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.
Similar content being viewed by others
Notes
Anharmonic chains evolving according to Hamiltonian dynamics have three conservation laws and correspondingly one heat mode and two reflection symmetric sound modes. The heat mode is the symmetric \(\tfrac{5}{3}\)-Lévy function and the prefactor is \(2 a_\mathrm{h}\) (compare for instance with [32, Eq. (50)]).
References
Ernst, M.H., Hauge, E.H., van Leeuwen, J.M.J.: Asymptotic time behavior of correlation functions. II. Kinetic and potential terms. J. Stat. Phys. 15, 7–22 (1976)
Forster, D., Nelson, D.R., Stephen, M.J.: Large-distance and long-time properties of a randomly stirred fluid. Phys. Rev. A 16, 732–749 (1977)
Lepri, S., Livi, R., Politi, A.: Heat conduction in chains of nonlinear oscillators. Phys. Rev. Lett. 78, 1896–1899 (1997)
Lepri, S., Livi, R., Politi, A.: Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377, 1–80 (2003)
Dhar, A.: Heat transport in low-dimensional systems. Adv. Phys. 57, 457–537 (2008)
van Beijeren, H.: Exact results for anomalous transport in one-dimensional Hamiltonian systems. Phys. Rev. Lett. 108, 180601 (2012)
Spohn, H.: Nonlinear fluctuating hydrodynamics for anharmonic chains. J. Stat. Phys. 154, 1191–1227 (2014)
Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986)
Corwin, I.: The Kardar-Parisi-Zhang equation and universality class. Random Matrices Theory Appl. 1, 113001 (2012)
Borodin, A., Gorin, V.: Lectures on integrable probability. arXiv:1212.3351 (2012)
Borodin, A., Petrov, L.: Integrable probability: from representation theory to Macdonald processes. arXiv:1310.8007 (2013)
Quastel, J., Remenik, D.: Airy processes and variational problems. arXiv:1301.0750 (2013)
Bernardin, C., Stoltz, G.: Anomalous diffusion for a class of systems with two conserved quantities. Nonlinearity 25, 1099–1133 (2012)
Hairer, M.: Solving the KPZ equation. Ann. Math. 178, 559–664 (2013)
Funaki, T., Quastel, J.: KPZ equation, its renormalization and invariant measures. arXiv:1407.7310 (2014)
Borodin, A., Corwin, I., Ferrari, P., Vetö, B.: Height fluctuations for the stationary KPZ equation. arXiv:1407.6977 (2014)
Imamura, T., Sasamoto, T.: Stationary correlations for the 1D KPZ equation. J. Stat. Phys. 150, 908–939 (2013)
Prähofer, M.: Exact scaling functions for one-dimensional stationary KPZ growth. http://www-m5.ma.tum.de/KPZ
Prähofer, M., Spohn, H.: Exact scaling functions for one-dimensional stationary KPZ growth. J. Stat. Phys. 115, 255–279 (2004)
Ferrari, P., Spohn, H.: Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. Commun. Math. Phys. 265, 1–44 (2006)
Ertaş, D., Kardar, M.: Dynamic relaxation of drifting polymers: a phenomenological approach. Phys. Rev. E 48, 1228–1245 (1993)
Mendl, Ch.B., Spohn, H.: Dynamic correlators of Fermi-Pasta-Ulam chains and nonlinear fluctuating hydrodynamics. Phys. Rev. Lett. 111, 230601 (2013)
Bernardin, C., Gonçalves, P., Jara, M.: \(3/4\)-superdiffusion in a system of harmonic oscillators perturbed by a conservative noise. arXiv:1402.1562 (2014)
Ferrari, P., Sasamoto, T., Spohn, H.: Coupled Kardar-Parisi-Zhang equations in one dimension. J. Stat. Phys. 153, 377–399 (2013)
Popkov, V., Schmidt, J., Schütz, G.M.: Superdiffusive modes in two-species driven diffusive systems. Phys. Rev. Lett. 112, 200602 (2014)
Popkov, V., Schmidt, J., Schütz, G.M.: Universality classes in two-component driven diffusive systems. arXiv:1410.8026 (2014)
Kulkarni, M., Lamacraft, A.: Finite-temperature dynamical structure factor of the one-dimensional Bose gas: from the Gross-Pitaevskii equation to the Kardar-Parisi-Zhang universality class of dynamical critical phenomena. Phys. Rev. A 88, 021603(R) (2013)
Kulkarni, M., Spohn H., Huse, D.: Nonlinear fluctuating hydrodynamics for the 1D Bose gas, draft
Mendl, Ch.B., Spohn, H.: Nonlinear lattice Schrödinger equation at low temperatures (in press)
Kac, M., van Moerbeke, P.: On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices. Adv. Math. 16, 160–169 (1975)
Toda, M.: Theory of Nonlinear Lattices (second enlarged edition). Solid-State Sciences, vol. 20. Springer, Berlin (1988)
Mendl, Ch.B., Spohn, H.: Equilibrium time-correlation functions for one-dimensional hard-point systems. Phys. Rev. E 90, 012147 (2014)
Das, S.G., Dhar, A., Saito, K., Mendl, Ch.B., Spohn, H.: Numerical test of hydrodynamic fluctuation theory in the Fermi-Pasta-Ulam chain. Phys. Rev. E 90, 012124 (2014)
