Transcript Grand canonical ensemble

Cold atoms
Lecture 2.
4th October, 2006
Preliminary plan for the whole fall term
Lecture 1
…
Something about everything (see next slide)
The textbook version of BEC in extended systems
Sep 22
Lecture 2
…
thermodynamics, grand canonical ensemble, extended
gas: ODLRO, nature of the BE phase transition
Oct 4
Lecture 3
…
atomic clouds in the traps – theory. Confined independent bosons, what is BEC?, interactions, GP equation
Oct 18
Lecture 4
…
atomic clouds in the traps – experiment. Cooling,
trapping, monitoring and measuring the clouds
Lecture 5
…
Infinite systems: Bogolyubov-de Gennes theory, BEC
and symmetry breaking, coherent states
Lecture 6
…
Time dependent GP theory Finite systems: BEC theory
preserving the particle number
???
2
Back to Lecture 1.
A few important points from the last class
 Why extreme low temperature physics: new phenomena connected with
freezing out the thermal noise and with quantum coherence
 Example of NDR in rhodium: temperature record and nuclear antiferromagnetism. Details in two attached documents
 BEC in atomic clouds – a multidisciplinary task: cryogenics&LT physics,
atomic physics, laser physics& spectroscopy, …
 Crucial point: identical particles are quantum correlated. This changes
statistics even for non-interacting particles
 BEC in an extended system according to Einstein: below a critical
temperature a macroscopic occupancy of the lowest one-particle level
4
Bose-Einstein condensation:
elementary approach
L1: Ideal quantum gases at a finite temperature
n  e  (   ) Boltzmann distribution
high temperatures, dilute gases
fermions
N
FD
n 
bosons
N
1
n 
e  (   )  1
T 0
F
 1 ,1 ,
,1 , 0 ,
B
1
BE
n 
e  (  )  1
1
e   1
T 0
T 0
?
freezing out
 N , 0 ,0 ,
,0 ,
vac
6
L1: Ideal quantum gases at a finite temperature
n  e  (   ) Boltzmann distribution
high temperatures, dilute gases
fermions
N
FD
n 
bosons
N
1
n 
e  (   )  1
T 0
F
 1 ,1 ,
,1 , 0 ,
B
1
BE
n 
e  (  )  1
1
e   1
T 0
T 0
?
freezing out
 N , 0 ,0 ,
,0 ,
vac
7
L1: Ideal quantum gases at a finite temperature
n  e  (   ) Boltzmann distribution
high temperatures, dilute gases
fermions
N
FD
n 
bosons
N
1
n 
e  (   )  1
T 0
1
BE
n 
e  (  )  1
T 0
1
e   1
T 0
out
Equation for the chemical potential closes
problem:
? the equilibriumfreezing
F
 1 ,1 ,
,1 , 0 ,
B
 N , 0 ,0 ,
,0 ,
vac
8
L1: Ideal quantum gases at a finite temperature
n  e  (   ) Boltzmann distribution
high temperatures, dilute gases
fermions
N
FD
n 
F
1
e  (   )  1
bosons
N
n 
1
e  (  )  1
BE
n 
1
e   1
for the chemical potential T
closes
T Equation
0
 0the equilibrium problem:
T 0
1
out
N  N (T ,  )   n(?
 j )    ( j  freezing
)
1
j
j e
 1 , 1 , ,1 , 0 ,
B  N , 0 ,0 , ,0 ,
vac
9
L1: A gas with a fixed average number of atoms
Ideal boson gas (macroscopic system)
p2
atoms: mass m, dispersion law  ( p) 
2m
system as a whole:
volume V, particle number N, density n=N/V, temperature T.
Equation for the chemical potential closes the equilibrium problem:
N  N (T ,  )   n( j )  
j
j
1
e
 ( j   )
1
Always  < 0. At high temperatures, in the thermodynamic limit,
the continuum approximation can be used:

N  V  d
0
1
e  (   )  1
The balance equation for
D ( )  N (T ,  )
h2
T  TC 
4 mk B
 N 


2,612
V


T  TC is
1

1
N  N (T ,  )   (   )
 V  d   (  0 )
D ( )
0
e
1
e
1
0
10
2
3
L1: Condensate concentration
3
2
T 
N (TC ,0)
nG 
 BT = n   for T  TC
V
 TC 
3
3


