Minimum Bias Events at ATLAS

Transcription

1 Camille Bélanger-Champagne Lehman McGill College University City University of ew York Thermodynamics Charged Particle and Correlations Statistical Mechanics in Minimum Bias Events at ATLAS Statistical Mechanics I October 014 February 6 th 01, WPPC 01 Physics A statistical motivationview Microstate and macrostate Minbias event and track selection Thermodynamic probability Azimuthal Two-state correlation particles results Forward-Backward Many-state particles correlation results Quantum states and energy levels Density of states 1

2 Consider mechanical system A ball is released from top A STATISTICAL VIEW which cascades consecutively through Details of each ball s motion levels are governed by ewton s laws of motion To predict where any given ball will end up in bottom row is difficult because ball s trajectory depends sensitively on its initial conditions and may even be influenced by random vibrations of entire apparatus We therefore abandon all hope of integrating equations of motion and treat the system statistically We assume that ball moves to right with probability p and to left with probability If there is no bias in system p = q = 1 q =1 p

3 Position after X = OE-DIMESIOAL RADOM WALK steps may be written X j =+1 j j=1 if ball moves to right at level j = 1 if ball moves to left at level j j (1) A teach level probability for these two outcomes P = p,+1 + q, 1 = ( p if =+1 q if = 1 () Multivariate distribution for all steps P( 1,..., )= Y P ( j ) j=1 Our system is equivalent to a one-dimensional random walk Inebriated pedestrian on sidewalk taking steps to right and left at random After steps pedestrian s location is 3 X

4 VARIACE AD ROOT MEA SQUARE X hxi = h j=1 X hx i = h j=1 ji = h i = X =±1 X j 0 =1 j j 0 i = h i = X =±1 P( )=(p q) = (p 1) P ( )=(p + q) = Variance Var (X) =h( X ) i h(x hxi) i = hx i hxi Var(X) = 1 (p 1) = (4p 4p )=4p(1 p) =4pq Root mean square deviation X rms = p Var(X) hxi / and X rms / 1/ lim X rms/hxi vanishes as 1/!1 4

5 PROBABILITY DISTRIBUTIO P (X) = p R q L R (3) R/L are numbers of steps taken to right/left = R + L and X = R L There are many independent ways to take How many possibilities are there? R steps to right For each of these independent possibilities probability is R =! R! L! p R q L since ± X = R/L P (X) =! +X X!! p(+x)/ q ( X)/ (4) 5

6 is large STIRLIG S APPROXIMATIO!1 x X/ x may be considered continuous variable Consider the limit but with finite Using Stirling s asymptotic expansion ln! ' ln + O (ln ) n P (X) ' ln 1 apple 1 (1 + x)ln (1 + x) + 1 (1 + x) 1 (1 x) apple 1 ln (1 x) + 1 (1 x)+1 (1 + x)lnp + 1 (1 x)lnq apple apple 1+x 1+x 1 x 1 x 1+x = ln + ln + 1 x ln p + ln q Terms proportional to f(x) = ln h 1+x 1+x ln have all cancelled leaving quantity linear in ln P (X) = f(x)+o(ln ) + 1 x 1 ln 6 x i h 1+x ln p + 1 x (5) ln q i

7 PROBABILITY DISTRIBUTIO I LARGE LIMIT Since f 0 (x) =0 is by assumption large probability is dominated by minima of f 0 (x) = 1 ln q p.1+x 1 x P (X) =Ce f(x/) 1+x 1 x = p q f 00 (x) = ) x = p q 1 1 x f(x) f(x) =f( x) + 1 f 00 ( x)(x x) +... apple apple (x x) (X X) P (X) =C exp = C exp (6) 8pq 8pq X = hxi = (p q) = x Invoking Taylor s theorem C =(8 pq) 1/ 1X P (X) X= 1 determined by normalization condition Z +1 1 apple (X X) dx C exp 8pq = C p 8 pq 7

8 GAUSSIA DISTRIBUTIO Comparison of exact distribution of (4) (red squares) with the Gaussian distribution of (6) (blue line) 8

9 MICROSTATE From quantum mechanics follows that states of system do not change continuously but are quantized Huge number of discrete quantum states with corresponding energy values being the main parameter characterizing these states We formulate quantum statistics for systems of noninteracting particles then results can be generalized In absence of interactions each particle has its own set of quantum states and for identical particles these sets of states are identical Particles can be distributed over their own quantum states in a great number of different ways called realizations Each realization of this distribution is a microstate of the system Main assumption of statistical mechanics all microstates occur with same probability 9

