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 (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 d = dx + dy =0 @x @y ☟ @f/@x @f/@y = @/@x @/@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 @ @ + ↵ + =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 + 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 are related to physical properties of the system Multiply (43) by N j and sum over j ↴ N ln g N ln N + ↵ N + 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 U (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 U simplifying ☛ ln ! = C U Identification with Boltzmann entropy yields ☛ S = k ln ! = S kU (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) ☛ = k @U ✓ ◆V 1 giving ☛ = (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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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 + 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 ☛ = 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

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=67f f NXNLYf>KfP_(L\`GfY\(]2Yfb?]9f2L2V5ef 
 new microstates obtained by shuffling lines and dots while keeping g j and N j fix ☟ binomial problem of coin-tossing experiment ☛ for j -th level ↴(Nj + gj 1)! Total # of microstates for allowable configuration !j = Nj!(gj 1)! n (N + g ↴1)! ! (N ,N , ,N )= j j BE 1 2 ··· n N !(g 1)! j=1 j j Y C.Luis B.-Champagne Anchordoqui 2 Thursday, October 30, 14 13 BOSE-EISNTEIN & LAGRANGE ln ! = ln(N + g 1)! ln N ! ln(g 1)! BE i iOverview i i i i i X X X using Stirling’s ↴ ln ! = [(N + g 1) ln(N + g 1) (N + g 1) N ln N BE i i i i i i i i i X + N (g 1) ln(g 1) + (g 1)] i i i i = [(N + g 1) ln(N + g 1) N ln N (g 1) ln(g 1)] i i i i i i i i i X using ☛ Ni = N Ni"i = U i i X ☟ X @ @ @ (N + g 1) ln(N + g 1) N ln N + ↵ N + 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 N + g 1 N ↴ln(N + g 1) + j j ln N j = ↵ " j j N + g 1 j N j j j j 1 1 C.Luis B.-Champagne Anchordoqui 2 Thursday, October 30, 14 | {z } |{z} 14 BOSE-EINSTEIN DISTRIBUTION

N + g +1 Overview ln j j = ↵ " N j ✓ j ◆ ☟ N 1 neglecting unity compared to ☛ j = g e ↵ "j 1 i 1 µ Using = and ↵ = kT kT ↴ N 1 Bose-Eisntein distribution ☛ j fj = = (" µ)/kT gi e j 1 1 continuous energy spectrum ☛ f(")= e(" µ)/kT 1

C.Luis B.-Champagne Anchordoqui 2 Thursday, October 30, 14 15 MAXWELL-BOLTZMANN STATISTICS Dilute gas ☛ for all energy levels occupations numbers are very small Overviewcompared with available number of quantum states Extremely unlikely more than 1 particle will occupy given state whether or not particles obey Pauli exclusion principle becomes irrelevant FD and BE statistics should be approximately identical in dilute gas limit

gj! (gj + Nj 1)! ! = !BE = FD N !(g N )! N !(g 1)! j j j j j j j Y Y For N g j ⌧ j ↴ g ! g (g 1)(g 2) (g N + 1)(g N )! j = j j j ··· j j j j gN (g N )! (g n )! ⇡ j j j j j (g + N 1)! = (g + N 1)(g + N 2) (g + N N )(g 1)! j j j j j j ··· j j j j Nj g (g 1)! N ⇡ j j g j ! ! j FD ⇡ BE ⇡ N ! j j Y C.Luis B.-Champagne Anchordoqui 2 Thursday, October 30, 14 16 MAXWELL-BOLTZMANN DISTRIBUTION

Difference between BoltzmannOverview and Maxwell-Boltzmann statistics Boltzmann statistics assumes distinguishable (localizable) particles gNj ! = j ! = N! ! MB N ! ) B MB j j Y Much larger Boltzmann probability includes permutation ☛ N! of N identifiable particles giving rise to additional microstates Distribution of particles among energy levels ☛ found using Lagrange multipliers