2
 T 2


T
n  nG  nBE  n    n 1    
  TC  
 TC 


3
2
f
r
a
c
t
i
o
n
GAS
T / TC
11
L1: Closer look at BEC
• Thermodynamically, this is a real phase transition, although unsual
• Pure quantum effect
• There are no real forces acting between the bosons, but there IS a real
correlation in their motion caused by their identity (symmetrical wave functions)
• BEC has been so difficult to observe, because other (classical G/L or G/S)
phase transitions set on much earlier
• BEC is a "condensation in the momentum space", unlike the usual liquefaction
of classical gases, which gives rise to droplets in the coordinate space.
• This is somewhat doubtful, especially now, that the best observed BEC takes
place in traps, where the atoms are significantly localized
• What is valid on the "momentum condensation": BEC gives rise to quantum
coherence between very distant places, just like the usual plane wave
• BEC is a macroscopic quantum phenomenon in two respects:
 it leads to a correlation between a macroscopic fraction of atoms
 the resulting coherence pervades the whole macroscopic sample
12
L1: Closer look at BEC
• Thermodynamically, this is a real phase transition, although unsual
• Pure quantum effect
• There are no real forces acting between the bosons, but there IS a real
correlation in their motion caused by their identity (symmetrical wave functions)
• BEC has been so difficult to observe, because other (classical G/L or G/S)
phase transitions set on much earlier
• BEC is a "condensation in the momentum space", unlike the usual liquefaction
of classical gases, which gives rise to droplets in the coordinate space.
• This is somewhat doubtful, especially now, that the best observed BEC takes
place in traps, where the atoms are significantly localized
• What is valid on the "momentum condensation": BEC gives rise to quantum
coherence between very distant places, just like the usual plane wave
• BEC is a macroscopic quantum phenomenon in two respects:
 it leads to a correlation between a macroscopic fraction of atoms
 the resulting coherence pervades the whole macroscopic sample
13
L1: Closer look at BEC
• Thermodynamically, this is a real phase transition, although unsual
• Pure quantum effect
• There are no real forces acting between the bosons, but there IS a real
correlation in their motion caused by their identity (symmetrical wave functions)
• BEC has been so difficult to observe, because other (classical G/L or G/S)
phase transitions set on much earlier
• BEC is a "condensation in the momentum space", unlike the usual liquefaction
of classical gases, which gives rise to droplets in the coordinate space.
• This is somewhat doubtful, especially now, that the best observed BEC takes
place in traps, where the atoms are significantly localized
• What is valid on the "momentum condensation": BEC gives rise to quantum
coherence between very distant places, just like the usual plane wave
• BEC is a macroscopic quantum phenomenon in two respects:
 it leads to a correlation between a macroscopic fraction of atoms
 the resulting coherence pervades the whole macroscopic sample
14
Off-Diagonal Long Range Order
Analysis on the one-particle level
Coherence in BEC: ODLRO
Off-Diagonal Long Range Order
Without field-theoretical means, the coherence of the condensate may be
studied using the one-particle density matrix.
Definition of OPDM for non-interacting particles: Take an additive
observable, like local density, or current density. Its average value for the
whole assembly of atoms in a given equilibrium state:
X    X  n
double average, quantum and thermal

=   X     n

 

insert unit operator




   | X 
n  X  change the summation order
define the one-particle density matrix

 Tr  X
 =   n 

16
Coherence in BEC: ODLRO
Without field-theoretical means, the coherence of the condensate may be
studied using the one-particle density matrix.
Definition of OPDM for non-interacting particles: Take an additive
observable, like local density, or current density. Its average value for the
whole assembly of atoms in a given equilibrium state:
X    X  n
double average, quantum and thermal

=   X     n

 

insert unit operator




   | X 
n  X  change the summation order
define the one-particle density matrix

 Tr  X
 =   n 

17
Coherence in BEC: ODLRO
Without field-theoretical means, the coherence of the condensate may be
studied using the one-particle density matrix.
Definition of OPDM for non-interacting particles: Take an additive
observable, like local density, or current density. Its average value for the
whole assembly of atoms in a given equilibrium state:
X    X  n
double average, quantum and thermal

=   X     n

 

insert unit operator




   | X 
n  X  change the summation order
define the one-particle density matrix

 Tr  X
 =   n 

18
Coherence in BEC: ODLRO
Without field-theoretical means, the coherence of the condensate may be
studied using the one-particle density matrix.
Definition of OPDM for non-interacting particles: Take an additive
observable, like local density, or current density. Its average value for the
whole assembly of atoms in a given equilibrium state:
X    X  n
double average, quantum and thermal