10 MACROSTATE Information contained in microstates is excessive only meaningful information is how many particles i which specify a macrostate Each macrostate k can be realized by a very large number w k of microstates are in energy level thermodynamic probability i Probability of k-th macrostate is simply proportional to w k p k = w k X k w k (7) True probability is normalized 1= X k p k (8) A microstate is specified by number of particles in each energy state Degeneracy: In general more than 1 energy state (quantum state) for energy level 10

11 AVERAGE OCCUPATIO UMBER Both w k and p k depend on whole set of p k = p k ( 1,,...) (9) For isolated system number of particles and energy are conserved X j = j " j energy of particle in level j X j " j = U j umber of particles averaged over all macrostates k Pk (k) j! k Pk (k) j! k j = j P k! k = (k) j number of particles in microstate j corresponding to macrostate 11 j U = X k (k) j p k k (10) (11) (1)

12 MAXIMUM PROBABILITY For each macrostate thermodynamic probabilities differ by a large amount Macrostates having small p k practically never occur and state of system is dominated by macrostates with largest p k Consideration of particular models shows that: maximum of p k is very sharp for large number of particles whereas (k) is a smooth function of k j In this case (6) becomes k max j ' (k max) j X corresponds to maximum of k p k = (k max) j p k (13) For large dominating true probability p k is found by maximization with respect to all j obeying two constraints above 1

13 TWO-STATE PARTICLES A tossed coin can land in two positions head up or tail up Considering coin as a particle has two quantum states 1 head If coins are tossed this can be considered as a system of = 1 tail particles with quantum states each Microstates of system are specified by states occupied by each coin As each coin has states there are total = microstates (14) Macrostates of this system are defined by numbers of particles in each state: 1 and 13

14 POSSIBLE OUTCOMES OF COI-TOSSIG EXPERIMET =4 = 5X! k = 16 k=1 1 = 1 16 = [(4 1) + (3 4) + ( 6) + (1 4) + (0 1)] = 1 + =4= 14

15 THERMODYAMIC PROBABILITY We need a way of computing thermodynamic probability This formula can be derived Pick 1 particles to be in state 1 and all others will be in state How many ways are there to do this? ways to pick first coin leaving ways for the second factorial Binomial coefficient! = 1 Total number of ways of picking 1 heads is! ( 1 )!! ( 1) ! = 1 ( 1) ( )... ( 1 + 1) = order of coins is not important divide by without tabulating actual state of each coin ways for the third... 1! permutations (15) (16) (17) (16) isn t yet thermodynamical probability because contains multiple counting of same microstates 15! 1!( 1 )! = 1

16 (14) is satisfied Thermodynamic probability To prove that MOST PROBABLE MACROSTATE = X k! k = w X 1 =0 has a maximum at half of coins head and half of coins tail 1 = / is maximum of!! =! is symmetric around 1 = /! 1!( 1 )! = 1 = / rewrite (15) in terms of new variable! (/+p)!(/ p =0 p)! (17) p = 1 / Working out ratio! /±1! / = (/)!(/)! (/ + 1)!(/ 1)! = / / +1 < 1 16

17 STIRLIG FORMULA Analysis of expressions with large factorials is simplified by Stirling formula (13) becomes! ' p e (18)! ' p (/e) p (/+p)[(/+p)/e] /+p p (/ p)[(/ p)/e] / p = = r 1 q 1 ( p ) q 1 ( p ) (1 + p (/+p) /+p (/ p) / p! / )/+p (1 p )/ p (19) where! / ' r (0) is maximal value of the thermodynamic probability 17

18 MAXIMUM VALUE OF THERMODYAMIC PROBABILITY (16) can be expanded for Since p p enters both bases and exponents careful must be taken expand rather than itself Square root term in (16) discarded gives negligible contribution ln!! O(p / ) ln! ' ln! / ' ln! / + p ln + p " p 1+ p 1 p # p ln 1 " p p p 1 p # ' ln! / p =ln!? p p + p + p p + p p! '! / exp (1) () 18

19 BIOIAL DISTRIBUTIO FOR TWO-STATE PARTICLES! becomes very small if p 1 / that is much smaller than for large! is small in most of the interval 0 apple 1 apple and is sharply peaked near 1 = / w( 1) / w(!/! / / ) = 4 = 16 = 1000 ( 1 / ) /( / ) 19

