  We study nonlinear response in weakly coupled hot $\phi^4$ theory. We obtain
an expression for a quadratic shear viscous response coefficient using two
different formalisms: transport theory and response theory. The transport
theory calculation is carried out by assuming a local equilibrium form for the
distribution function and expanding in the gradient of the local four
dimensional velocity field. By doing a gradient expansion on the Boltzmann
equation we obtain a hierarchy of equations for the coefficients of this
expansion.To do the response theory calculation we use Zubrave's techniques in
nonequilibrium statistical mechanics to derive a generalized Kubo formula.
Using this formula allows us to obtain the quadratic 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 theoryto show that this three-point function can be calculated using
equilibrium quantum field theory 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 of ladder and extended-ladder diagrams.The connection between transport
theory 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 the
gradient expansion of the distribution function.
