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Camille Bélanger-Champagne
McGill University
February 26
th
WNPPC 2012
Charged Particle Correlations in
Minimum Bias Events at ATLAS
● Physics motivation
● Minbias event and track selection
●
Azimuthal correlation results
● Forward-Backward correlation results
2 October 2014
Luis Anchordoqui
Lehman College
City University of New York
Thermodynamics and Statistical Mechanics
-
Equilibrium thermodynamics
Thermodynamics V
Kinetic Theory of Gases I
-
Phase transitions and phase equilibria
-
Basic assumptions of kinetic theory
C. B.-Champagne 2
Overview
Luis Anchordoqui
ENTROPY MAXIMUM
When 2 bodies with and are brought in thermal contact
T
1
T
2
Total entropy in process of equilibration increases
Investigating behavior of total entropy near its maximum
When equilibrium is reached should attain its maximal value
heat flows from hot to cold body so that temperatures equilibrate
This is ☛ Second Law of thermodynamics that follows from experiment
S = S
1
+ S
2
S
is subject of first part of this class
C. B.-Champagne 2
Overview
Luis Anchordoqui
C P
C V
C P
Extremum of corresponds to ☛ thermal equilibrium
THERMODYNAMIC STABILITY
Quadratic term in (159) shows that this extremum is a maximum
S
T 1
= T 2
would be unstable with respect to transfer of a small amount of energy
Small fluctuation would lead to avalanche-like further transfer of energy
in same direction since temperature on receiving side would decrease
is condition of thermodynamic stability
State with would be unstable ☛ heat flow from hot to cold body
☛ is also positive
C
V
C
V
T 1
= T C 2 V
For (^) < 0 ☛ initial state with
would lead to increase of T instead of equilibration
2
T
1
provided heat capacities are positive ☛
C. B.-Champagne 2
Overview
Luis Anchordoqui
(162) gives entropy decrease caused by deviation of system’s temperature
dT
If 2nd body is much larger than 1st one ☛ it can be considered as bath
Using and dropping index for bathed system
dS =
C
V
2 T
2
(dT )
2
MORE ON THERMODYNAMIC STABILITY
T
by a small amount from bath temperature
dU
1
= C
V 1
dT
1
and second fraction in (161) can be neglected
C
V 2
C
V 1
At equilibrium ☛ T (159) becomes
1
= T
2
= T
(160) complements condition (9) of mechanical stability
C
V
dS =
2 T
2
C
V 1
C
V 2
(dU
1
2
T
C. B.-Champagne 2
Overview
Luis Anchordoqui
Requiring that in (165) has three consequences:
AL QUE QUIERE CELESTE...QUE LE CUESTE
(i) Energy flows from hotter body to colder body
(ii) Body with a higher pressure expands
(iii) Particles diffuse from body with a higher chemical potential
The thermodynamic equilibrium is characterized by
to that with the lower
P
1
= P
2
(mechanical equilibrium)
T
1
= T
2
(thermal equilibrium)
1
2
(di↵usive equilibrium)
dS 0
at the expense of body with lower pressure
μ
C. B.-Champagne 2
Overview
Luis Anchordoqui
Thirdly ☛ diffusive stability condition should exist to the effect that
adding particles to the system at constant volume and internal energy
Total must have maximum with respect to all 3 variables at equilibrium
Investigating this requires adding second-order terms to (165)
Analysis is somewhat cumbersome but the results can be figured out
Firstly ☛ condition of thermal stability (160) should be satisfied
Secondly ☛ condition of mechanical stability (9) should be satisfied
COLLATERAL EFFECT
S
should increase its chemical potential
C. B.-Champagne 2
Overview
Luis Anchordoqui
THERMODYNAMIC PHASE DIAGRAMS
Typical thermodynamic phase diagram of system
Solid lines delineate boundaries between distinct thermodynamic phases
and thermodynamic potentials are singular
Along these lines we have coexistence of 2 phases
triple point ☛ 3 phase coexistence
temperature T
p
pressure
generic
substance
3
He
(a) (b) (c)
2.24: (a) Typical thermodynamic phase diagram of a si ngle component p-V - T
phase coexistence) and critical point. ( Source: Univ. of Helsinki.) Also shown:
e (c). What a difference a neutron makes! ( Source: Brittanica.)
