Fluctuation-response relation in turbulent systems

Nonequilibrium statistical mechanics of the turbulent energy cascade: Irreversibility and response functions

The statistical properties of turbulent flows are fundamentally different from those of systems at equilibrium due to the presence of an energy flux from the scales of injection to those where energy is dissipated by the viscous forces: a scenario dubbed "direct energy cascade." From a statistical mechanics point of view, the cascade picture prevents the existence of detailed balance, which holds at equilibrium, e.g., in the inviscid and unforced case. Here, we aim at characterizing the nonequilibrium properties of turbulent cascades in a shell model of turbulence by studying an asymmetric time-correlation function and the relaxation behavior of an energy perturbation, measured at scales smaller or larger than the perturbed one. We contrast the behavior of these two observables in both nonequilibrium (forced and dissipated) and equilibrium (inviscid and unforced) cases. Finally, we show that equilibrium and nonequilibrium physics coexist in the same system, namely, at scales larger and smaller, respectively, of the forcing scale.

Two-tag correlations and nonequilibrium fluctuation-response relation in ageing single-file diffusion

Spatiotemporally correlated motions of interacting Brownian particles, confined in a narrow channel of infinite length, are studied in terms of statistical quantities involving two particles. A theoretical framework that allows analytical calculation of two-tag correlations is presented on the basis of the Dean-Kawasaki equation describing density fluctuations in colloidal systems. In the equilibrium case, the time-dependent Einstein relation holds between the two-tag displacement correlation and the response function corresponding to it, which is a manifestation of the fluctuation-dissipation theorem for the correlation of density fluctuations. While the standard procedure of closure approximation for nonlinear density fluctuations is known to be obstructed by inconsistency with the fluctuation-dissipation theorem, this difficulty is naturally avoided by switching from the standard Fourier representation of the density field to the label-based Fourier representation of the vacancy field. In the case of ageing dynamics started from equidistant lattice configuration, the time-dependent Einstein relation is violated, as the two-tag correlation depends on the waiting time for equilibration while the response function is not sensitive to it. Within linear approximation, however, there is a simple relation between the density (or vacancy) fluctuations and the corresponding response function, which is valid even if the system is out of equilibrium. This non-equilibrium fluctuation-response relation can be extended to the case of nonlinear fluctuations by means of closure approximation for the vacancy field.

Response function of turbulence computed via fluctuation-response relation of a Langevin system with vanishing noise

For a shell model of the fully developed turbulence and the incompressible Navier-Stokes equations in the Fourier space, when a Gaussian white noise is artificially added to the equation of each mode, an expression of the mean linear response function in terms of the velocity correlation functions is derived by applying the method developed for nonequilibrium Langevin systems [Harada and Sasa, Phys. Rev. Lett. 95, 130602 (2005)]. We verify numerically for the shell-model case that the derived expression of the response function, as the noise tends to zero, converges to the response function of the noiseless shell model.

Direct-numerical-simulation-based measurement of the mean impulse response of homogeneous isotropic turbulence

A technique for measuring the mean impulse response function of stationary homogeneous isotropic turbulence is proposed. Such a measurement is carried out here on the basis of direct numerical simulation (DNS). A zero-mean white-noise volume forcing is used to probe the turbulent flow, and the response function is obtained by accumulating the space-time correlation between the white forcing and the velocity field. This technique to measure the turbulent response in a DNS numerical experiment is a research tool in that field of spectral closures where the linear-response concept is invoked either by resorting to renormalized perturbations theories or by introducing the well-known fluctuation-dissipation relation (FDR). Although the results obtained in the present work are limited to relatively low values of the Reynolds number, a preliminary analysis is possible. Both the characteristic form and the time scaling properties of the response function are investigated in the universal subrange of dissipative wave numbers; a comparison with the response approximation given by the FDR is proposed through the independent DNS measurement of the correlation function. Very good agreement is found between the measured response and Kraichnan's description of random energy-range advection effects.

Eulerian spectral closures for isotropic turbulence using a time-ordered fluctuation-dissipation relation

Procedures for time-ordering the covariance function, as given in a previous paper [K. Kiyani and W. D. McComb, Phys. Rev. E 70, 066303 (2004)], are extended and used to show that the response function associated at second order with the Kraichnan-Wyld perturbation series can be determined by a local (in wave number) energy balance. These time-ordering procedures also allow the two-time formulation to be reduced to time-independent form by means of exponential approximations and it is verified that the response equation does not have an infrared divergence at infinite Reynolds number. Last, single-time Markovianized closure equations (stated in our previous paper) are derived and shown to be compatible with the Kolmogorov distribution without the need to introduce an ad hoc constant.

Time-ordered fluctuation-dissipation relation for incompressible isotropic turbulence

The Kraichnan-Wyld perturbation expansion is used to justify the introduction of a renormalized response function connecting two-point covariances at different times. The resulting relationship was specialized by a suitable choice of initial conditions to the form of a fluctuation-dissipation relation (FDR). This was further developed to reconcile the time symmetry of the covariance with the causality of the response by the introduction of time ordering along with a counterterm. This formulation provides a solution to an old problem, that of representing the time dependence of the covariance and response by exponential forms. We show that the derivative (with respect to difference time) of the covariance now vanishes at the origin. This allows one to study the relationships between two-time spectral closures and time-independent theories such as the self-consistent field theory of Edwards or the more recent renormalization group approaches. We also show that the renormalized response function is transitive with respect to intermediate times and report a different Langevin equation model for turbulence. We note the potential value of this time-ordering procedure in all applications of the FDR.

Relaxation of finite perturbations: beyond the fluctuation-response relation

We study the response of dynamical systems to finite amplitude perturbation. A generalized fluctuation-response relation is derived, which links the average relaxation toward equilibrium to the invariant measure of the system and points out the relevance of the amplitude of the initial perturbation. Numerical computations on systems with many characteristic times show the relevance of the above-mentioned relation in realistic cases.

Linear response of the Lorenz system

The present numerical study provides strong evidence that at standard parameters the response of the Lorenz system to small perturbations of the control parameter r is linear. This evidence is obtained not only directly by determining the response in the observable A(x)=z, but also indirectly by validating various implications of the assumption of a linear response, like a quadratic response at twice the perturbation frequency, a vanishing response in A(x)=x, the Kramers-Kronig relations, and relations between different response functions. Since the Lorenz system is nonhyperbolic, the present results indicate that in contrast to a recent speculation the large system limit (thermodynamic limit) need not be invoked to obtain a linear response for chaotic systems of this type.