A good example of nonlinear electromagnetics is in high energy dense plasmas, where vortical phenomena occur which seemingly violate the second law of thermodynamics by increasing the energy gradient within the electromagnetic field and violate Maxwell's laws by creating ion currents which capture and concentrate their own and surrounding magnetic fields. In particular Lorentz force law, which elaborates Maxwell's equations is violated by these force free vortices. These apparent violations are due to the fact that the traditional conservation laws in classical and quantum electrodynamics (QED) only display linear U(1) symmetry (in particular, by the extended Noether theorem, conservation laws such as the laws of thermodynamics need not always apply to dissipative systems, which are expressed in gauges of higher symmetry). The second law of thermodynamics states that in a closed linear system entropy flow can only be positive (or exactly zero at the end of a cycle). However, negative entropy (i.e. increased order, structure or self-organisation) can spontaneously appear in an open nonlinear thermodynamic system that is far from equilibrium, so long as this emergent order accelerates the overall flow of entropy in the total system. The 1977 Nobel Prize in Chemistry was awarded to thermodynamicist Ilya Prigogine for his theory of dissipative systems that described this notion. Prigogine described the principle as "order through fluctuations" or "order out of chaos". It has been argued by some that all emergent order in the universe from galaxies, solar systems, planets, weather, complex chemistry, evolutionary biology to even consciousness, technology and civilizations are themselves examples of thermodynamic dissipative systems; nature having naturally selected these structures to accelerate entropy flow within the universe to an ever-increasing degree. For example, it has been estimated that human body is 10,000 times more effective at dissipating energy per unit of mass than the sun.
A GOOd exAmPle of nOnlIneAR elEcTroMaGneTics IS In high ENeRGy deNse plAsMaS, WhEre VoRtIcal pHeNomENA occur wHicH seemINgly ViolaTE The second LAw oF thErmodynamIcs by iNcreasing The eNergy gRadiEnt WIthin tHe ElECTroMagnETIc fIeld anD vioLate mAxwell'S lawS by CreatInG ion cUrrenTs wHiCh cAPture anD cOnCentrAtE TheIR own aNd SurrounDIng MagneTic fiElds. in particULAR loRentz force lAW, which elabOrates MAxwell'S equAtions is VIolaTed bY TheSE Force free vOrtices. these apPaRenT vioLationS arE dUe TO The FAct tHat tHe trADiTIONal coNservAtION laws In clAsSIcal aNd qUantUM eleCtrODynaMics (qed) onLy dIsPlay LineaR u(1) SyMmetrY (in particUlar, by the ExtENded NoethEr Theorem, conSeRvAtioN LaWs SUch As THe Laws oF thermodynAmics NEed NoT aLways AppLy to dIssiPAtIve sysTems, wHicH are expResseD In gauGES of HigHer SymmetrY). thE seCond lAw Of tHerModyNAmicS STAtes ThaT in A closED LineAr SysteM eNtrOpy FLow cAn OnlY bE poSItIVe (oR eXactlY Zero at The END Of a cycle). hOweVEr, neGative entRopy (I.e. inCreasED oRDER, structure Or self-OrganisatiOn) can spontaneoUsly apPEar iN aN opEN noNlInear thErmodYnamIC System That Is faR FRoM EQUilibRium, SO loNg as thIs emergeNt oRdeR aCceleRaTeS the overaLL flOw of eNTRopy in tHe total syStEM. tHe 1977 nObel PRIze in cheMIsTrY waS awarded to ThErMOdynAmicIst ilYa pRiGogiNe for HIs THeOrY of dISsipAtive sYStems that descRiBEd this NOtIoN. prigoGInE descrIbEd thE PRincIpLE as "OrdeR ThRoUgh flUctuaTiONS" or "order out of chAos". iT has Been aRgued by somE That All EmERgENt orDer In thE unIVeRSe frOm gaLaxies, sOlaR SyStEms, pLanEts, wEather, cOmplex cHEmistry, eVolUtiOnARy BIoLOGy to Even COnScIousneSs, tECHNoloGy and cIvilIzAtiONS are themseLves eXampleS of therMODYnaMIc diSSipatiVE sYsTems; NatURe hAViNg naTUrally SelecTED theSE struCTuRes to acCeLeRatE EntRoPy fLow WIThin the universe tO an EvEr-INCreaSIng dEgRee. for ExaMpLe, iT Has BEEN eStimateD THaT human Body is 10,000 TimEs moRe efFecTIve AT DissipatINg enERgY peR uNit of mass Than thE Sun.