Fluctuation-Response Theory of Non-Equilibrium Complex Fluids

TL;DR

This work develops a first-principles generalized hydrodynamic framework for non-equilibrium, memory-bearing complex fluids with chemo-mechanical coupling. By deriving an exact fluctuation–response relation for steady states and expressing transport through memory kernels, it unifies stress, strain, and chemical driving within isotropic fluids and chemical reaction networks. A key result is Active Viscoelastic Memory, where chemical reaction cycles renormalize viscous relaxation and can yield a negative storage modulus at finite frequency, revealing qualitatively new rheological behavior in active matter. The framework links transport coefficients to correlation functions, recovers Harada–Sasa and Green–Kubo relations in appropriate limits, and makes concrete, testable predictions about length-scale dependent relaxation and reciprocity breaking in non-equilibrium fluids.

Abstract

A fundamental challenge in soft matter physics is to describe materials, such as the living cytoplasm and tissues, that are simultaneously active, chemically driven, and exhibit long-lasting memory of mechanical stresses. Here, we construct a generalized hydrodynamic framework at finite wavevectors and frequencies that can be applicable to non-equilibrium fluids with memory. Our approach is based on a non-equilibrium fluctuation-response relation in a steady state using correlation function identities. This approach provides a general formalism to derive hydrodynamic constitutive equations that is distinct from Mori-Zwanzig projection formalism. As a corollary, we obtain a generalized fluctuation-response relation in non-equilibrium steady states similar to the relation obtained by Harada-Sasa. Applying our theory to chemically driven active fluids reveals Active Viscoelastic Memory, whereby chemical reaction cycles renormalize the system's viscous response. We find that this active viscoelastic memory can produce a negative storage modulus at finite frequency, behavior absent in ordinary viscoelastic fluids. Our first-principles framework provides a general basis for understanding memory-dependent dynamics across a wide range of biological and synthetic active systems.

Figures (2)

• Figure 1: Generalized transport coefficients in the time domain. Time-dependent kernels $\zeta(q,t)$ and $\Lambda(q,t)$ are shown, normalized by $\zeta_c(\varepsilon)$ and $\Lambda_c(q)$, respectively. Blue circles with solid line and red triangles with solid line show $\epsilon=1$ (Onsager reciprocity) and $\epsilon=10$ (broken Onsager reciprocity), respectively. (a) Long-wavelength regime, $q\xi_\mu = 0.2$: for $\varepsilon = 1$, $\zeta(q,t)$ exhibits a monotonic, overdamped relaxation. Breaking reciprocity ($\varepsilon = 10$) generates a weak underdamped transient with small oscillations. (b) Short-wavelength regime, $q\xi_\mu = 5.0$: even in the reciprocal case ($\varepsilon = 1$), $\zeta(q,t)$ develops oscillatory relaxation. Increasing $\varepsilon$ amplifies these oscillations. (c–d) The coupling kernel $\Lambda(q,t)$ is oscillatory at both wavelengths ($q\xi_\mu=0.2$ and $5.0$) for $\varepsilon=1$ and $\varepsilon=10$, with larger $\varepsilon$ enhancing the oscillation amplitude. Model parameters: $\rho_0 = 1$, $\eta^{\parallel} = 2$, $\lambda = 1$, $\xi_\mu = 1$, and $\sqrt{C/C_\mu} = 1$ (arbitrary units).
• Figure 2: Complex moduli. Frequency-dependent complex moduli $\tilde{G}(q,\omega)$ are shown for two length scales, $q\xi_\mu=0.2$ and $q\xi_\mu=5.0$, under Onsager reciprocity ($\varepsilon=1$, blue circles with solid line) and broken reciprocity ($\varepsilon=10$, red triangles with solid line). (a–b) Reactive components $\tilde{G}'(q,\omega)$ for $q\xi_\mu=0.2$ and $5.0$. $\tilde{G}'$ becomes negative, reflecting the reactive chemo–mechanical coupling. Breaking reciprocity alters both magnitude and sign, revealing non-equilibrium reactive behavior. At shorter wavelength, the response shifts to higher frequencies, indicating faster microscopic relaxation, with enhanced amplitude under broken reciprocity. (c–d) Dissipative components $\tilde{G}"(q,\omega)$ for the same $q$. For $\varepsilon=1$, $\tilde{G}"(\omega)\ge0$ as required by thermodynamic passivity, whereas $\varepsilon=10$ introduces non-equilibrium contributions that locally reverse the sign, corresponding to active energy injection. The inset in (d) shows the magnified plot for $\varepsilon =10$ showing the sign change. Model parameters are same as Fig. \ref{['fig:relaxation']}.