On the Speed of Heat Wave. Mihály Makai
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1 On h Spd of Ha Wa Mihály Maai maai@ra.bm.hu
2 Conns Formulaion of h problm: infini spd? Local hrmal qulibrium (LTE hypohsis Balanc quaion Phnomnological balanc Spd of ha wa Applicaion in plasma ranspor
3 1. Th Problm Diffusion and ha conducanc: infini spd In racor physics: diffusion quaion or lgraphr s quaion (Wignr Th moiaion of Zwanzig: mmory ffcs. L α i sand for h diaion of h ih sa ariabl from is quilibrium forc hn dαi d whr F is a hrmodynamic forc L i -ranspor cofficn 3 L i F ( α 1... α n (1
4 1. Th problm (con.-1 According o Eq. (1 h rspons of a sysm o an applid forc is simulanous wih h applicaion of h forc. As a gnral rul such simulaniy in a macroscopic hory urns ou o b an approimaion o a casual bhaior whr h rspons o a forc coms afr h applicaion of h forc. 4
5 1. Th Problm (con.- Josph and Prciosi in conncion wih ha conducanc and diffusion: Two problms ar h sourc of his sram: h problm of infini wa spd and h problm of scond sound. 5
6 1. Th Problm (con.-3 How o sima h spd of a ha wa in ou-of-quilibrium sa? Local hrmal quilibrium limiaions Balanc quaions fini spd Onsagr s linar modl fini spd can b obaind 6
7 . Local Thrmal Equilibrium (LTE Considr a Σ ou of quilibrium bu is pars b in quilibrium. Thn hr ar such small olums in Σ ha in ach olum p T is. Thn h firs law of hrmodynamics holds in ach small olum: du ( r T ( r ds( r P( r dv µ dm Such a small olum is infinisimal bu larg nough o b in quilibrium (larg numbr of collisions!. 7
8 . LTE (con.-1 Whn h numbr of paricls is consan hn du ( r T ( r ds( r P( r dv ( r Th firs law applis in ach subsysm hus and U(r and S(r ar no indpndn funcions h Wronsian is zro: U S U S 0 8
9 . LTE (con.- 9 r r 3 ( ( ln d f f S r 3 ( 1 d f m U ( ( ( f f r r r C A hyprbolic quaion wih on characrisics: r C d d In LTE any chang in h disribuion funcion has a limid spd.
10 3. Balanc quaions Balanc quaions ar gnrally applicabl in nar quilibrium sa. Basic ariabls ar h nsi X ( X1... X n quaniis pr uni mass and h currns ar wrin as J(X 1...X n. Componn ind c ; 10
11 3. Balanc quaions (con.- Mass balanc: c J m m o J 0 J m o J Equaion of moion ( m Di( p J o p J P F Enrgy J 0 / ψ u inic ponial inrnal 11
12 3. Balanc quaion (con.-3 m J P ψ J J q J P ψ J Th firs law conncs wor ha and inrnal nrgy. Th ha balanc: q s dq d J q 0 ds d m J q J µ q m J T T 1 q 1 m µ 1 J T J T F T T T T q s P : Grad W nd a balanc quaion wih nsis only: quaion of sa P P( T 1
13 3. Balanc quaions (con.-4 Th srucur of balanc quaion is Whn M M y and M z commu hr iss a common igncor s X 0 and w ha o sol X d X J( X q( X d J i ( X m J X i m X X X X M M y M z q y z m 13
14 3. Balanc quaions (con ( 0 r f X z X M y X M X M X z y W ha o sol Th soluion is: Φ X d f X X 0 ( ' ' ( ' ( ( r r r Hr ( y z is an ignalu of h M marics. Tha soluion propagas wih fini spd.
15 4. Phnomnological balanc J ( X J ( X J ( X o Currns dpnd on (r. Conduci currn is n in conci currn is odd in. J Λ Y ( X ΛN X J o( X X N ij Y X i j X X ΛN X q This is h lgraphr s quaion has wa fron and fini spd. 15
16 5. Spd of Ha Wa Considr h inrnal nrgy: uc T u L qq 1 T c u c u c u 0 u u u 1 u J q L qq c 3 u u ( u u L u c qq u 0 J q u Fini spd!!! To g h Infin spd: c Lqq u u 16
17 5. Spd of ha wa Ha conducion: w sar from h balanc of h inrnal nrgy. L 0 ψ0 hn nrgyu h nrgy balanc is (slid 1 : u J q Conduci currn is a linar prssion of gradins of innsis (now only u: q 1 1 c c J L qq u ct u T u u T Subsiu ha ino h balanc o g 0 17
18 5. Spd of ha wa (con.-1 Nw noaion: u u L u c qq L qq c u 0 u c Lqq u u q J and wih hm h balanc as h form of u ( u 0 No ha u is fiini so h ha conducion is no insananous! u u u 18
19 5. Spd of ha wa-(con.- How o g h radaional ha quaion? Subsiuing hr J q w g: u L qq c 1 u 3 u u ( u Nglcing h firs rm w arri a h radiional ha quaion: u λ T Tha quaion accords wih Onsagr s principl: highr han linar gradins ar nglcd. 19
20 6. Plasma sabiliy Plasma is a srongly non-quilibrium sysm. Is dscripion is basd on h balanc quaion. 7 characrisic spds ar obaind. Th sps: 1 marial balanc momnum balanc 3 nrgy balanc 4 Mawll quaions (for B and dib0 7 quaions 0
21 Plasma sabiliy-(con.-1 Th balanc quaion as h form: X M( X X Hr ( M( X M ( X M y ( X M z ( X q For simpliciy s sa assum ha h plasma is isoropic and M M y M z commu. Thir ignalu problm is b( X M ( X β ( X b( X igncor ignalu of dimnsion lociy!!! On ignalu for ach nsi! 1
22 Plasma sabiliy-(con.- Our plasma quaions: ( B B 8π ( B 8 quaions for B. q ( P π ( Β B 1 P 1 q B ( B 4π B 0
23 Plasma sabiliy-(con.-3 Th plasma nsi cor: 3 ( z y z y B B B U W ha o sol 0 z F y F F U z y Whr h F s ar h nsi currn componns U U F F Ths ar h M marics on slid 13
24 Plasma sabiliy-(con.-4 Th ignalus of M marics gi h aailabl spds in h plasma: 4 s f s f c b c c b c 0 ( ; 4 4 i b a c b a c B b π No daild hr Alfn was: b magnoacousic was: c s ; c f
25 Concluding rmars Diffusion quaion is only an approimaion.g. In racor physics h corrc quaion is h lgraphr s quaion; From h undrlying saisical physics fini spd is obaind; Th LTE hypohsis ss a limi o h spd of ranspor procsss; Non-quilibrium saisical physics prdics fini spd n in far from quilibrium sas. In pracical calculaions h diffusion is a good approimaion. 5
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