Fluctuation-response relation as a probe of long-range correlations in nonequilibrium quantum and classical fluids

The absence of a simple fluctuation-dissipation theorem is a major obstacle for studying systems that are not in thermodynamic equilibrium. We show that for a fluid in a nonequilibrium steady state characterized by a constant temperature gradient the commutator correlation functions are still related to response functions; however, the relation is to the bilinear response of products of two observables, rather than to a single linear response function as is the case in equilibrium. This modified fluctuation-response relation holds for both quantum and classical systems. It is both motivated and informed by the long-range correlations that exist in such a steady state and allows for probing them via response experiments. This is of particular interest in quantum fluids, where the direct observation of fluctuations by light scattering would be difficult. In classical fluids it is known that the coupling of the temperature gradient to the diffusive shear velocity leads to correlations of various observables, in particular temperature fluctuations, that do not decay as a function of distance, but rather extend over the entire system. We investigate the nature of these correlations in a fermionic quantum fluid and show that the crucial coupling between the temperature gradient and velocity fluctuations is the same as in the classical case. Accordingly, the nature of the long-ranged correlations in the hydrodynamic regime also is the same. However, as one enters the collisionless regime in the low-temperature limit the nature of the velocity fluctuations changes: they become ballistic rather than diffusive. As a result, correlations of the temperature and other observables are still singular in the long-wavelength limit, but the singularity is weaker than in the hydrodynamic regime.

Figure 1

A fluid subject to a constant temperature gradient in the $z$ direction between two parallel confining plates a distance $L$ apart. The three vectors $\widehat{\mathit{k}}, {\widehat{\mathit{k}}}_{\perp }$, and ${\widehat{\mathit{k}}}_{\perp }^{\left(2\right)}$ form a right-handed system that spans the wave-vector space.

Figure 2

Relevant frequency/energy regimes.

Figure 3

The three vectors spanning $\mathit{k}$-space and the temperature gradient.