A minimizing principle for the Poisson-Boltzmann equation

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1、AminimizingprincipleforthePoisson-BoltzmannequationA.C.MaggsLaboratoirePCT,GulliverCNRS-ESPCIUMR7083,10rueVauquelin,75231ParisCedex05.ThePoisson-Boltzmannequationisoftenpresentedviaavariationalformulationbasedontheelectrostaticpotential.However,thefunctionalhasthedefectofbeingnon-convex.Itc

2、annotbeusedasalocalminimizationprinciplewhilecoupledtootherdynamicdegreesoffreedom.Weformulateaconvexdualfunctionalwhichisnumericallyequivalentatitsminimumandwhichismoresuitedtolocaloptimization.Introductionvariationalfield.Ifwedosothenwefindthat(∇φ)2XThePoisson-Boltzmanntreatmentoffreeionsis

3、widelyf=ρφ−ǫ+kBT(cjln(cj/cj0)−cj)(4)2usedinsimulationswithimplicitsolvents.Itsuccessfullyjdescribesthemeanfieldscreeningpropertiesofionicso-lutions.AtthemostphenomenologicallevelitisfoundWeseeatoncethatwehavegainedinlocalityofthebybalancingtwocrucialfeaturesoftheionicsystem:Theformulation;th

4、eotheradvantageisthatitisvalidforelectrostaticenergycomingfromCoulomb’slaw,plusthearbitraryspatialvariationinthedielectricproperties,idealentropyofmixingoftheions.Writtenintermsǫ(r).Unfortunately,thecounterpartisthattheresult-ofionconcentrationscj,orthechargeconcentrationsingfreeenergyisno-

5、longerconvex.Wedonothaveaρj=cjqjwefindthefreeenergyofasolutionfromminimizingprinciple,ratherastationaryprinciple.ThisZexcludessomeofthesimplestoptimizationstrategiesthat33′1′onemightwantapply–suchassimultaneousannealingF=drdrρ(r)ρ(r)+4πǫ0

6、r−r′

7、ofconformationalandelectrostaticdegreesoffreedom

8、.ZXFromeq.(4)onestudiesthedifferential3kBTdr(cjln(cj/cj0)−cj)(1)Xjdf=dcj(qjφ+kBTln(cj/cj0))+dφ(ρ+∇·ǫ∇φ)Pwherethetotalchargedensityρ=ρj+ρf.ρfisanjjexternalfixedchargedensity–associatedwithsurfacesWenowimposethatthecoefficientofdcjiszerosothatormolecularsources,cj0isthereferencedensityofcom-ponen

9、tj.Ifweminimizethisfunctionalofcjthenweqjφ+kBTln(cj/cj0)=0(5)findaneffectivefreeenergyforthesourcefieldρf,[1].TheprobleminthisformulationistheappearanceoftheSubstitutingbackintoeq.(4)wefindthestandardformlong-rangedCoulombinteraction[2]whichrendersthefortheP