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arXiv:2511.12574 · Physical Review Letters

Symmetry-Based Nonlinear Fluctuating Hydrodynamics in One Dimension

Yuki Minami, Hiroyoshi Nakano, Keiji Saito

Published Mar 31, 2026• 0 citations
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Abstract

We present a symmetry-based formulation of nonlinear fluctuating hydrodynamics (NFH) for one-dimensional many-particle systems with generic homogeneous nearest-neighbor interactions. We derive the hydrodynamic equations solely from symmetry and conservation principles, ensuring full consistency with thermalization. Using the dynamic renormalization group, we identify a Kardar-Parisi-Zhang (KPZ)-type fixed point, characterized by the dynamical exponent z=3/2 for both the sound and heat modes. Extensive numerical simulations of the derived NFH equations confirm this exponent and further reveal that both modes are close to the universal KPZ scaling function-the Prähofer-Spohn function. These findings establish a unified, symmetry-based framework for understanding universal transport and fluctuation phenomena in one-dimensional nonequilibrium systems, independent of microscopic details.

DOI
10.1103/2z9s-5d7w

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We present a symmetry-based formulation of nonlinear fluctuating hydrodynamics (NFH) for one-dimensional many-particle systems with generic homogeneous nearest-neighbor interactions. We derive the hydrodynamic equations solely from symmetry and conservation principles, ensuring full consistency with thermalization. Using the dynamic renormalization group, we identify a Kardar-Parisi-Zhang (KPZ)-type fixed point, characterized by the dynamical exponent z=3/2 for both the sound and heat modes. Extensive numerical simulations of the derived NFH equations confirm this exponent and further reveal that both modes are close to the universal KPZ scaling function-the Prähofer-Spohn function. These findings establish a unified, symmetry-based framework for understanding universal transport and fluctuation phenomena in one-dimensional nonequilibrium systems, independent of microscopic details.
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We construct multimode viscous hydrodynamics for one-dimensional spinless electrons. Depending on the scale, the fluid has six (shortest lengths), four (intermediate, exponentially broad regime), or three (asymptotically long scales) hydrodynamic modes. Interaction between hydrodynamic modes leads to anomalous scaling of physical observables and waves propagating in the fluid. In the four-mode regime, all modes are ballistic and acquire Kardar-Parisi-Zhang (KPZ)-like broadening with asymmetric power-law tails. "Heads" and "tails" of the waves contribute equally to thermal conductivity, leading to ω^{-1/3} scaling of its real part. In the three-mode regime, the system is in the universality class of a classical viscous fluid [O. Narayan and S. Ramaswamy, Anomalous Heat Conduction in One-Dimensional Momentum-Conserving Systems, Phys. Rev. Lett. 89, 200601 (2002).PRLTAO0031-900710.1103/PhysRevLett.89.200601, H. Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains, J. Stat. Phys. 154, 1191 (2014).JSTPBS0022-471510.1007/s10955-014-0933-y]. Self-interaction of the sound modes results in a KPZ-like shape, while the interaction with the heat mode results in asymmetric tails. The heat mode is governed by Levy flight distribution, whose power-law tails give rise to ω^{-1/3} scaling of heat conductivity.
Hydrodynamic projections, the projection onto conserved charges representing ballistic propagation of fluid waves, give exact transport results in many-body systems, such as the exact Drude weights. Focussing one one-dimensional systems, I show that this principle can be extended beyond the Euler scale, in particular to the diffusive and superdiffusive scales. By hydrodynamic reduction, Hilbert spaces of observables are constructed that generalise the standard space of conserved densities and describe the finer scales of hydrodynamics. The Green-Kubo formula for the Onsager matrix has a natural expression within the diffusive space. This space is associated with quadratically extensive charges, and projections onto any such charge give generic lower bounds for diffusion. In particular, bilinear expressions in linearly extensive charges lead to explicit diffusion lower bounds calculable from the thermodynamics, and applicable for instance to generic momentum-conserving one-dimensional systems. Bilinear charges are interpreted as covariant derivatives on the manifold of maximal entropy states, and represent the contribution to diffusion from scattering of ballistic waves. An analysis of fractionally extensive charges, combined with clustering properties from the superdiffusion phenomenology, gives lower bounds for superdiffusion exponents. These bounds reproduce the predictions of nonlinear fluctuating hydrodynamics, including the Kardar-Parisi-Zhang exponent 2/3 for sound-like modes, the Levy-distribution exponent 3/5 for heat-like modes, and the full Fibonacci sequence.
Describing emergent macroscopic phenomena in low spatial dimensions is known to be notoriously challenging, primarily due to strong interactions that render perturbative approaches inapplicable. On the other hand, low-dimensional systems host a wealth of unorthodox phenomena. A prominent example is the emergence of superdiffusive transport in one-dimensional interacting systems, traditionally studied in the framework of nonlinear fluctuating hydrodynamics. After identifying and discussing internal inconsistencies in the previous formulations, in this work we develop a general and systematic approach for constructing effective field theories of one-dimensional hydrodynamic systems in the form of coupled stochastic Langevin-type equations compatible with the physical requirements of local equilibrium states such as the thermodynamic Maxwell relation and fluctuation-dissipation symmetry. We implemented a general numerical integration scheme and exemplified our construction on a simple model of two interacting hydrodynamic modes with a non-Gaussian stationary equilibrium measure.
Anomalous transport in one-dimensional translation invariant hamiltonian systems with short range interactions is shown to belong in general to the Kardar-Parisi-Zhang universality class. Exact asymptotic forms for density-density and current-current time correlation functions and their Fourier transforms are given in terms of the Prähofer-Spohn scaling functions, obtained from their exact solution for the polynuclear growth model. The exponents of corrections to scaling are found as well, but not so the coefficients. Mode coupling theories developed previously are found to be adequate for weakly nonlinear chains but in need of corrections for strongly anharmonic interparticle potentials. A simple condition is given under which Kardar-Parisi-Zhang behavior does not apply, sound attenuation is only logarithmically superdiffusive, and heat conduction is more strongly superdiffusive than under Kardar-Parisi-Zhang behavior.