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Basics Experiment Statistics Summary
Search for the critical point of QCD
Sourendu Gupta
SERC School in EHEPDelhi University28 April, 2016
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
-
Basics Experiment Statistics Summary
Thermodynamics and statistical mechanics
Experiment sees thermodynamicsObserving nearly Gaussian fluctuationsDetailed test of thermalizationWhat could happen near the critical point
Error analysis for ratios
Summary
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
-
Basics Experiment Statistics Summary
Outline
Thermodynamics and statistical mechanics
Experiment sees thermodynamicsObserving nearly Gaussian fluctuationsDetailed test of thermalizationWhat could happen near the critical point
Error analysis for ratios
Summary
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary
Can experiments see the critical point?
T
µ
freezeout
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
-
Basics Experiment Statistics Summary
Can experiments see the critical point?
T
µ
freezeout
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
-
Basics Experiment Statistics Summary
Can experiments see the critical point?
T
µ
freezeout
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary
Thermodynamic fluctuations
Thermodynamics deals with matter in bulk: Avogadro number ofparticles (NA = 0.602× 1024). Short range interactions, so energyis additive. Particle number, charge, etc are all additive. All giverise to thermodynamic extensive quantities.
Equilibrium system: all extensive quantities have a precise andmeasurable value. Disturb the system a little and let it settle toequilibrium: new precise and measurable values.
Thermal fluctuations deduced by Boltzmann, Maxwell.Observation (1827) by Robert Brown of position fluctuation oftracer particles in water explained in 1905 by Einstein as due tothermal fluctuations of molecules.
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary
Grand canonical ensemble (GCE)
To study fluctuations, divide material into small part which isobserved (system) and large unobserved part (heat bath).Extensive thermodyamic variables of system fixed; but microscopicvariables can still fluctuate. Need a collection (ensemble) ofdifferent realizations of the microscopic states in thermodynamicequilibrium.
Grand canonical ensemble: all conserved quantities allowed tofluctuate by exchanging energy and material with the heat bath.Possible to realize by observing a small portion of the fireball byputting angular cuts.
Which ensemble is created in an experiment? Different realizationsof a system may be same system at different times: normal intabletop experiments. More natural in colliders to usemeasurements in different collisions to have many realizations ofthe same system.
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary
Fluctuations and correlations
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary
Fluctuations and correlations
ξ
For almost ideal gases ξ ≃ λ.
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary
Fluctuations and correlations
ξ
For almost ideal gases ξ ≃ λ.
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
-
Basics Experiment Statistics Summary
Fluctuations and correlations
ξ
For almost ideal gases ξ ≃ λ.
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
-
Basics Experiment Statistics Summary
Fluctuations and correlations
ξ
For almost ideal gases ξ ≃ λ.
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary
Gaussian fluctuations
When Vobs ≫ ξ3 then distribution of any extensive quantity isGaussian: central limit theorem. Mean value of Gaussian is theextensive thermodynamic equilibrium value. Thermodynamicquantities alone have little connection to microscopic properties.
Study of microscopic properties therefore requires study offluctuations. Width of Gaussian related to specific heat. Widthproportional to
√NA, therefore usually negligible, with respect to
mean.
When system is small enough then deviations from Gaussian arevisible. These are reflections of microscopic physics.
Central limit theorem fails when ξ ≃ L, i.e., ξ3 ≃ Vobs . Then nodifference between microscopic and macroscopic.
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary
Non-linear susceptibilities
Taylor expansion of the pressure in µB
P(T , µB +∆µB)/T4 =
∑
n
1
n!
[
χ(n)(T , µB)Tn−4]
(
∆µBT
)n
has Taylor coefficients called non-linear susceptibilities (NLS).When µB = 0 they can be computed directly on the lattice,otherwise reconstructed from such computations.Gavai, SG: 2003, 2010
Cumulants of the event-to-event distribution of baryon number aredirectly related to the NLS:
[B2] = T 3V
(
χ(2)
T 2
)
, [B3] = T 3V
(
χ(3)
T
)
, [B4] = T 3Vχ(4).
V unknown, can be removed by taking ratios.(SG: 2009)
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary
What are the cumulants?
[B] = 〈B〉[B2] = 〈(B − 〈B〉)2〉[B3] = 〈(B − 〈B〉)3〉
[B4] = 〈(B − 〈B〉)4〉 − 3[B2]2
[B5] = 〈(B − 〈B〉)5〉 − 10[B2] [B3][B6] = 〈(B − 〈B〉)6〉 − 15[B2] [B4]− 10[B3]2 + 30[B2]3
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary
Tests and assumptions
m1 :[B3]
[B2]=
χ(3)(T , µB)/T
χ(2)(T , µB)/T 2
m2 :[B4]
[B2]=
χ(4)(T , µB)
χ(2)(T , µB)/T 2
m3 :[B4]
[B3]=
χ(4)(T , µB)
χ(3)(T , µB)/T
Also for cumulants of electric charge, Q, and strangeness, S .
