Lecture notes
Linear response theory
- Institution
- University College London (UCL)
Providing methods to use the known past behaviour of a stochastic process in order to predict its response to a perturbation
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LinearresponsetheoryA DIE IE at E IE dBt IE EIR
perturbed P DIE KCIE EE IE t http XI die Occers
Define an observable gex
Reigle ECgCIE g IE
P mightbe a climatemodel
It is thestatevectore IR n large
gift couldbethe meansurfacetemp
E could be a perturbationdueto a violent eruptiongreenhousegasemissionsunspots etc
ASSUMPTIONS
1 x has a smooth invariantDensity psix polydefining unperturbedclimate
2 Xt has a TRANSITIONDENSITY pics tly t defining thepolyof IE given Is y
3 Xt satisfies the ERGODICPROPERTY longtimeaveragesandexpectations are equivalent
tiny 7Itg IE at Ecg IE seediscussionin notes
4 Theforcefunction FCI E in CP is SEPARABLE
FCI E L.CIget
structurestimestructure
spatial
DEFINITION
Let Xt Ye be stochasticprocesses The CORRELATIONFunction is defined to be
Cxyets ElXtYs
Applyto Xt Xe C Ii
Ye d XE
Then Cadets ECCCAE ICID
cat dex observables of x
A is autonomous timelag property
Cad C ace ECC IE d CI E Es
Infactdlypcty dealy
jointpayof CIE and Ii
By definitionofconditional probability peay piece.tlye psly polyversionof PlanB PAlBPCB
PickElyo is transitiondensity
Correlation function density
for IE given Ii y
Cad t fnf.ccDdly Picx.tly.opslyelxdy
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