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arXiv:2607.02318

Hydrodynamics, Renormalization Group, and Universality Classes Far from Equilibrium

Patrick Jentsch, Chiu Fan Lee

Published Jul 2, 2026• 0 citations
arXivHTMLPDF

Abstract

Universality is one of the central organising principles of modern physics, explaining why systems with vastly different microscopic constituents can exhibit identical large-scale behaviour. While the classification of equilibrium critical phenomena through hydrodynamics and the renormalization group (RG) is now well established, our understanding of universality far from equilibrium remains far less developed. In recent years, however, rapid progress - driven in large part by developments in active and living matter - has uncovered a growing range of genuinely nonequilibrium universality classes (UCs) with no equilibrium counterparts. In this review, we present a pedagogical and unified introduction to hydrodynamic and RG approaches to nonequilibrium many-body systems. We first show how hydrodynamic theories can be systematically constructed from symmetry and conservation laws alone. We then introduce perturbative dynamic RG methods and demonstrate how hydrodynamic theories are organised into distinct UCs according to their scaling behaviour. Building on these foundations, we review the diverse nonequilibrium UCs uncovered since 2015, while emphasizing the conceptual connections and unifying physical principles underlying their emergence. We conclude by discussing open theoretical and experimental challenges for the field.

DOI
10.48550/arxiv.2607.02318

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Universality is one of the central organising principles of modern physics, explaining why systems with vastly different microscopic constituents can exhibit identical large-scale behaviour. While the classification of equilibrium critical phenomena through hydrodynamics and the renormalization group (RG) is now well established, our understanding of universality far from equilibrium remains far less developed. In recent years, however, rapid progress - driven in large part by developments in active and living matter - has uncovered a growing range of genuinely nonequilibrium universality classes (UCs) with no equilibrium counterparts. In this review, we present a pedagogical and unified introduction to hydrodynamic and RG approaches to nonequilibrium many-body systems. We first show how hydrodynamic theories can be systematically constructed from symmetry and conservation laws alone. We then introduce perturbative dynamic RG methods and demonstrate how hydrodynamic theories are organised into distinct UCs according to their scaling behaviour. Building on these foundations, we review the diverse nonequilibrium UCs uncovered since 2015, while emphasizing the conceptual connections and unifying physical principles underlying their emergence. We conclude by discussing open theoretical and experimental challenges for the field.
One of the most challenging tasks in physics has been understanding the route an out-of-equilibrium system takes to its thermalized state. This problem can be particularly overwhelming when one considers a many-body quantum system. However, several recent theoretical and experimental studies have indicated that some far-from-equilibrium systems display universal dynamics when close to a so-called non-thermal fixed point (NTFP), following a rescaling of both space and time. This opens up the possibility of a general framework for studying and categorizing out-of-equilibrium phenomena into well-defined universality classes. This paper reviews the recent advances in observing NTFPs in experiments involving Bose gases. We provide a brief introduction to the theory behind this universal scaling, focusing on experimental observations of NTFPs. We present the benefits of NTFP universality classes by analogy with renormalization group theory in equilibrium critical phenomena.
Originally, the term ‘dynamic critical phenomena’ was coined for time-dependent properties near second-order phase transitions in thermal equilibrium . The kinetics of phase transitions in magnets, at the gas–liquid transition, and at the normal- to superfluid phase transition in helium 4 were among the prominent examples investigated already in the 1960s. The dynamic scaling hypothesis , generalizing the scaling ansatz for the static correlation function and introducing an additional dynamic critical exponent, successfully described a variety of these experiments. Yet only the development of the systematic renormalization group (RG) approach for critical phenomena in the subsequent decade provided a solid conceptual foundation for phenomenological scaling theories. Supplemented with exact solutions for certain idealized model systems, and guided by invaluable input from computer simulations in addition to experimental data, the renormalization group now provides a general framework to explore not only the static and dynamic properties near a critical point, but also the large-scale and low-frequency response in stable thermodynamic phases. Scaling concepts and the renormalization group have also been successfully applied to phase transitions at zero temperature driven by quantum rather than thermal fluctuations. It is to be hoped that RG methods may help to classify the strikingly rich phenomena encountered in far-from-equilibrium systems as well. Recent advances in studies of simple reaction-diffusion systems, active to absorbing state phase transitions, driven lattice gases, and scaling properties of moving interfaces and growing surfaces, among others, appear promising in this respect.