Result can be written down immediately observing that

!B and ! MB differ only by a constant

"j /kT Nj Ne Maxwell-Boltzmann distribution ☛ fj = (49) ⌘ gj Z

C.Luis B.-Champagne Anchordoqui 2 Thursday, October 30, 14 17 CONNECTION BETWEEN CLASSICAL AN STATISTICAL THERMODYNAMICS when matter is added orOverview taken from open system change of internal energy is↴ dU = TdS PdV + µdN define Helmholtz function F = U TS ☛ dF = SdT PdV + µdN @F µ = (50) @N TV ✓ ◆ N gj Nj N " /kT ! = , = e j Calculate S and F for MB statistics ☛ MB N ! g Z j j j Y S = k ln ! = k N ln g ln N ! 2 j j j 3 j j X X 4 5 = k N ln g N ln N + N 2 j j j j j3 j j j X X X 4 5 N = k N N ln j 2 j g 3 j j X ✓ ◆ 4 5 C.Luis B.-Champagne Anchordoqui 2 Thursday, October 30, 14 18 CONNECTION BETWEEN CLASSICAL AN STATISTICAL THERMODYNAMICS II Overview 1 S = k N ln N N +lnZ N + N " 2 j j kT j j3 j j j X X X U4 5 = + Nk(ln Z ln N + 1) T

F = U TS = NkT(ln Z ln N + 1) nkT Using (50) ☛ µ = kT(ln Z ln N + 1) + N N = kT ln Z ✓ ◆

N µ/kT Nj 1 = e fj (" µ)/kT Z ) ⌘ gj e j

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C.Luis B.-Champagne Anchordoqui 2 Thursday, October 30, 14 20 STATISTICAL THERMODYNAMICS OF IDEAL GAS Consider ideal gas in a sufficientlyOverview large container Levels of system are quantized so finely ☛ introduce density of states ⇢ ( " ) Number of particles in quantum states within energy interval d " # of particles in one state f ( " ) times # of states dn " in this energy interval With f ( " ) given by (49)

N " dN = f(")dn = e ⇢(✏)d" " " Z

For finely quantized levels ☛ replace summation by integration

... d"⇢ (")... ) i X Z " Partition function ☛ Z = d" ⇢ (") e Z C.Luis B.-Champagne Anchordoqui 2 Thursday, October 30, 14 21 MAXWELL SPEED DISTRIBUTION For quantum particles inOverview a rigid box V 2m 3/2 1 V 2m 3/2 p⇡ mk T 3/2 p " (51) Z = 2 d" "e = 2 2 3/2 = V 2 (2⇡) ~ 0 (2⇡) ~ 2 2⇡~ ⇣ ⌘ Z ⇣ ⌘ ⇣ ⌘ 2 Using " = mv / 2 and d " = mvdv number of particles in speed interval dv ☛ dN v = N(v)dv = Nf(v) dv ☟

2 3/2 3/2 1 " 1 2⇡~ V 2m m " f(v)= e ⇢ (") mv = 2 2 vmve Z V mkBT (2⇡) ~ 2 ⇣ ⌘ ⇣ ⌘ r m 3/2 mv2 = 4⇡v2exp 2⇡kBT 2kBT ⇣ ⌘ ⇣ ⌘ coincides with result from kinetic theory of gases

Plank’s constant ~ ☛ link to quantum mechanics disappeared from final result

C.Luis B.-Champagne Anchordoqui 2 Thursday, October 30, 14 22 EQUATION OF STATE OF IDEAL GAS Internal energy of ideal gas is its kinetic energy U = N"¯ 2 Overview "¯ = mv¯ /2 being average kinetic energy of an atom f From kinetic theory ☛ "¯ = k T 2 B f =3 corresponding to three translational degrees of freedom n n N " N @Z Same result obtained from (51) and U = " N = e i = i i Z Z @ i=1 i=1 X X @ ln Z @ m 3/2 3 @ 3 1 3 U = N = N V ln = N ln = N = NkBT @ @ 2⇡~2 2 @ 2 2 ⇣ ⌘ @F P is defined by thermodynamic formula ☛ P = @V T With help of (51) ⇣ ⌘ @ ln Z @ ln V Nk T P = Nk T = Nk T = B B @V B @V V

that amounts to equation of state of ideal gas PV = NkBT

C.Luis B.-Champagne Anchordoqui 2 Thursday, October 30, 14 23