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1、InformationInequalityInterpretationofUncertaintyRelationinStochasticProcessYoshihikoHasegawa∗andTanVanVu†DepartmentofInformationandCommunicationEngineering,GraduateSchoolofInformationScienceandTechnology,TheUniversityofTokyo,Tokyo113-8656,Japan(Dated:September11,2018)Thermodynamicuncertaintyr
2、elationisaninequalityforprecisionofthermodynamicquantities,statingthatitisimpossibletoattainhigherprecisionthanthebounddefinedbytheentropyproduction.Suchtrade-offrelationsbetween“cost”and“quality”existinseveralfields.Particularly,instatisticalinferencetheory,theinformationinequalitiesassertthati
3、tisinfeasibleforanystatisticalestimatorstoachievesmallererrorthantheprescribedbound.Inspiredbythesimilaritybetweenthethermodynamicuncertaintyrelationandtheinformationinequalities,weapplytheinformationinequalitiestosystemsdescribedbyLangevinequationsandderivetheboundforthevarianceofthermodynam
4、icquantities.WhenapplyingtheCram´er–Raoinequality,theobtainedinequalityisshowntoreducetothefluctuation-responseinequality,whichrelatesthefluctuationwiththelinearresponseofsystemsagainstperturbations.WealsoapplyHammersley–Chapman–Robbinsinequalitytothesystemsandfindarelation,givingthelowerboundof
5、theratiobetweenthevarianceandthesensitivityofthesystemsinresponsetoarbitraryperturbations.Toconfirmthesecondinequality,weapplyittoastochasticlimitcycleoscillatorandnumericallyshowthattheratiobetweenthephasevarianceandthephasesensitivityisboundedfrombelowbythereciprocalofthePearsondivergence.In
6、troduction.—Inthesetwodecades,byvirtueofad-tionforone-dimensionalLangevinequationtoattainthevancementofstochasticthermodynamics,substantialequalityofTUR.Furthermore,weapplyHammersley–progresseshavebeenmadeinuniversalrelationsamongChapman–Robbinsinequality(HCRI),whichisagener-thermodynamicquan
7、tities,suchasfluctuationtheoremsalizationofCRI,tothesystemstoshowthat,foranyandgeneralizedsecondlaws[1–4].Oneofthefundamen-perturbation,theratiobetweenthevarianceandthetalachievementsisthethermodynamicuncertaintyrela-sensitivityisboundedfrombe