=   X     n

 

insert unit operator




   | X 
n  X  change the summation order
define the one-particle density matrix

 Tr  X
 =   n 

19
Coherence in BEC: ODLRO
Without field-theoretical means, the coherence of the condensate may be
studied using the one-particle density matrix.
Definition of OPDM for non-interacting particles: Take an additive
observable, like local density, or current density. Its average value for the
whole assembly of atoms in a given equilibrium state:
X    X  n
double average, quantum and thermal

=   X     n

 

insert unit operator




   | X 
n  X  change the summation order
define the one-particle density matrix

 Tr  X
 =   n 

20
OPDM for homogeneous systems
In coordinate representation
 (r , r ')   r k nk k r'
k

1
ei k ( r  r ') nk

V k
• depends only on the relative position (transl. invariance)
• Fourier transform of the occupation numbers
• isotropic … provided thermodynamic limit is allowed
• in systems without condensate, the momentum distribution is smooth and
the density matrix has a finite range.
CONDENSATE
lowest orbital with
k0
21
OPDM for homogeneous systems: ODLRO

CONDENSATE
lowest orbital with
1
3
k0  O(V )  0
1 i k0 ( r  r ')
1
 (r  r ')  e
n0   ei k ( r  r ') nk
V
V k  k0
coherent across
the sample
  BE (r  r ')
FT of a smooth function
has a finite range
 G (r  r ')
22
OPDM for homogeneous systems: ODLRO

CONDENSATE
lowest orbital with
1
3
k0  O(V )  0
1 i k0 ( r  r ')
1
 (r  r ')  e
n0   ei k ( r  r ') nk
V
V k  k0
coherent across
the sample
  BE (r  r ')
FT of a smooth function
has a finite range
 G (r  r ')
DIAGONAL ELEMENT r = r'
 (0 )   BE (0 )
 nBE
 G (0 )
+ nG
23
OPDM for homogeneous systems: ODLRO

CONDENSATE
lowest orbital with
1
3
k0  O(V )  0
1 i k0 ( r  r ')
1
 (r  r ')  e
n0   ei k ( r  r ') nk
V
V k  k0
coherent across
the sample
  BE (r  r ')
FT of a smooth function
has a finite range
 G (r  r ')
DIAGONAL ELEMENT r = r'
 (0 )   BE (0 )
 nBE
 G (0 )
+ nG
DISTANT OFF-DIAGONAL ELEMENT | r - r' |
|r  r '|
 BE (r  r ') 
 nBE
|r  r '|
G (r  r ') 
 0
|r  r '|

 nBE
 (r  r ')
Off-Diagonal Long Range Order
ODLRO
24
From OPDM towards the macroscopic wave function

CONDENSATE
k0  O(V )  0
lowest orbital with
 (r  r ') 
1
3
1 i k0 ( r  r ')
1
e
n0   ei k ( r  r ') nk
V
V k  k0
coherent across
the sample
FT of a smooth function
has a finite range
 (r )  (r' )

dyadic
1
i k ( r  r ')
e
nk

V k  k0
MACROSCOPIC WAVE FUNCTION
 (r )  nBE  ei( k r + ) , 
0
an arbitrary phase
• expresses ODLRO in the density matrix
• measures the condensate density
• appears like a pure state in the density matrix, but macroscopic
• expresses the notion that the condensate atoms are in the same state
• is the order parameter for the BEC transition
25
From OPDM towards the macroscopic wave function

CONDENSATE
k0  O(V )  0
lowest orbital with
 (r  r ') 
1
3
1 i k0 ( r  r ')
1
e
n0   ei k ( r  r ') nk
V
V k  k0
coherent across
the sample
FT of a smooth function
has a finite range
 (r )  (r' )

dyadic
1
i k ( r  r ')
e
nk

V k  k0
MACROSCOPIC WAVE FUNCTION
 (r )  nBE  ei( k r + ) , 
0
an arbitrary phase
• expresses ODLRO in the density matrix
• measures the condensate density


• appears like a pure state in the density matrix, but macroscopic

• expresses the notion that the condensate atoms are in the same state
• is the order parameter for the BEC transition
26
From OPDM towards the macroscopic wave function

CONDENSATE
k0  O(V )  0
lowest orbital with
 (r  r ') 
1
3
1 i k0 ( r  r ')
1
e
n0   ei k ( r  r ') nk
V
V k  k0
coherent across
the sample
FT of a smooth function
has a finite range
 (r )  (r' )

dyadic
1
i k ( r  r ')
e
nk

V k  k0
MACROSCOPIC WAVE FUNCTION
 (r )  nBE  ei( k r + ) , 
0
an arbitrary phase
• expresses ODLRO in the density matrix
• measures the condensate density

? why bother?

