Mass and charge renormalization for classical processes with a fluctuation-dissipation theorem

The fluctuation-dissipation theorem (FDT) valid both for canonical systems in equilibrium and for purely irreversible stochastic processes with detailed balance is shown to imply that the renormalized values of all coefficients in the field equation ("mass" and "charges") are simply related to static cumulants of the field. As a consequence, the problem of determining the static behavior of the system (which reduces to quadratures when the FDT holds) is completely separated from the dynamic problem; the perturbation theory for the time dependence of the two-point cumulant has, if properly renormalized, the full statics of the system incorporated exactly; the renormalized perturbation theory (RPT) will, in general, be much more useful than the unrenormalized one since in contrast to the latter it does not necessarily require the system to be nearly Gaussian. If the nonlinearity in the field equation involves only a quartic charge matrix, the renormalized charge matrix is $\frac{C^{\mathrm{}\left(4\right)\mathrm{}}}{C^{\mathrm{}\left(2\right)4}}$, where $C^{\mathrm{}\left(2\right)\mathrm{}}$ and $C^{\mathrm{}\left(4\right)\mathrm{}}$ are the static cumulants of order 2 and 4, respectively, whereupon the expansion parameter in the RPT becomes $\frac{C^{\mathrm{}\left(4\right)\mathrm{}}}{C^{\mathrm{}\left(2\right)2}}$; when the static cumulants are known, the smallness of this expansion parameter (which does not necessarily require $C^{\mathrm{}\left(4\right)\mathrm{}}\approx 0$, i.e., near-Gaussian statics) can be checked explicitly. The usefulness and necessity of mass and charge renormalization is demonstrated by a perturbative calculation of the linewidth of the Van der Pol oscillator; although this oscillator is far from Gaussian near and above threshold, the result (below, near, and above threshold) is quite satisfactory already in second and excellent in fourth order; the unrenormalized perturbation expansion, on the other hand, yields nonsense in any finite order except very far below threshold where the oscillator is indeed nearly Gaussian. For many-body systems the static cumulants defining the renormalized mass and charge(s) can, in general, not be evaluated exactly; however, in many cases sufficiently accurate approximate values or experimental information are available to render the RPT practical.