This chapter introduces the reader to the proper language in which to express with precision results of both theory and experiment concerning the response of a many-particle system to macroscopic and microscopic external probes. It shows why the Landau theory is rigorous in the long wavelength limit: as we pointed out earlier, the Landau theory is equivalent to neglecting multipair excitations in the calculation of the response function. The chapter suggests that the Landau theory succeeds because in the limit of small q and ω, the contribution of multipair excitations is usually negligible. By studying the structure of the linear response function, one may describe explicitly the transition from the collisionless high-frequency regime (where the system response is essentially microscopic), to the low-frequency hydrodynamic regime (where the response is governed by macroscopic average laws). The chapter considers a simple physical example of probe-liquid interaction, the inelastic scattering of an external particle beam by a neutral quantum liquid.
This chapter introduces the reader to the proper language in which to express with precision results of both theory and experiment concerning the response of a many-particle system to macroscopic and microscopic external probes. It shows why the Landau theory is rigorous in the long wavelength limit: as we pointed out earlier, the Landau theory is equivalent to neglecting multipair excitations in the calculation of the response function. The chapter suggests that the Landau theory succeeds because in the limit of small q and ω, the contribution of multipair excitations is usually negligible. By studying the structure of the linear response function, one may describe explicitly the transition from the collisionless high-frequency regime (where the system response is essentially microscopic), to the low-frequency hydrodynamic regime (where the response is governed by macroscopic average laws). The chapter considers a simple physical example of probe-liquid interaction, the inelastic scattering of an external particle beam by a neutral quantum liquid.
For real fermion systems, the particle interaction and the exclusion principle act simultaneously; we are thus led to study degenerate Fermi liquids, in which both effects are important. This chapter explores Landau's semiphenomenological point of view and the applications of the theory to neutral systems such as degenerate liquid. It focuses on the Landau theory of Fermi liquids, which shows clearly all the important features brought about by the interaction. Landau's theory corresponds to an expansion in powers of the fraction of excited particles. Within the two restrictions, the theory is rigorous. It lends itself to a large number of problems: equilibrium properties, transport properties, response to long wavelength external perturbations. Landau's theory is not a "universal panacea": its well-defined limitations should always be kept in mind. Moreover, it uses (and provides) far more information than any experiment will ever sample. The chapter concludes with a brief summary of the range of validity and main achievements of Landau's theory.
The macroscopic theory of Fermi liquids we have developed in the preceding chapters possesses the advantage of being exact for any normal liquid, irrespective of its density or law of interaction, so long as the temperature is sufficiently low and the phenomena under discussion involve both long wavelengths and low frequencies. It is, of course, incomplete, since it can provide no account of microscopic phenomena involving short wavelengths and large excitation energies, or of equilibrium properties such as the ground state energy and pair distribution function. In contrast, microscopic theories are able to describe both macroscopic and microscopic phenomena. They are, however, subject to one of the following limitations: They are exact for only a limited range of particle density or interaction, one which rarely corresponds to physical reality. The extent to which they provide an approximate account of a real physical system can rarely be estimated with any precision.
The dynamics of a confined fluid of Bose atoms is treated within the linear response regime, with a view to establishing a current-density functional formalism for an inhomogeneous superfluid state. After evaluating in full detail a simplified case of an external coupling to the density and phase of the condensate, the theory is extended to include the coupling to the total current density. The Kohn-Sham response functions of the condensate and all the exchange-correlation kernels for the superfluid are introduced from the microscopic equations of motion and are expressed in a physically transparent way through functional derivatives of correlation functions. A microscopic formula for the superfluid density is derived and used to introduce a generalized hydrodynamic approach for a weakly inhomogeneous two-fluid model in isothermal conditions. Local-density expressions are thereby derived for the velocities of first and second sound in the weakly inhomogeneous superfluid and for visco-elastic functions describing the transition from the hydrodynamic to the collisionless regime. Landau's hydrodynamic theory and known results in Green's functions language are recovered in the limiting case of a homogeneous superfluid.
There exists a regime of overlap between the equilibrium and non-equilibrium behavior of a system, the non-equilibrium behavior of weakly perturbed states. When a system is perturbed ever so slightly, its response will be linear in the perturbation, say the current of the conduction electrons in a metal will be proportional to the strength of the applied electric field. This regime is called the linear response regime, and though the system is in a non-equilibrium state all its characteristics can be inferred from the properties of its equilibrium state. In the next chapter we shall go beyond the linear regime by showing how to obtain quantum kinetic equations. The kinetic-equation approach to transport is a general method, and allows in principle nonlinear effects to be considered. However, in many practical situations one is interested only in the linear response of a system to an external force. The linear response limit is a tremendous simplification in comparison with general non-equilibrium conditions, and is the subject matter of this chapter. In particular the linear response of the density and current of an electron gas are discussed. The symmetry properties of response functions, and the fluctuation–dissipation theorem are established. Lastly we demonstrate how correlation functions can be measured in scattering experiments, as illustrated by considering neutron scattering from matter. Needless to say, in measurements of (say) the current in a macroscopic body, far less information in the current correlation function is probed. Linear response In this section we consider the response of an arbitrary property of a system to a general perturbation.
Landau (1958) gave a rigorous microscopic derivation of his phenomenological Fermi -liquid theory starting from the first principles Hamiltonian of strongly interacting Fermions. The derivation is based on an asymptotic expansion in small parameters such as q/kF, bc.>/EF, kT/EF, where q and c.> represent the wave numbers and frequencies of real excitations of the system. Landau showed that the dynamics of these low-lying excitations (quasiparticles and collective modes) is governed by classical transport equations with all quantum many-body effects absorbed into effective interaction parameters. Landau's approach has been generalized to superfluid Fermi liquids and superconductors by Leggett (1965) and, going beyond linear response, by Eliashberg (1971). Their ideas merged into a recent review on the quasiclassical approach to superfluid 3 He by Serene and Rainer (1983). I will use the notation of this review throughout this paper. Landau's transport theory together with its offsprings will be called 'quasiclassical theory in the following. In this context it is important to mention that Eilenberger (1968) and Larkin and Ovchinnikov (1968) have shown that the complete theory of superconductors can be formulated as a quasiclassical transport theory. Surprisingly, this theory permits calculating microscopic properties, e.g. the quasiparticle excitation spectrum, which are predominantly governed by quantum effects.
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.