• appears like a pure state in the density matrix, but macroscopic

• expresses the notion that the condensate atoms are in the same state
• is the order parameter for the BEC transition
? what is it?
? how?
27
F.Laloë: Do we really understand Quantum mechanics,
Am.J.Phys.69, 655 (2001)
28
Thermodynamics of BEC
Capsule on thermodynamics
Grand canonical ensemble
Thermodynamic functions of an ideal gas
BEC in an ideal gas
Comparison with real gases/vdW equation
Homogeneous one component phase:
boundary conditions (environment) and state variables
T P  dual variables, intensities
"intensive"
SV
S V N isolated, conservative
open S V 
S P N isobaric
isothermal T V N
S P  not in use
grand T V 
T P N isothermal-isobaric
not in use T P 
30
Homogeneous one component phase:
boundary conditions (environment) and state variables
T P  dual variables, intensities
"intensive"
SV
S V N isolated, conservative
The important four
isothermal T V N
grand T V 
T P N isothermal-isobaric
31
Thermodynamic potentials and all that
Basic thermodynamic identity (for equilibria)
dU  T d S  P dV   d N
For an isolated system,
 
T  T ( S ,V , N )  U
S
, etc.
V ,N
For an isothermic system, the independent variables are T, V, N.
The change of variables is achieved by means of the Legendre
transformation. Define Free Energy
F  U  TS , U  U (T ,V , N ), S  S (T ,V , N ),
d F  S d T  P dV   d N
 
S  F
T
, etc.
V ,N
32
Thermodynamic potentials and all that
Basic thermodynamic identity (for equilibria)
dU  T d S  P dV   d N
For an isolated system,
 
T  T ( S ,V , N )  U
S
, etc.
V ,N
For an isothermic system, the independent variables are T, V, N.
The change of variables is achieved by means of the Legendre
transformation. Define Free Energy
F  U  TS , U  U (T ,V , N ), S  S (T ,V , N ),
d F  S d T  P dV   d N
 
S  F
T
V ,N
, etc.
New variables:
perform the substitution
everywhere; this shows in
the Maxwell identities
(partial derivatives)
33
Thermodynamic potentials and all that
Basic thermodynamic identity (for equilibria)
dU  T d S  P dV   d N
For an isolated system,
 
T  T ( S ,V , N )  U
S
, etc.
V ,N
For an isothermic system, the independent variables are T, V, N.
The change of variables is achieved by means of the Legendre
transformation. Define Free Energy
F  U  TS , U  U (T ,V , N ), S  S (T ,V , N ),
d F  S d T  P dV   d N
 
S  F
, etc.
Legendre transformation:
T V , N
subtract the relevant
product of
conjugate (dual) variables
New variables:
perform the substitution
everywhere; this shows in
the Maxwell identities
(partial derivatives)
34
Thermodynamic potentials and all that
Basic thermodynamic identity (for equilibria)
dU  T d S  P dV   d N
For an isolated system,
 
T  T ( S ,V , N )  U
S
, etc.
V ,N
For an isothermic system, the independent variables are T, S, V.
The change of variables is achieved by means of the Legendre
transformation. Define Free Energy
F  U  TS , U  U (T ,V , N ), S  S (T ,V , N ),
d F  S d T  P dV   d N
 
S  F
T
, etc.
V ,N
35
A table
isolated system
U
internal energy
isothermic system
canonical ensemble
F  U  TS
free energy
isothermic-isobaric system
free enthalpy
isothermic open system
grand potential
How comes
Thus,
S ,V , N
dU T dS  P dV   dN
microcanonical ensemble
T ,V , N
d F S dT  P dV   dN
T , P, N
  U  TS  PV   N d  S dT  V d P  dN
isothermic-isobaric ensemble
T ,V , 
  U TS  N   PV d   S dT  P dV  Nd
  PV ? 
grand canonical ensemble
is additive, V is the only additive independent variable.
  
