Camille Bélanger-ChampagneLuis Anchordoqui LehmanMcGill College University City University of New York ThermodynamicsCharged Particle and CorrelationsStatistical Mechanics in Minimum Bias Events at ATLAS • Boltzmann● Physics motivation distribution Fermi-Dirac distribution • ● Minbias event and track selection Statistical Mechanics II Bose-Einstein distribution th • OctoberFebruary 2014 26 2012,• Boltzmann-Maxwell● Azimuthal correlation distribution results WNPPC 2012 • Statistical● Forward-Backward thermodynamics correlation of theresults ideal gas Thursday, October 30, 14 1 BOLTZMANN STATISTICS Overview Equilibrium configuration for system of N distinguishable noninteracting particles n n (40) subject to constraints ☛ Nj = N and Nj"j = U j=1 j=1 X X # of ways of selecting N 1 particles from total of N to be place in j =1 level N N! = N1 N !(N N )! ✓ ◆ 1 − 1 # of ways these N 1 particles can be arranged if there are g 1 quantum states N1 for each particle there are g 1 choices ☛ ( g 1 ) possibilities in all # of ways to put N 1 particles into a level containing g 1 distinct options N N!gN1 = 1 N1 N !(N N )! ✓ ◆ 1 − 1 C.Luis B.-Champagne Anchordoqui 2 Thursday, October 30, 14 2 BOLTZMANN STATISTICS II For j =2 ☛ same situationOverview except that there are only ( N N ) particles remaining to deal with − 1 (N N )!gN2 − 1 2 N !(N N N )! 2 − 1 − 2 Continuing process ↴ N!gN1 (N N )!gN2 ! (N ,N ,N )= 1 − 1 2 B 1 2 n N !(N N )! ⇥ N !(N N N )! 1 − 1 2 − 1 − 2 (N N N )!gN3 − 1 − 2 3 ⇥ N !(N N N N )! ··· 3 − 1 − 2 − 3 n Nj gN1 gN2 gN3 g = N! 1 2 3 ··· = N! j N !N !N ! N ! 1 2 3 j=1 j ··· Y We now have to maximize ! B subject to constraints (40) C.Luis B.-Champagne Anchordoqui 2 Thursday, October 30, 14 3 LAGRANGE MULTIPLIERS Maximization of f ( x, yOverview ) subject to constraint φ(x, y) = constant @f @f df = dx + dy =0 @x @y @f @f If dx and dy were independent ☛ = =0 @x @y @ @ However ☛ subject to constraint equation dφ = dx + dy =0 @x @y ☟ @f/@x @f/@y = @φ/@x @φ/@y For constant ratio ↵ @f @ @f @ + ↵ =0 and + ↵ =0 @x @x @y @y expressions we would get if we attempt to maximize f + ↵ without constraint For n variables and two constraint relations @f @ @ + ↵ + β =0,i=1, 2, 3 n @xi @xi @xi ··· C.Luis B.-Champagne Anchordoqui 2 Thursday, October 30, 14 4 BOLTZMANN DISTRIBUTION Task ☛ find maximum ofOverview ! B with respect to all N that satisfy constraints (40) In practice ☛ more convenient to maximize ln ! than w itself n n ln ! =lnN!+ N ln g ln N ! i i − i i=1 i=1 X X We are concerned with N 1 ☛ use Stirling’s asymptotic expansion i ln N! N ln N N +lnp2⇡N + (41) ' − ··· Neglecting relatively small last term in (41) n n ln ! =lnN!+ N ln g N ln N + N (42) i i − i i i i=1 i=1 i X X X Search for maximum of target function using Lagrange multipliers ↴ @ @ @ N ln g N ln N + N + ↵ N + β N " =0 @N i i − i i i @N i @N i i j " i i i # j i ! j i ! X X X X X C.Luis B.-Champagne Anchordoqui 2 Thursday, October 30, 14 5 (46) BOLTZMANN DISTRIBUTION II In working out the derivativesOverview ☛ only contribution comes from terms with j = i Nj ln gj ln Nj +1+↵ + "j =0 (43) − −Nj For every energy level =0 # of particles per quantum state for equilibrium of the system | {z } Nj ↵+"j = e = fj("j) (44) gj Constants ↵ and β are related to physical properties of the system Multiply (43) by N j and sum over j ↴ N ln g N ln N + ↵ N + β N " =0 (45) j j − j j j j j j j j j X X ☟ X X N ln g N ln N = ↵N βU (46) j j − j j − − j j X X C.Luis B.