1. Two sides of the equation equal if there is thermal equilibriumand no other sources of fluctuations.
2. Right hand side computed in the grand canonical ensemble(GCE). Can observations simulate a grand canonicalensemble? What T and µB?
3. Why should hydrodynamics and diffusion be neglected?
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary
Thermodynamics and fluctuations
Observations
In a single heavy-ion collision, each conserved quantity (B , Q, S) isexactly constant when the full fireball is observed. In a small partof the fireball they fluctuate: from part to part and event to event.
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary
Thermodynamics and fluctuations
Observations
In a single heavy-ion collision, each conserved quantity (B , Q, S) isexactly constant when the full fireball is observed. In a small partof the fireball they fluctuate: from part to part and event to event.
Thermodynamics
If ξ3 ≪ Vobs ≪ Vfireball , then fluctuations can be explained in thegrand canonical ensemble: energy and B, Q, S allowed to fluctuatein one part by exchange with rest of fireball (diffusion: transport).
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary
Thermodynamics and fluctuations
Observations
In a single heavy-ion collision, each conserved quantity (B , Q, S) isexactly constant when the full fireball is observed. In a small partof the fireball they fluctuate: from part to part and event to event.
Thermodynamics
If ξ3 ≪ Vobs ≪ Vfireball , then fluctuations can be explained in thegrand canonical ensemble: energy and B, Q, S allowed to fluctuatein one part by exchange with rest of fireball (diffusion: transport).
Comparison
When Vobs ≪ Vfireball , Gaussian as Vobs/ξ3 → ∞. Finite sizeeffects are mainly controlled by NLS. Otherwise system is in thecritical regime.
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary
Ensuring control
Ensembles used in tabletop experiments keep intensive quantitiesfixed by controlling the heatbath: fixed temperature, fixedcomposition etc.
Collider experiments find this harder. Events may be classified byimpact parameter, but there may still be fluctuations in the totalenergy content of the fireball. Similarly, the chemical compositionof the fireball may need to be controlled. Compositional controlmay be harder to achieve at low densities.
LHC will play an important role in ensuring that these controlmechanisms are implemented properly.
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary
Why thermodynamics and not dynamics?
Chemical species may diffuse on the expanding background of thefireball, so why should we neglect diffusion and expansion?
First check whether the system size, L, is large enough comparedto the correlation length ξ: Knudsen’s number K = L/ξ. IfK ≫ 1, ie, L ≫ ξ then central limit theorem will apply.Next, compare the relative importance of diffusion and flowthrough a dimensionless number (Peclet’s number):
W = L2
tD =Lvflow
D =Kξvflowξcs
=Kvflow
cs= MK ,
where M is the Mach number. When W ≪ 1 diffusion dominates.Near chemical freeze-out K ≃ 1 and M ≃ 1, so flow dominates:fluctuations are frozen in. Detector observes thermodynamicfluctuations at chemical freeze out.(Bhalerao, SG: 2009)
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary
Peclet phenomenology
leng
ths
time
V
V1/3
1/3obs
ξ
fireball
λ p
Define Peclet length λp ≃ ξ/M where W = 1. λp remains fairlyconstant until time R0/cs then falls rapidly as flow becomes fully3d. So freezeout time is not very strongly dependent on rapiditywindow.
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality
Outline
Thermodynamics and statistical mechanics
Experiment sees thermodynamicsObserving nearly Gaussian fluctuationsDetailed test of thermalizationWhat could happen near the critical point
Error analysis for ratios
Summary
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
-
Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality
Outline
Thermodynamics and statistical mechanics
Experiment sees thermodynamicsObserving nearly Gaussian fluctuationsDetailed test of thermalizationWhat could happen near the critical point
Error analysis for ratios
Summary
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality
Event by event fluctuations
)pN∆Net Proton (-20 -10 0 10 20
Num
ber
of E
vent
s
1
10
210
310
410
510
610 0-5%30-40%70-80%
Au+Au 200 GeV
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Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality
Grand canonical thermodynamics?
When Vobs/ξ3 → ∞ and Vfireball/Vobs → ∞, then thermodynamics
in the grand canonical ensemble works; all distributions ofconserved quantities are Gaussian. For a Gaussian the onlynon-vanishing cumulants are the mean, [B], and the variance [B2].
Observation of any other non-vanishing cumulant [Bn] means thefluctuations are non-Gaussian. Is it because the system is not largeenough for thermodyamics to be applied? If it is a finite size effect,we are still ok.
If finite size effect, then trivial volume dependence of cumulants,i.e., all cumulants scale as V .