We review the progress made in dynamic bulk critical behaviour in equilibrium in the last 25 years since the review of Halperin and Hohenberg. We unify the presentation of the theoretical background by restricting ourselves to the field-theoretic renormalization group method. The main results obtained in the different universality classes are presented. This contains the critical dynamics near the gas–liquid transition in pure fluids (model H), the plait point and consolute point in mixtures (model H � ), the superfluid transition in 4 He (model F) and 4 He– 3 He mixtures (model F � ), the Curie point (model J) and Neel point (model G) in Heisenberg magnets and the superconducting transition. In comparison with experimental results, it became clear that in most cases one has to consider apart from the universal asymptotic critical behaviour also the non-universal effective behaviour. Either because it turned out to be inevitable due to a small dynamical transient exponent inhibiting the system to reach the asymptotics (e.g., at the superfluid transition) or because one is interested in the region further away from the phase transition like in pure fluids and mixtures at their gas–liquid or demixing transition. The calculation of the critical dynamics is adequate in most cases only in two-loop order. We review these results and present the solution to unreasonable features found for some models. Thus, we consider model C where relaxational and diffusive dynamics are coupled and the scaling properties and the limit to a purely relaxational model (model A) have not been understood. In general for models where the order parameter couples to other conserved densities time scale ratios between the kinetic coefficients of the order parameter and the conserved densities play an important role. Their fixed-point values and the approach to the fixed point are changed considerably in two-loop order compared to their values in one-loop order. These considerations are relevant for the explanation of the dynamical critical shape functions of systems such as superfluid helium (model F) and the isotropic antiferromagnet (model G). As far as possible, the comparison of results obtained by the renormalization group theory with numerical simulations has been made.
It is explained how field-theoretic methods and the dynamic renormalisation group (RG) can be applied to study the universal scaling properties of systems that either undergo a continuous phase transition or display generic scale invariance, both near and far from thermal equilibrium. Part 1 introduces the response functional field theory representation of (nonlinear) Langevin equations. The RG is employed to compute the scaling exponents for several universality classes governing the critical dynamics near second-order phase transitions in equilibrium. The effects of reversible mode-coupling terms, quenching from random initial conditions to the critical point, and violating the detailed balance constraints are briefly discussed. It is shown how the same formalism can be applied to nonequilibrium systems such as driven diffusive lattice gases. Part 2 describes how the master equation for stochastic particle reaction processes can be mapped onto a field theory action. The RG is then used to analyse simple diffusion-limited annihilation reactions as well as generic continuous transitions from active to inactive, absorbing states, which
The study of far-from-equilibrium quantum systems is a rapidly growing field of condensed matter physics. Notions of quantum information theory have emerged as important tools in this context, central to understanding the quantum thermalization process. In this thesis, we study the dynamics of such quantities, and uncover many of their interesting universal features. We achieve this by combining the study of simple minimal models, with extensive numerical simulations, developing a `hydrodynamics of quantum information'.
An introductory review of the central ideas in the modern theory of dynamic critical phenomena is followed by a more detailed account of recent developments in the field. The concepts of the conventional theory, mode-coupling, scaling, universality, and the renormalization group are introduced and are illustrated in the context of a simple example---the phase separation of a symmetric binary fluid. The renormalization group is then developed in some detail, and applied to a variety of systems. The main dynamic universality classes are identified and characterized. It is found that the mode-coupling and renormalization group theories successfully explain available experimental data at the critical point of pure fluids, and binary mixtures, and at many magnetic phase transitions, but that a number of discrepancies exist with data at the superfluid transition of 4He^{4}\mathrm{He}4He.