-Champagne Anchordoqui 2 Thursday, October 30, 14 6 TEMPERATURE AS A LAGRANGE MULTIPLIER Substituting (42) ☛ lnOverview! =lnN!+N ↵N βU − − simplifying ☛ ln ! = C βU − Identification with Boltzmann entropy yields ☛ S = k ln ! = S kβU (47) 0 − dU PdV @S @S From classical theory ☛ dS = + = dU + dV dT T @U @V ✓ ◆V ✓ ◆U @S 1 giving ☛ = @U V T @S ✓ ◆ From (47) ☛ = kβ @U − ✓ ◆V 1 giving ☛ β = (48) −kT 2/3 Constancy of V is implied in (40) ☛ because ✏j V − / C.Luis B.-Champagne Anchordoqui 2 Thursday, October 30, 14 7 PARTITION FUNCTION ↵ "j /kT Substituting (48) into (44)Overview ☛ Nj = gje e− ↵ "j /kT so ↵ can be easily found from (40) ☛ N = Nj = e gje− j j X ☟ X N e↵ = "j /kT j gje− P "j /kT Nj Ne− Boltzmann distribution becomes ☛ fj = = "j /kT gj j gje− partition function (German Zustandssumme) ↴ P n ✏ /kT Z g e− j ⌘ j j=1 X C.Luis B.-Champagne Anchordoqui 2 Thursday, October 30, 14 8 ENERGY LEVELS CROWDED TOGETHER VERY CLOSELY degeneracy g j replaced Overviewby ⇢ ( " ) d " ☛ # of states in energy range ("," + d") correspondingly Nj replaced by N(")d" ☛ # of particles in range ("," + d") "/kT N(") Ne− f(") = "/kT ⌘ ⇢(") ⇢(")e− d" Continuous distribution function is analogousR to discrete case " /kT Z = ⇢(") e− j d" Z Occupation numbers are fully determined by temperature and volume Set of occupation numbers that maximize ! specify equilibrium macrostate Conclusion: Two states variables define a thermodynamic state exactly as in classical theory! 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Total # of microstates for allowable configuration !j = Nj!(gj Nj)! n ↴ g ! − ! (N ,N , ,N )= j FD 1 2 ··· n N !(g N )! j=1 j j j Y − C.Luis B.-Champagne Anchordoqui 2 Thursday, October 30, 14 10 FERMI-DIRAC & LAGRANGE ln !FD = Overviewln gi! ln Ni! ln(gi Ni)! − − − i i i usingX Stirling’sX X ln ! = [g ln g g N ln↴N + N (g N )ln(g N )+(g N )] FD i i − i − i i i − − i i − i i − i i X = [g ln g N ln N (g N )ln(g N )] i i − i i − i − i i − i i X using ☛ Ni = N Ni"i = U i i X ☟ X @ @ @ N ln N + (g N )ln(g N ) + ↵ N + β N " =0 −@N i i i − i i − i @N i @N i i j " i i # j i ! j i ! X X X X Note that ↴ @ g ln g =0 Nj (gj Nj) −@N i i and ln Nj +ln(gj Nj) − ( 1) = ↵ "j j i ! − − N − − (g N ) − − − X j j − j 1 1 − C.Luis B.-Champagne Anchordoqui |{z} | {z } 2 Thursday, October 30, 14 11 FERMI-DIRAC DISTRIBUTION Overview gj Nj 1 ln 1 = ↵ "j = N − − − ) g e ↵ "j +1 ✓ j ◆ j − − 1 once again ☛ β = −kT µ provisionally define ☛ ↵ = kT ↴ Nj 1 Fermi-Dirac distribution ☛ fj = = (" µ)/kT gj e j − +1 1 continuous energy spectrum ☛ f(")= (" µ)/kT e − +1 C.Luis B.-Champagne Anchordoqui 2 Thursday, October 30, 14 12 f +6 f $..%6)6 4)24&6 32.2.6 V]} c} +28V\2} r2} c2f} 56 !6 r/q/] [8/f 03f3\PBS'fBVS} V8} e>3} Vf>2\} (<`'T<3}QnLfAWKA3^}*u} cnQQAT<} Vq3_} f?3} c} 0V3cUf} sV`J} 9V`} f?Ac}0Acg`F*nz fAVT }!`VqAcAVT'KKu}s3}(ccV,B'f2} rBh>}f>3},>3QB,'L}XVf4SfB'M}f0BqB030}*v}!6 (T1}`2c3b3}:V`}K(f3`}f?3}Y?wcA,)K}CTi3`X`3f(fDVS}V9}f>Ec},VSS3,fBVS }$oc}s3}c5f} cV}f>(f}6*3,VP2c} %2} ,V\\2cXVS0BS=}T/W^@Zf8V]} Z8/f-VKfBK^V^Wf3U2]=x}WO/,ZS^EfBd} f f #f ""#f "#!&# f Vf} 'ABf 2L3Q2Tf'\x} O'QZ;+A/Wf V*/x} j>3} "'pNB} 3t.LocBVU} X]BU-BXL3} X>fVUc} 'R/f Z8/fEMWZfIMZ')A/f/c+/OZ:MIf$^Wfr6}I//-f WZZBcZB+Wf8V\} ',0'1",#6X'\fB{ ,L2c}'Sx} I^F)/Sf V8} r>B,>} -'S} V,,pXx} '} =Bq7S} [o'TfoP} cf'f3 } #n,>}