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality
Is the fluctuation Gaussian?
STAR: QM 2009, Knoxville
◮ Higher cumulantsscale down withlarger powers of V .
◮ Npart is a proxy forV .
◮ Cumulants observedto scale correctly asNpart .
◮ Can one connect toQCD?
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality
Same data, different plot
0
2
4
6
8
10
12
0 50 100 150 200 250 300 350 400
Cum
ulan
ts
N part
from STAR data [naive error propagation, no systematic errors]200 GeV Au Au
[B ]1
[B ]2
[B ]3
[B ]4
Central limit theorem follows. Scaling implies ξ3 ≪ Vobs at some√S .
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality
Same data, different plot
0
2
4
6
8
10
12
0 50 100 150 200 250 300 350 400
Cum
ulan
ts
N part
from STAR data [naive error propagation, no systematic errors]62.4 GeV Au Au
[B ]1
[B ]2
[B ]3
[B ]4
Central limit theorem follows. Scaling implies ξ3 ≪ Vobs at some√S .
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality
Same data, different plot
0
2
4
6
8
10
12
0 50 100 150 200 250 300 350 400
Cum
ulan
ts
N part
from STAR data [naive error propagation, no systematic errors]19.6 GeV Au Au
[B ]1
[B ]3
[B ]2
[B ]4
Central limit theorem follows. Scaling implies ξ3 ≪ Vobs at some√S .
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality
Very important deduction
The cumulants are linear in the volume, so the higher orderfluctuations will scale away in the limit when the volume becomesinfinite. The fluctuations will then become Gaussian.
This happens at almost all√S . This is consistent with
thermodynamics at an ordinary point.
This means that departure from Gaussian at one√S could be
significant, and should be investigated as a signal of a criticalpoint.
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
-
Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality
Outline
Thermodynamics and statistical mechanics
Experiment sees thermodynamicsObserving nearly Gaussian fluctuationsDetailed test of thermalizationWhat could happen near the critical point
Error analysis for ratios
Summary
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality
We are so lucky
0.7
0.8
0.9
1
1.1
0 1 2 3 4 5
T/T
c
/TBµ
Freezeout curve
10 GeV
20 GeV30 GeV Mumbai Nt=4
Mumbai Nt=6Mumbai Nt=8
Look at m1, m2 and m3 along the freeze-out curve.Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality
m1 = S/σ
0.001
0.01
0.1
1
10
1 10 100 1000 10000
m1
S (GeV)NN
Nt=4Nt=6
Gavai, SG: 2010
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality
m1 = S/σ
0.001
0.01
0.1
1
10
1 10 100 1000 10000
m1
S (GeV)NN
HRG
power law
Gavai, SG: 2010
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality
m2 = Kσ2
-1
0
1
2
3
4
5
1 10 100 1000 10000
m2
S (GeV)NN
Nt=4Nt=6
Gavai, SG: 2010
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality
m2 = Kσ2
-1
0
1
2
3
4
5
1 10 100 1000 10000
m2
S (GeV)NN
HRG
Gavai, SG: 2010
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality
m3 = Sσ/K
0.1
1
10
100
1000
1 10 100 1000 10000
m3
S (GeV)NN
Nt=4Nt=6
Filled circles: negative values
Gavai, SG: 2010
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality
m3 = Sσ/K
0.1
1
10
100
1000
1 10 100 1000 10000
m3
S (GeV)NN
power lawFilled circles: negative values
HRG
Gavai, SG: 2010
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality
Comparison with lattice predictions
STAR
Collaboration,1004.4959(2010)
Surprising agreementwith lattice QCD:
◮ impliesnon-thermalsources offluctuations arevery small
◮ T does not varyacross thefreezeout surface.
◮ tests QCD innon-perturbativethermal region
Gavai, SG, 1001.3796
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality
Experiment vs lattice QCD
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
-
Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality
Outline
Thermodynamics and statistical mechanics
Experiment sees thermodynamicsObserving nearly Gaussian fluctuationsDetailed test of thermalizationWhat could happen near the critical point
Error analysis for ratios
Summary
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality
Finite size effects damp divergences
1. System size limits correlation lengths near the critical point:ℓ ≃ ξ. The Knudsen number is never small near the CEP, socentral limit theorem will stop working. Check the scaling ofσ2, S and K and see whether there are violations of thecentral limit theorem. (SG: 2009)
2. As a result, the Peclet number need not be large, anddiffusion may play an important role even close to kineticfreeze-out. Then fluctuations of conserved quantities may notbe comparable to thermal equilibrium values at chemicalfreeze-out!
