On the solution of the Heaviside - Klein - Gordon thermal equation for heat transport in graphene
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1 On he solion o he eaviside - lein - Gordon hermal eqaion or hea ranspor in graphene Magdalena Pel Insie o Physis Maria Crie - Sklodowska Universiy Lblin Poland
2 Absra We repor sdies o he solion o he eaviside - lein - Gordon hermal eqaion. As he resl i is shown ha he solion onsiss o wo omponens: he as hermal wave and slow dision or very large ompared o relaaion ime ime period. We arge ha he as hermal wave an be reognized as he indiaion o he ballisi hea ranspor. As an eample we onsider he ballisi hea ranspor in graphene. ey words: hea ranspor eqaion ballisi ranspor graphene.. Inrodion In he desripion o he evolion o any physial sysem i is mandaory o evalae as araely as possible he order o magnide o dieren haraerisi ime sales sine heir relaionship wih he ime sale o observaion he ime dring whih we assme or desripion o he sysem is valid will deermine along wih he relevan eqaion paern. he adven o aoseond laser plses opens he new ield o invesigaion o he qanm phenomena. As all measred relaaion imes are mh longer he aoseond laser plses observe he generi qanm nare o he phenomena no averaged over ime. In monograph [] he heoreial ramework or ranspor proesses generaed by aoseond laser plses was ormlaed. I was shown ha he maser eqaion is he hyperboli ranspor eqaion: r Δ r q r. υ Vm mυ q. υ α where α /37 is he eleromagnei opling onsan and is he ligh veloiy m - hea arrier mass. Depending on he sign o he q he eqaion is he eaviside eqaion q < or lein - Gordon eqaion q >. or q Eq.. is he wave eqaion whih desribes he ballisi qasi-ree propagaion o arriers.
3 Reen measremen o he elerial and hermal properies o graphene [}shed new ligh on he appliaion o he eqaion. o he invesigaion o he ranspor phenomena. As an eample we invoke he ballisi ranspor in graphene whih as an be seen rom Eq.. is he resl o he q moreover i ors ha he veloiy o ermions in graphene is o he order o 6 m/s whih agrees wih υ α. he aim o his paper is he solion o he eqaion. or he lein - Gordon and eaviside branhes.. he model eqaions We will desribe he hermal energy ranspor in graphene [] wihin he heory o he hyperboli ranspor eqaion []. Given he iniial vale problem or hyperboli hermal eqaion < < > wih he iniial ondiions. g. < < he solion a an arbirary poin is given by [] dd g d.3 where < and is he solion o he eqaion. As an be seen rom eqaion.4.4 is he Green nion or he eqaion. and lils he ormla or < or >.5 and y is he Bessel nion o he order. Inroding he eaviside sep nion z we epress ormlae.5 as
4 [ ] [ ].6 Sine we see ha.6 redes o he Green nion or he one dimensional wave eqaion i we se. We now show ha saisies.4. o ha aim we se.7 where is he Green nion or he wave eqaion and saisies.8 wih replaed by. hen.9 or he irs erm in.9 is eqal zero. Also we have [ ] [ ] [ ] [ [ ] [ ] [. ' ] ]. he epression. vanishes sine [ ] [ ] [ ] [. ]. [ ] [ ] [ ] [. ]. on sing inally we have.
5 [ ]..3 I may be noed ha wih i where i in he eqaion. we obain eaviside eqaion..4 Sine i I he modiied Bessel nion o zero order we obain or in plae.6 [ ] [ I ].5 is he Green nion. In he doble inegral.3 we have or > so ha he limi in he inegral eends only p o. Also rom.5 we onlde ha vanishes nless < and his is eqivalen o. < <.6 hereore we obain. dd dd.7 rher we have [ ] [ ].8 so ha. d g d g.9 Sine he prod o he eaviside nion vanishes oside he inerval. inally
6 [ ] [ ] [ ] [ ] [ ] [. ' ]. In view o he sbsiion propery o he dela nion he las wo erms in. rede o [ ] [ ] sine and. hereore we obain ' d d. Combining hese resls and noing ha he Bessel nion o order one gives he solion o he iniial vale problem. as '. dd d d g. his solion ormla redes o ha or he Cahy problem or he inhomogenos wave eqaion i we se. Also i we se i in. and noe ha iz I z and iz ii z we obain as he solion eaviside eqaion. < < > wih he iniial ondiion.3
7 . dd I d I d I g.4 3. Ballisi hea ranspor in graphene Very imporan reason or he ineres in graphene is a niqe nare o is harge arriers. In ondensed maer physis he Shrödinger eqaion rles he world sally being qie siien o desribe eleroni properies o maerials. Graphene is an eepion: is harge arriers mimi relaivisi pariles and are easier and more naral o desribe saring wih he relaivisi eqaions: lein - Gordon and Dira raher han he Shrödinger eqaion. alhogh here is nohing parilarly relaivisi abo elerons moving arond arbon aoms heir ineraion wih a periodi poenial o graphene laie gives rise o new qasipariles ha a low energies are araely desribed by dimensional eqaion wih an eeive speed o ligh m/s. hese qasipariles alled massless Dira ermions an be seen as elerons ha los heir res mass m 6 υ or as nerinos ha aqired he eleron harge. In he monograph [] he qanm hermal eqaion or he hea ranspor was ormlaed.. Vm m υ 3. he solion o eqaion 3. an be wrien as e. 3. Aer sbsiing ormla 3. ino 3. we obain new eqaion e q υ 3.3 and
8 Vm mυ q. 3.4 Eqaion 3.3 an be wrien as where υ qυ G G υ e. 3.5 Eqaion 3.5 has he same orm as he eqaion.. As an be seen rom Eq.3.5 or q we obain ballisi hea ranspor wih veloiy υ α 6 m/s as in graphene. Reerenes [] A Casro e al.m arxiv. ond-ma:79.63 [] M.ozlowski.Mariak-ozlowska hermal proesses sing aoseond laser plses Springer 6
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