   (T ,  )  V ,  (T ,  )  
  P
 V T , 
Similar consideration holds for
   (T , P)  N
36
Digression: which environment to choose?
THE ENVIRONMENT IN THE THEORY SHOULD CORRESPOND
TO THE EXPERIMENTAL CONDITIONS
… a truism difficult to satisfy
 For large systems, this is not so sensitive for two reasons
•
System serves as a thermal bath or particle reservoir all by itself
•
Relative fluctuations (distinguishing mark) are negligible

Adiabatic system
SB heat exchange – the slowest
S
B
Real system
medium fast
process
• temperature lag
• interface layer
Isothermal system
the fastest
S
B
 Atoms in a trap: ideal model … isolated. In fact: unceasing energy exchange
(laser cooling). A small number of atoms may be kept (one to, say, 40). With
107, they form a bath already. Besides, they are cooled by evaporation and
they form an open (albeit non-equilibrium) system.
 Sometime, N =const. crucial (persistent currents in non-SC mesoscopic rings)
37
Grand canonical ensemble
Definition following Gibbs
General treatment for independent particles
Thermodynamic functions of an ideal gas
BEC in an ideal gas
Comparison with real gases/vdW equations
Grand canonical ensemble - definition
Grand canonical ensemble admits both energy and particle number exchange
between the system and its environment.
The statistical operator (many body density matrix)
̂
acts in the Fock space
External variables are T ,V ,  . They are specified by the conditions
ˆ ˆ  U
Hˆ  Tr 
V  sharp
ˆ Nˆ  N
Nˆ  Tr 
ˆ ln ˆ  max
S  kB Tr 
Grand canonical statistical operator has the Gibbs' form
ˆ
ˆ
ˆ  Z 1 e  ( H   N )
ˆ
ˆ
Z (  ,  ,V )  Tr e   ( H   N )  e   (  ,  ,V ) statistical sum
 (  , ,V )  kBT ln Z (  ,  ,V )
grand canonical potential
39
Grand canonical ensemble – general definition
Grand canonical ensemble admits both energy and particle number exchange
between the system and its environment.
The statistical operator (many body density matrix)
̂
acts in the Fock space
External variables are T ,V ,  . They are specified by the conditions
ˆ ˆ  U
Hˆ  Tr 
V  sharp
ˆ Nˆ  N
Nˆ  Tr 
ˆ ln ˆ  max
S  kB Tr 
Grand canonical statistical operator has the Gibbs' form
ˆ
ˆ
ˆ  Z 1 e  ( H   N )
ˆ
ˆ
Z (  ,  ,V )  Tr e   ( H   N )  e   (  ,  ,V ) statistical sum
 (  , ,V )  kBT ln Z (  ,  ,V )
grand canonical potential
volume … for an extended homogeneous system
V
… generic for generalized coordinates of external fields whose change
40
is connected with the mechanical work done by the system
Grand canonical ensemble – finite systems
Grand canonical ensemble admits both energy and particle number exchange
between the system and its environment.
The statistical operator (many body density matrix)
̂
acts in the Fock space
External variables areT ,   ,. They are specified by the conditions
ˆ ˆ  U
Hˆ  Tr 
  
sharp
ˆ Nˆ  N
Nˆ  Tr 
ˆ ln ˆ  max
S  kB Tr 
In fact, the external variables T ,  are properties of the bath/reservoir.
They are imposed on the system and have meaning even for small systems .
.
one particle has a temperature – that of the bath
41
Fluctuations I. – global quantities
Fluctuations of the total number of particles around the mean value
First derivative of the grand potential
  ( Hˆ   Nˆ )
ˆ



Tr
N
e
ˆ
ˆ

  Nˆ
 kBT ln Z   kBT ln Tr e   ( H   N )  
ˆ   Nˆ )


(
H



Tr e
Second derivative of the grand potential
ˆ
ˆ
 2
 ˆ
 Tr Nˆ e   ( H   N )

N 

2
  ( Hˆ   Nˆ )


 Tr e
ˆ
ˆ
ˆ
ˆ
Tr Nˆ 2 e   ( H   N ) (Tr Nˆ e   ( H   N ) ) 2
ˆ 2  Nˆ




N
ˆ
ˆ
ˆ
ˆ
Tr e   ( H   N )
(Tr e   ( H   N ) ) 2
2
Final estimate for the relative fluctuation
Nˆ
2
 Nˆ
Nˆ
2
2
 ˆ
N