3. Another way of saying this is: critical divergences are limiteddue to finite size effects: no singularities, hence no directmeasurement of the critical exponents. System drops out ofequilibrium due to finite lifetime. (Stephanov: 2008; Berdnikov,Rajagopal: 1998)
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
-
Basics Experiment Statistics Summary
Outline
Thermodynamics and statistical mechanics
Experiment sees thermodynamicsObserving nearly Gaussian fluctuationsDetailed test of thermalizationWhat could happen near the critical point
Error analysis for ratios
Summary
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
-
Basics Experiment Statistics Summary
The meaning of errors and error propagation
One usually uses the central limit theorem to say that the error ina measurement, ∆x , is related to the variance of the measurement,σ2(x), and the number of measurements, N, by the formula
(∆x)2 =σ2(x)
N.
This works if the distribution of the estimates of the measurementsis Gaussian. The error is the 65% confidence limit.
It is very tempting to use standard error analysis for products andratios of numbers. If t = x/y and then one usually writes
(
∆t
t
)2
=
(
∆x
x
)2
+
(
∆y
y
)2
.
What does this mean?
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary
The error formula for ratios is meaningless
One can prove that the ratio (or product) of two Gaussiandistributed numbers is not Gaussian. In fact the ratio has infinitevariance! So the formula for error propagation gives some ∆twhich has no meaning as a confidence limit. In fact, since thevariance is infinite, such a ∆t is perfectly meaningless. (Geary: 1930)
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary
The error formula for ratios is meaningless
One can prove that the ratio (or product) of two Gaussiandistributed numbers is not Gaussian. In fact the ratio has infinitevariance! So the formula for error propagation gives some ∆twhich has no meaning as a confidence limit. In fact, since thevariance is infinite, such a ∆t is perfectly meaningless. (Geary: 1930)
You do not believe this?
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
-
Basics Experiment Statistics Summary
The error formula for ratios is meaningless
One can prove that the ratio (or product) of two Gaussiandistributed numbers is not Gaussian. In fact the ratio has infinitevariance! So the formula for error propagation gives some ∆twhich has no meaning as a confidence limit. In fact, since thevariance is infinite, such a ∆t is perfectly meaningless. (Geary: 1930)
You do not believe this?
Do a Monte Carlo experiment. Draw two numbers x and y eachfrom a Gaussian distribution. Take the ratio t. Repeat this processN times so you have N values of x , y and t. Now compute thevariance of each quantity. As you increase N you find σ2(x) andσ2(y) are finite numbers but σ2(t) grows without bound.
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
-
Basics Experiment Statistics Summary
The error formula for ratios is meaningless
One can prove that the ratio (or product) of two Gaussiandistributed numbers is not Gaussian. In fact the ratio has infinitevariance! So the formula for error propagation gives some ∆twhich has no meaning as a confidence limit. In fact, since thevariance is infinite, such a ∆t is perfectly meaningless. (Geary: 1930)
You do not believe this?
Do a Monte Carlo experiment. Draw two numbers x and y eachfrom a Gaussian distribution. Take the ratio t. Repeat this processN times so you have N values of x , y and t. Now compute thevariance of each quantity. As you increase N you find σ2(x) andσ2(y) are finite numbers but σ2(t) grows without bound.
Do not use the error propagation formula for ratios. Othermethods are possible.
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary
The variance of a ratio grows with statistics
10
100
1000
100 1000 10000
σ (ra
tio)/
N1/
2
N
Result of a simple Monte Carlo test.Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary
Alternatives to variance
half−widthmodehalf−width t
Measures of error exist even when variance is infinite.Phase transitions and statistics are very closely linked!
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
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Basics Experiment Statistics Summary
Alternatives to variance
33% of area 33% of area
mean t
Measures of error exist even when variance is infinite.Phase transitions and statistics are very closely linked!
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
-
Basics Experiment Statistics Summary
Outline
Thermodynamics and statistical mechanics
Experiment sees thermodynamicsObserving nearly Gaussian fluctuationsDetailed test of thermalizationWhat could happen near the critical point
Error analysis for ratios
Summary
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
-
Basics Experiment Statistics Summary
Summary
1. The ensemble of collider events can be used to simulate thegrand canonical ensemble provided
◮ the fireball is thermalized◮ one observes only a small part of the fireball◮ this part is much larger than the correlation volume
GCE fluctuations can then be measured through event byevent fluctuations.
2. Crucial check: cumulants of B , Q, and S , must scale linearlywith system volume (proxy: Npart).
3. If the ratios of cumulants equal the results of the lattice, thenthe fireball is in thermal equilibirum.
4. Near a critical point the fireball cannot come to thermalequilibrium.
Sourendu Gupta QGP@SERC-EHEP: Lecture 4
Thermodynamics and statistical mechanicsExperiment sees thermodynamicsObserving nearly Gaussian fluctuationsDetailed test of thermalizationWhat could happen near the critical point
Error analysis for ratiosSummary