Straka, M.: KPZ scaling in the one-dimensional FPU-model. Master Thesis, University of Florence, Italy (2013)
Zwillinger, D.: CRC Standard Mathematical Tables and Formulae, vol. 31. CRC Press, Boca Raton (2003)
Jara, M., Komorowski, T., Olla, S.: Superdiffusion of energy in a chain of harmonic oscillators with noise. arXiv:1402.2988 (2014)
Uchaikin, V., Zolotarev, V.: Chance and Stability. Stable Distributions and Applications, Modern Probability and Statistics Series. De Gruyter, Utrecht (1999)
Acknowledgments
We thank Christian Mendl for numerous instructive discussions, as well as Günther Schütz for stimulating comments on a preliminary version of this manuscript. H.S. is grateful for the support through the Institute for Advanced Study, Princeton, where the first steps in this project were accomplished and thanks David Huse for insisting on a complete classification.
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix 1: Scaling Functions for Two Cross-Coupled Modes
We study the asymptotic behavior of two cross-coupled Burgers equations of the form
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
Initially \(f_\sigma (x,0) = \delta (x)\) and the normalization is preserved,
Furthermore, by symmetry of the equations,
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,
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],
Taking the spatial Fourier transform of (6.2) leads to
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,
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
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
so that, introducing the new variables \(w = k^\gamma t\) and \(u = q s^\beta \),
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
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
one arrives at
In the limit \(k\rightarrow 0\),
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
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])
with
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
which forces \(h(w) = \overline{g(w^\beta )}\) and
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 \(\tan (\pi \beta /2) = -1/\tan (\pi \gamma /2)\),
with
Inserting (6.12) in the latter expression, it follows that
Thereby we finally obtain the scaling function
with
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
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
\(c > 0\), while \(f_1\) evolves according to
compare with (2.5)–(2.6). Through Fourier transform in space one obtains
As in Appendix 1.1, it suffices to consider \(k>0\). Following the scheme in [7, Sect. 4], we make the ansatz
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
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
Working out the integrals yields
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
Combining (7.2) and (7.3) one arrives at the fixed point equation for the scaling function \(F\),
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
1.1 Appendix 3.1: Matrix \(R\) and Sound Velocity
The right eigenvectors of the matrix \(A\) are proportional to
with, respectively, associated eigenvalues 0 and
The corresponding left eigenvectors are proportional to
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
as well as
Moreover,
with
Finally,
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
and
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,
and using the expressions (8.3) for the partial derivatives \(\partial _h \tau ,\partial _e \tau \), as well as the rules
The elements of the \(G\) matrices are then computed as
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
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
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
so that \(c = -2(1+\kappa \tau )\), \(H_h = 0\),
In addition,
so that \(H_e \psi _2 = 0\). The only non-zero coefficient of \(G^1\) therefore is \(G_{11}^1\), which reads
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 \(c = -5.28\),
and
For the KvM potential (4.2) with \(\kappa =1\), we obtain \(c=-4\),
and
Rights and permissions
About this article
Cite this article
Spohn, H., Stoltz, G. Nonlinear Fluctuating Hydrodynamics in One Dimension: The Case of Two Conserved Fields. J Stat Phys 160, 861–884 (2015). https://doi.org/10.1007/s10955-015-1214-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-015-1214-0