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arXiv:hep-lat/0609064v128Sep2006LatticeFieldTheoryMethodsinModernBiophysicsAnthonyDuncan∗DepartmentofPhysicsandAstronomy,UniversityofPittsburgh,Pittsburgh,PA15260E-mail:tony@dectony.phyast.pitt.eduAneffectivefieldtheoryexistsdescribingaverylargeclassofbiophysicallyinterestingCoulombgassystems:thelowestorder(mean-field)versionofthistheorytakestheformofageneral-izedPoisson-Boltzmanntheory.Interactiontermsdependondetails(finite-sizeeffects,multipoleproperties,etc).Convergenceoftheloopexpansionholdsonlyifmutualinteractionsofmobilechargesaresmallcomparedtotheirinteractionwiththefixed-chargeenvironment,whichisfre-quentlynotthecase.Problemswiththestrongly-coupledeffectivetheorycanbecircumventedwithanalternativelocallatticeformulation,withrealpositiveaction.Inrealisticsituations,withvariabledielectric,adeterminantofthePoissonoperatormustbeinsertedtogeneratecorrectelectrostatics.MethodsadoptedfromunquenchedlatticeQCDdothisveryefficiently.XXIVthInternationalSymposiumonLatticeFieldTheoryJuly23-28,2006Tucson,Arizona,USA∗Speaker.cCopyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence.fixedelectriccharges(ormoregenerally,particleswithhighermultipolemomentsaswell),inabackground(typ-icallywater,butalsoelectricallyneutralinteriorsofproteinpolypeptides,inwhichfixedchargesmaybeembedded)ofvariabledielectric.DirectmoleculardynamicssimulationsofCoulombgasproblemsareimpracticalforlargesystems,asthelong-rangecharacteroftheCoulombinteractionmeansthattheelectrostaticenergyofeverypairofparticlesinthesystemhastobecomputed.Insystemswithuniformdielectric,thisproblemcanbeamelioratedbyFourier(Ewald)techniques,butinthegeneralcase,thedielectric“constantinfactvariesspatially(andalsodynamically,inthecourseofthesimulation,ifmacroionsorpolymercomponentsareallowedtochangeconfor-mation),makingFouriermethodsimpractical.OfcoursethescreeningeffectsinahomogeneousmediumwithmobilechargeshavebeenunderstoodforalongtimeonthebasisofDebye-Hückeltheory[1],providedthechargeconcen-trationsarenottoohigh.HereonestartsfromaPoisson-Boltzmannequationwhichsummarizesthemean-fieldeffectsofthemobileionsoneachother(andwithanyfixedcharges).Startinginthe1940s,agreatdealwasaccomplishedwithalinearizedversionofthisequation(theDLVOmethodintroducedbyDerjaguin,Landau,VerweyandOverbeek[2]).However,suchmethodsdonotpro-videanyintrinsicprocedureforsystematicimprovementofthemean-fieldresult,andfrequentlyfailcompletelyintheregimeofhighconcentrations.AmoregeneralformalismfordealingwithCoulombgasesunderverygeneralcircumstanceswasintroducedintheearly90’s[3]:theinitialemphasiswasindealingwithsystemsoffixedchargedmacroionssurroundedbyagasofsmallmobileions.Thegrandcanonicalpartitionfunc-tionforsuchsystemscanbeconvertedintoapath-integralregularizedonaspatiallattice,andthesaddle-pointexpansionofthefunctionalintegralthenleadstoa(discretized)Poisson-Boltzmannequationwhichcanberapidlysolvednumerically.Moreover,thebasictechniqueallowsforstraightforwardgeneralizationtosystemswithshort-rangerepulsiveforcesbetweenthemobileions[4],multipolarions[5],chargedpolymerinteractions[6,7,8],amongothers.Inallofthesecasesthehigherorderfluctuation(“loop)effectsareclearlydefined,andtheleadingcorrectionstothemean-fieldresultcomputable.Despitetheformallyattractivenatureofthiseffectivefieldapproach,therestillremainsthedifficultythatinmanycasesthesefluctuationcorrectionsareverylarge,soaperturbativesaddle-pointexpansiondoesnotyieldusefulresults.Furthermore,theeffec-tiveactionforthesetheoriesisalwayscomplex,sodirectMonteCarlosimulationofthefunctionalintegralisimpracticalduetotheinfamoussigneffect.Recently,Maggsandcollaboratorssuggested[9]analternativeapproachtoCoulombgasesinwhichthelong-rangeCoulombinteractionislocalizedbywritingthepathintegralfortheparti-tionfunctionintermsoflocalelectricfieldvariables(essentially,onetakestheHamiltonianpathintegralforfinitetemperatureMaxwellelectrodynamicsandneglectsmagneticterms).Aconsid-erableamountofworkhasnowbeendevotedtostreamliningandimprovingtheefficiencyofthisapproach[10,11].TheoriginalalgorithmofMaggsetal.onlyhandlessystemsofuniformdielec-tric,however,forwhichonemayarguethatFourieracceleratedmoleculardynamicssimulationsarecompetitive.Forsystemsinwhichthedielectricmediumisdynamical,theMaggsetal.func-2LatticeFieldTheoryMethodsinModernBiophysicsAnthonyDuncantionalintegralproducesaspuriousinteractionforcebetweentheparticles,whichmustberemovedtoobtainthecorrectelectrostaticenergy.Recentwork[12,13]hasshownthatthiscanbedoneexactlybyintroducingthedeterminantofthegeneralizedPoissonoperatorintothepathintegral,incompleteanalogytothewaythedeterminantofthequarkDiracoperatormustbeintroducedinthepathintegralofunquenchedQCD.NumericalsimulationscanbeperformedusingavarietyofmethodsimportedfromlatticeQCD:inthefollowing,wedescriberesult