O Nˆ
2
Nˆ

1

42
Useful identities

Calculate T in two ways.




 kBT ln Z   kB ln Z  kBT ln Z
T
T
T
1 
 1 
  kB ln Z  kBT
Z   kB ln Z  kBT

Z
Z T
T Z 


1

  Hˆ   Nˆ

T
1
1
ˆ ( Hˆ   Nˆ )  
   Tr 


T
T
 kB Tr 
ˆ  ln Z  1 ( Hˆ   Nˆ )    S

kBT



  
  Hˆ  TS   Nˆ , S   

 T   ,V
ln ̂
43
Grand canonical statistical sum for independent bosons
Recall
ˆ
ˆ
Z (  ,  ,V )  Tr e   ( H   N )  e   (  ,  ,V ) statistical sum
= e  ( E

e
 N )
   (   ) n

n 

TRICK!!    e

n
  (   )
eigenstate label


n
 e
n 
  (   )




n
 n  with
 n N

up to here trivial
1
1  e   (   )
44
Grand canonical statistical sum for independent bosons
Recall
ˆ
ˆ
Z (  ,  ,V )  Tr e   ( H   N )  e   (  ,  ,V ) statistical sum
= e  ( E

e
 N )
   (   ) n

n 

TRICK!!    e

n
  (   )
eigenstate label


n
 e
n 
  (   )




n
 n  with
 n N

up to here trivial
1
1  e   (   )
45
Grand canonical statistical sum for independent bosons
Recall
ˆ
ˆ
Z (  ,  ,V )  Tr e   ( H   N )  e   (  ,  ,V ) statistical sum
= e  ( E

e
 N )
   (   ) n

n 

TRICK!!    e

Z (  ,  ,V )  

  (   )
eigenstate label


n
 e
n 
1
1 e
  (    )



n


  (   )

n
 n  with
 n N

up to here trivial
1
1  e   (   )
1
e

z
activity
fugacity
1  z e  
46
Grand canonical statistical sum for independent bosons
Recall
ˆ
ˆ
Z (  ,  ,V )  Tr e   ( H   N )  e   (  ,  ,V ) statistical sum
= e  ( E

e
 N )
   (   ) n

n 

TRICK!!    e

Z (  ,  ,V )  

  (   )
eigenstate label


n
 e
n 
1
1 e
  (    )



n


  (   )

n
 n  with
 n N

up to here trivial
1
1  e   (   )
1
e

z
activity
fugacity
1  z e  
47
Grand canonical statistical sum for independent bosons
Recall
ˆ
ˆ
Z (  ,  ,V )  Tr e   ( H   N )  e   (  ,  ,V ) statistical sum
= e  ( E

e
 N )
   (   ) n

n 

TRICK!!    e

Z (  ,  ,V )  

  (   )
eigenstate label


n
 e
n 
1
1 e
  (    )



n


  (   )

n
 n  with
 n N

up to here trivial
1
1  e   (   )
e

z
1
activity
fugacity
1  z e  
 (  , ,V )  kBT ln Z (  ,  ,V )


 +k T  ln 1  z e

grand canonical potential
= +kBT  ln 1  e   (   )

  
B

48
Grand canonical statistical sum for independent bosons
Recall
ˆ
ˆ
Z (  ,  ,V )  Tr e   ( H   N )  e   (  ,  ,V ) statistical sum
= e  ( E

e
 N )
   (   ) n

n 

TRICK!!    e

Z (  ,  ,V )  

  (   )
eigenstate label


n
 e
n 
1
1 e
  (    )



n


1  e   (   )

up to here trivial
e

z
1
activity
fugacity
1  z e  


 +k T  ln 1  z e

= +kBT  ln 1  e   (   )

  


n
 n N
1
 (  , ,V )  kBT ln Z (  ,  ,V )
B
  (   )
 n  with
grand canonical potential
valid for
- extended "ïnfinite" gas
- parabolic traps
just the same
49
Thermodynamic functions for an extended Bose gas
For Born-Karman periodic boundary conditions, the lowest level is
 (k  0 )  0
Its contribution has to be singled out, like before:

 
  (T ,  ,V )  ln(1  z )  V  d  ln(1  z e ) D ( )

