So far in this book we have studied properties of systems in thermal equilibrium. We now consider transport phenomena in fluids out of equilibrium. There are two distinct methods to calculate the transport coefficients. One is based on the Boltzmann equation in relativistic kinetic theory [1]. The other finds the response of the system to the energy–momentum tensor [2]. Here we take up the latter procedure, leaving the former to a brief review in Problem 9.1.

Assuming the system to be not far from equilibrium, we consider only responses linear in the perturbation causing non-equilibrium in the fluid medium. The phenomenological form of its energy–momentum tensor involves gradients of fluid velocity and temperature. The terms in this tensor without these gradients describe perfect fluid, while terms containing them represent dissipative effects in real fluids. The latter terms have (transport) coefficients to be determined. The method of linear response calculates these coefficients from thermal and quantum fluctuations arising in loop graphs of quantum field theory.

The first five sections are reviews leading to field theoretical expressions for the transport coefficients. Following Weinberg [3] we first describe elements of relativistic hydrodynamics to get the phenomenological expression for the energy–momentum tensor of imperfect fluids. Besides defining transport coefficients, it offers physical insight into the problem. We then present Zubarev's formulation of non-equilibrium statistical theory [2] and find the thermal expectation value of the components of the corresponding tensor operator. The transport coefficients so obtained are written as standard equilibrium correlation functions of the retarded type as in Hosoya et al. [4]. In the last section we evaluate these coefficients for the pion gas using the real time method of thermal field theory developed in this book. As already mentioned in Section 5.4, this method has the advantage over the imaginary time method in that it does not require any analytic continuation from discrete frequencies.

Relativistic Hydrodynamics

Consider a fluid having at each point a velocity three-vector v in a lab frame of reference (at rest in the lab). In a comoving frame (moving with the fluid) at some space–time point, the fluid will be at rest at that point. A perfect or ideal fluid is one which appears isotropic around this point from the comoving (denoted by tilde) frame.