{pmatrix} \right] \nonumber \\ \end{aligned}$$

(8.24)

with

$$\begin{aligned} \begin{array}{ll} a_1 &{}= 2 (-h')^{-1/2} \big ( h h'' - h'^2 - 2 h^2 h' \big ), \\ a_2 &{}= 4 h (-h')^{-1}\big ( h h'' - 2 h'^2 \big ),\\ a_3 &{}= 4 h^3 - 6 h h' + h'' . \end{array} \end{aligned}$$

(8.25)

As above, the signs of some entries in $G_h^{\sigma }$ are flipped compared to [28], due to different sign conventions for the eigenvectors of A.