20 THERMODYAMIC THERMODYAMIC PROBABILITY LIMIT AD ETROPY!1 For Main postulate of statistical mechanics observed macrostate is realized by greatest number of microstates most nearly random configuration (macrostate) is the one almost always occurs We have seen in thermodynamics that isolated system initially in a non-equilibrium state ordered regions evolves to almost the equilibrium never occurstate characterized by the maximal entropy S!!! max Entropy and thermodynamic is extremely probability small should compared be related to one being a monotonic function of the other Form of this function can be found We are led to a very important conclusion: f noticing that entropy is additive while thermodynamic probability is multiplicative If a system consists of two subsystems that weakly interact with each other total number of microstates is very nearly equal to maximum number S = S 1 + S whereas! =! 1! Boltzmann entropy S = k = B ln X! k 1 =! k! max k B = JK 1 (3) 0

21 THERMODYAMIC PROBABILITY AD ETROPY Main postulate of statistical mechanics observed macrostate is realized by greatest number of microstates We have seen in thermodynamics that isolated system initially in a non-equilibrium state evolves to equilibrium state characterized by maximal entropy Entropy S and thermodynamic probability! should be related one being a monotonic function of the other Form of this function can be found f noticing that entropy is additive while thermodynamic probability is multiplicative If system consists of two subsystems that weakly interact with each other S = S 1 + S whereas! =! 1! Boltzmann entropy S = k B ln! k B = JK 1 (4) 1

22 MAY-STATE PARTICLES umber of ways to distribute particles over boxes so that there are particles in i box umber of microstates in macrostate described by numbers i Thermodynamic temperature! = n! 1!!... n! This formula can be obtained by using (15) successively = ith! Q n i=1 i! (5) 1 1 umber of ways to put particles in box 1 and other in other boxes is given by (15) umber of ways to put particles in box is given by (15) with replacement! 1 and 1! 1 ( 1 )! =!( 1 )! = ( 1)!! 3! Iteration continues for box 3 until last box

23 ! = MAY-SATE PARTICLES (cont d) Resulting number of microstates is! 1!( 1 )! ( 1)!!( 1 )! ( 1! 3!( 1 3 )! ( n + n 1 + n )! n!( n 1 + n )! ( n 1 + n )! n 1! n! n! n!0! (6) All numerators in (6) except for first one and all second terms in denominators cancel each other so that (5) follows 3

24 .. STATIOARY SCHRODIGER EQUATIO In formalism of quantum mechanics = (r) (r) Ĥ = ˆp quantized states and their energies are solutions of eigenvalue problem complex function called wavefunction probability for a particle to be found near space point m + U(r) classical momentum 1= p Ĥ =" Z d 3 r (r) Hamilton operator or Hamiltonian is replaced by operator r ˆp (7) (8) d umber of measurements of total measurements in which particle is found in elementary volume d = (r) d 3 r d 3 r = dxdydz around r (9) 4

25 OE-DIMESIOAL RIGID BOX Consider particle in a one-dimensional rigid box momentum becomes ˆp = i~ d dx 0 apple x apple L (30) (7) takes form ~ m d (x) =" (x) dx (31) rewritten as d dx (x) +k (x) =0 (3) Solution of this equation satisfying boundary conditions (0) = (L) =0 n(x) =A sin(k n x) k n = n L n =1,, 3, (33) Eigenstates are labeled by index and normalization (8) yields n A = p /L 5

26 EERGY LEVELS OF A PARTICLE I A BOX Energy eigenvalues are " Energy is quadratic in momentum (as it should be) Energy levels are discrete because of quantization For very large box and energy levels become quasicontinuous Lowest-energy level with For a three-dimensional box with sides ground state energy levels parametrized by three quantum numbers ground state L!1 n =1 " nx,n y,n z = ~ n x m L x n x = n y = n z =1 We order states in increasing " n = ~k n m = ~ n ml p = ~k de Broglie relation L x,l y,l z + n y L y + n z L Z similar calculation yields " and number them by index j ground state being (34) (35) j =1 6! n x,n y,n z

27 DEGEERACY If L x = L y = L z = L Same value of " j = " nx,n y,n z = ~ ml (n x + n y + n z)= ~ ml n j " j can be realized for different sets of " j g j (n x,n y,n z ) (36) number of different sets having same is called degeneracy First three states of a three-dimensional infinite potential well 7

28 DESITY OF QUATUM STATES For systems of a large size and very finely quantized states (") dn " d" we define density of states as number of energy levels in interval dn " = (") d" (37) It is easily seen that dn " = 4 V m ~ 3/ p "d" (38) (") = V m 3/ p " ( ) ~ (39) 8