In physics, two types of mechanics are usually examined: classical mechanics and quantum mechanics. For both types of mechanics, the standard mathematical approach is to consider two concepts: The complete state of the mechanical system at a given time, mathematically encoded as a phase point (classical mechanics) or a pure quantum state vector (quantum mechanics). An equation of motion which carries the state forward in time: Hamilton's equations (classical mechanics) or the Schrödinger equation (quantum mechanics) Using these two concepts, the state at any other time, past or future, can in principle be calculated. There is however a disconnect between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in.
IN phYSICs, two types of mechAnicS Are usuaLly eXamiNed: ClasSiCaL MechaNics and quanTuM mecHanics. for both tYpES of MecHAniCS, THe stanDArd MAthemaTiCAl aPPrOach Is To ConSideR twO conCEpTS: THe comPletE sTATE oF tHE meCHanicaL System AT a gIvEn TIme, matHEmatiCaLly encodeD as a phase poINt (classical MeChaNICs) or A PuRe qUantUM staTe vEcToR (quantum MechAnIcs). aN eQuation of motiOn which carRIEs The StaTe forWard in time: hAMilTon's equatioNs (clAssicAl mEchanics) or THE schRöDingeR equatIoN (quantuM MechANicS) uSiNg ThEse twO CoNcepTS, the staTE at anY otHeR tIme, PasT or FUTure, Can In PriNCiplE Be calculateD. tHere Is However a disconneCt beTweeN tHese laws aNd eveRyday lIFe eXpeRienCes, as we do not find it neceSsAry (Nor Even thEoReTically possible) tO kNoW eXACtly at a microscopic level tHe simUltAneous poSItiOns AnD velOcities of eAch moLecUle WHile carrying oUt pRoceSseS at thE hUman SCaLe (fOr ExamplE, WheN PerfOrming A ChemicaL REactIon). sTaTiSTiCal meChANiCs filLS tHis DiSCoNnectION BEtween THE LAws of mechaniCs anD the PracTical experiEnCe Of InCompLeTe knoWlEdge, bY ADding SoMe uncERTaiNty aBouT whIch sTate The SysTeM iS iN.