P V T
C. B.-Champagne 2
Overview
Luis Anchordoqui
Equation of state for single component system may be written as
Single constraint on 3 state variables
f (P, V, T ) = 0
P-V-T SURFACES
This may in principle be inverted to yield
P = P (V, T ) V = V (T, P ) T = T (P, V )
f (P, V, T )
NSITIONS AND PHASE EQUILIBRIA 73
- T surface for a substance which contracts upon freezing. The red dot is the critical point and the
he critical isotherm. The yellow dot is the triple point at which there is three phase coexistence of
defines surface in space
{P, V, T }
C. B.-Champagne 2
Overview
Luis Anchordoqui CHAPTER 2. THERMODYNAMICS n of state for a substance which expands upon freezing, projected to the (v, T ) and (v, p) and ts an additional pressure of M g 20 kg × 9 .8 m/s 2 5 Real surfaces are much richer than that for ideal gas
P-V-T SURFACE OF REAL GASES
P v T
because real systems undergo phase transitions in which thermodynamic properties are singular
or discontinuous along certain curves on P v T surface
High temperature isotherms resemble those of ideal gas but as one cools below isotherms become singular
T
c
T = T
c
v = v
c which is critical molar volume P = P (v, T c @ isotherm )becomes perfectly horizontal @ Thursday, October 2, 14 13
C. B.-Champagne 2
Overview
Luis Anchordoqui
Figure 2.28: Projection of the p-v-T surface of Fig. 2.26 onto the (v, p) plane.
movie, directed by Sidney Lumet). My point here is that Hans’ crappy wooden skates can’t compare to the metal
ones, even though the surface melt between the ice and the air is the same. The skate blade material also makes a
difference, both for the interface energy and, perhaps more importantly, for the generation of friction as well.
2.12.4 Slow melting of ice : a quasistatic but irreversible process
Suppose we have an ice cube initially at temperature T < Θ ≡ 273 .15 K ( i.e. Θ = 0
◦
C) and we toss it into a pond
PROJECTION OF THE P-V-T SURFACE
lim
T !T C
T
= lim
T !T c
1
v
✓
@v
@P
◆
T
= 1
Below isotherms have a flat portion
corresponding to a two-phase region where liquid and vapor coexist
T
c
Thursday, October 2, 14 14
C. B.-Champagne 2
Overview
Luis Anchordoqui
SECOND ORDER PHASE TRANSITIONS
⌘
⌘ _ (T c
T )
> 0
Phases are described by order parameter
that is zero in one of phases and nonzero in other phase
Most of second-order transitions are controlled by temperature
High-temperature (symmetric) phase ☛
For
T < T
c
with
For chemical potential in form
there are boundaries between regions with different values of
μ(⌘)
which are associated to different values of
⌘
μ
Particles migrate from phase with higher to that with lower
spatial boundary between phases moves to reach equilibrium state
μ μ
Since can change continuously
it can adjust in uniform way without any phase boundaries
decreasing its chemical potential everywhere
⌘
C. B.-Champagne 2
Overview
Luis Anchordoqui
SMART MATERIALS
Tetragonal phase expands more rapidly in 2 directions than the 3rd one
becomes cubic phase that expands uniformly in 3 directions as is raised
There is no rearrangement of atoms at transition temperature
Ferromagnetic ordering below the Curie point
T
C. B.-Champagne 2
Overview
Luis Anchordoqui
BASIC ASSUMPTIONS OF THE MOLECULAR THEORY
Because of large number of particles
Characteristic distance between molecules largely exceeds molecular size
This assumption allows to consider gas as ideal
10
19
1 cm
3
In describing equilibrium properties of ideal gas
There are about molecules in at normal conditions
impacts of individual particles on walls merge into time-independent pressure
and typical radius of intermolecular forces
with internal energy dominated by kinetic energy of molecules
collisions between molecules can be neglected
Molecules are uniformly distributed within container
Directions of velocities of molecules are also uniformly distributed
C. B.-Champagne 2
Overview
Luis Anchordoqui
CHARACTERISTIC LENGTHS OF GAS
Concentration of molecules n is defined by
n ⌘
N
V
r 0
=
1
n
1
3
Characteristic distance between molecules can be estimated as
volume of container
total number of molecules
There are also long-range attractive forces between molecules
Let be radius of molecule ☛ (^) assumption (2) requires
r
0
a a^ ⌧^ r
0
but they are weak and do not essentially deviate molecular trajectories
if temperature is high enough and gas is ideal
