Nonlinear response from transport theory and quantum field theory at finite temperature - art. no. 025001

We study the nonlinear response in weakly coupled hot phi (4) theory. We ob tain an expression for a quadratic shear viscous response coefficient using two different formalisms: transport theory and response theory. The transp ort theory calculation is done by assuming a local equilibrium form for the distribution function and expanding in the gradient of the local four dime nsional velocity field. By performing a Chapman-Enskog expansion on the Bol tzmann equation we obtain a hierarchy of equations for the coefficients of the expanded distribution function. To do the response theory calculation w e use Zubarev's techniques in nonequilibrium statistical mechanics to deriv e a generalized Kubo formula. Using this formula allows us to obtain the qu adratic shear Viscous response from the three-point retarded Green function of the viscous shear stress tensor. We use the closed time path formalism of real time finite temperature field theory to show that this three-point function can be calculated by writing it as an integral equation involving a four-point vertex. This four-point vertex can in turn be obtained from an integral equation which represents the resummation of an infinite series o f ladder and extended-ladder diagrams. The connection between transport the ory and response theory is made when we show that the integral equation for this four-point vertex has exactly the same form as the equation obtained from the Boltzmann equation for the coefficient of the quadratic term of th e gradient expansion of the distribution function. We conclude that calcula ting the quadratic shear viscous response using transport theory and keepin g terms that are quadratic in the gradient of the velocity field in the Cha pman-Enskog expansion of the Boltzmann equation is equivalent to calculatin g the quadratic shear viscous response from response theory using the next- to-linear response Kubo formula, with a vertex given by an infinite resumma tion of ladder and extended-ladder diagrams.