ABSTRACT
Forexample,inviscousmechanicaldamping,orcapacitativeelectricalsmoothing,
f=e(t'-1)/T.....(5) whereTisthetimeconstant(responsetime).Foranysuchinvariantaveraging operatorwhichinadditionhas'finitememory',meaningthatf(- oo)A(- oo)is
zero,itiseasytoshowbydirectcalculationthat
Nowconsidertheeffectsoftime-averagingonthework-energyrelation. Therelationitselfis
wheretheforcevectorFandthevelocityvectorvareconsideredinthesystem phasespace,sothattheproductF·vincludessummationoverallcoordinates ofallparticles.Forinvariantfinitememorytimeaveragingwehave,combining
equations6and7:
Ontheotherhand,wecalculatethemacroscopicrateofworkdoneonthe
systemas dW dt=<F)·<v),...(9)
sothatwerecognizethedynamiccrosscorrelation(<F·v)- <F)·(v))asthe rateofheatflowdQ/dtintothesystem,basedonthefirstlawdefinition
Inthiscasethetime-averagingisperformedbythemeasuringdevicesandthe
multiplicationisperformedbytheobserver.