N (T ,  ,V ) 
z
1
 V  d  1 
D ( )
1 z
z e 1
0
U (T ,  ,V )  V  d 
0
z
activity
fugacity
3D DOS
0



e


1

z e 1
3
2
 2m 
D ( )  2  2   
h 
D ( )
HOW TO PROCEED
Start from . This we did already. Below TC,
z
 N BE  z  1  0, ln 1  z   ln N BE  O  ln V 
1 z
Thus, the singular term in  is negligible for V   , there is none in 
50
Thermodynamic functions for an extended Bose gas
The following equation of state results
* P(T ,  ,V )  2  2mh
3
2 2


kBT  d  ln(1  z e   )  
1
2
0
Integrating by parts, Eqs.  and  are found as nearly identical
* PV  23 U
This is an extension of the result known for classical gases
Series expansion
P  2  2mh
3
2 2


kBT  d  ln(1  z e
 
1
2
)  
0

1
1
  n
2
d

(
z
e
)



0
n


**
1
zn
 n  d  (e  )n  2
0
P
  3 g5 / 2 ( z )
kBT
  h(2 mkBT )

1
2

g5 / 2 ( z )  
1
zp
p5 / 2
51
Thermodynamic functions for an extended Bose gas
Similar expansions:
N
z

  3 g3 / 2 ( z )
V 1 z
U 3
 kBT   3 g5 / 2 ( z )
V 2
  h(2 mkBT )

1
2

g3 / 2 ( z )  
1
  h(2 mkBT )

1
2

g5 / 2 ( z )  
1
zp
p3 / 2
zp
p5 / 2
Suitable for numerical studies above the critical temperature
Results simplify in the condensation region:
z  1,  3     kBT 
3/ 2
and all temperature dependences are explicit.
N N BE
3/ 2

  g3 / 2 (1)   kBT 
V
V
U 3
5/ 2
  g5 / 2 (1)   kBT 
V 2
P   g5 / 2 (1)   kBT 
5/ 2
52
Specific heat
CV 
 U / N 
T
V ,N
A weak singularity …
what decides is the coexistence of two phases
53
Isotherms in the P-V plane
For a fixed temperature, the specific
volume V / N can be arbitrarily small.
By contrast, the pressure is volume
independent …typical for condensation

P
vc  AT
3 / 2
T   vc / A 
2 / 3
P  BT 5 / 2
Pc  BA5 / 3  vc 5 / 3
v 
vc
N N BE
3/ 2

  g3 / 2 (1)   kBT 
V
V
P   g5 / 2 (1)   kBT 
5/ 2
54
Compare with condensation of a real gas
CO2
Basic similarity:
increasing pressure with compression
critical line
beyond is a plateau
Differences:
at high pressures
at high compressions
55
Compare with condensation of a real gas
CO2
Basic similarity:
increasing pressure with compression
critical line
beyond is a plateau
Differences:
no critical point
 at high pressures
at high compressions
Conclusion:
Fig. 151 Experimental isotherms of
carbon dioxide (CO2}.
BEC in a gas is a phase transition
of the first order
56
What will be the analogue for BEC?
57
The end
Problems
Problems to Lecture 1.
Some problems are expanding on the presented subject matter and are voluntary… (*)
The other ones are directly related to the theme of the class and are to be worked out within a
week. The solutions will be presented on the next seminar and posted on the web.
(1.1*) Problems with metastable states and quasi-equilibria in defining the temperature and
T
applying the 3rd law of thermodynamics c
(1.2*) Relict radiation and the Boomerang Nebula
(1.3) Work out in detail the integral defining
(1.4) Extend the resulting series expansion to the full balance equation (BE integral)
(1.5) Modify for a 2D gas and show that the BE condensation takes never place
(1.6) Obtain an explicit procedure for calculating the one-particle density matrix for an ideal
boson gas [difficult]

60
Problems to Lecture 2.
Some problems are expanding on the presented subject matter and are voluntary… (*)
The other ones are directly related to the theme of the class and are to be worked out within a
week. The solutions will be presented on the next seminar and posted on the web.
(2.1*) How was measured the record temperature of the order of 100 pK?
(2.2*) The BE distribution can be obtained from a combinatorial analysis of microstates of the
system with the subsequent use of the Stirling formula. Which is the least number of the
particles, for which the limit makes sense? (see the example for Boltzmann in Blatt).
(2.3) Extend the integrals from problem (1,4) to all thermodynamic quantities
(2.4) Obtain the entropy for BE gas. Below the transition, it should display the two phases
(1.5) Fluctuations of
n
(1.6) Phase diagram in p – V coordinates for BEC [difficult]

61