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We present a theory for the construction of renormalized kinetic equations todescribe the dynamics of classical systems of particles in or out ofequilibrium. A closed, self-consistent set of evolution equations is derivedfor the single-particle phase-space distribution function $f$, the correlationfunction $C=<\delta f\delta f >$, the retarded and advanced density responsefunctions $\chi^{R,A}=\delta f/\delta\phi$ to an external potential $\phi$, andthe associated memory functions $\Sigma^{R,A,C}$. The basis of the theory is aneffective action functional $\Omega$ of external potentials $\phi$ thatcontains all information about the dynamical properties of the system. Inparticular, its functional derivatives generate successively thesingle-particle phase-space density $f$ and all the correlation and densityresponse functions, which are coupled through an infinite hierarchy ofevolution equations. Traditional renormalization techniques are then used toperform the closure of the hierarchy through memory functions. The lattersatisfy functional equations that can be used to devise systematicapproximations. The present formulation can be equally regarded as (i) ageneralization to dynamical problems of the density functional theory of fluidsin equilibrium and (ii) as the classical mechanical counterpart of the theoryof non-equilibrium Green's functions in quantum field theory. It unifies andencompasses previous results for classical Hamiltonian systems with any initialconditions. For equilibrium states, the theory reduces to the equilibriummemory function approach. For non-equilibrium fluids, popular closures (e.g.Landau, Boltzmann, Lenard-Balescu) are simply recovered and we discuss thecorrespondence with the seminal approaches of Martin-Siggia-Rose and ofRose.and we discuss the correspondence with the seminal approaches ofMartin-Siggia-Rose and of Rose.

Renormalized kinetic theory of classical fluids in and out of equilibrium