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We study nonlinear response in weakly coupled hot $\phi^4$ theory. We obtainan expression for a quadratic shear viscous response coefficient using twodifferent formalisms: transport theory and response theory. The transporttheory calculation is carried out by assuming a local equilibrium form for thedistribution function and expanding in the gradient of the local fourdimensional velocity field. By doing a gradient expansion on the Boltzmannequation we obtain a hierarchy of equations for the coefficients of thisexpansion.To do the response theory calculation we use Zubrave's techniques innonequilibrium statistical mechanics to derive a generalized Kubo formula.Using this formula allows us to obtain the quadratic shear viscous responsefrom the three-point retarded green function of the viscous shear stresstensor. We use the closed time path formalism of real time finite temperaturefield theoryto show that this three-point function can be calculated usingequilibrium quantum field theory by writing it as an integral equationinvolving a four-point vertex. This four-point vertex can in turn be obtainedfrom an integral equation which represents the resummation of an infiniteseries of ladder and extended-ladder diagrams.The connection between transporttheory and response theory is made when we show that the integral equation forthis four-point vertex has exactly the same form as the equation obtained fromthe Boltzmann equation for the coefficient of the quadratic term of thegradient expansion of the distribution function.Comment: Revtex, 25 pages with 7 figure

Nonlinear response from transport theory and quantum field theory at finite temperature