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PHY 471, Statistical Physics, 2007. Lecture 09. Grand Canonical Ensemble. Mahn-Soo Choi (Korea University). F. Reif, Fundamentals of Statistical and Thermal Physics (1965). Chapter 6. Thermal and Chemical Equilibrium of the “System” A with the “Bath” B. "Universe". "System".

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PHY 471, Statistical Physics, 2007

Lecture 09.

Grand Canonical Ensemble

Mahn-Soo Choi (Korea University)

F. Reif, Fundamentals of Statistical and Thermal Physics (1965)

Chapter 6.


Thermal and Chemical Equilibrium

of the “System” A with the “Bath” B

"Universe"

"System"

"Environment (Bath)"

The probability PA (EA , NA ) that the “system” has the energy

EA and the particle number NA ?

The probability Pα for the “system” to be found in the

microstate α?


Equilibrium Condition

The probability distribution PA (EA , NA ):

Γ(EA , NA |E , N )

Γ(E , N )

PA (EA , NA ) =

ΓA (EA , NA )ΓB (E − EA , N − NA )

Γ(E , N )

=

PA (EA , NA ) and hence log PA (EA , NA ) has a sharp peak at

EA = E¯A and NA = N¯ A in equilibrium.

PA(EA)

ΓA(EA)

ΓB (E − EA)

EA


The equilibrium condition is thus

∂ EA

1

TA

1

TB

log PA

=0

=

EA =E¯A

∂ NA

µA = µB

log PA

=0

¯

NA =NA


Grand Canonical Ensemble

The probability Pα

ΓB (E − Eα , N − Nα )

Γ(E , N )

Pα =

kB log Pα

∂ SB

∂ EB

∂ SB

∂ NB

≈ SB (E , N ) −

EA −

NA

EB =E

NB =N

µNα

T

T

= SB (E , N ) −

+

Canonical distribution

Eα − µNα

kB T

1

ZG

exp −

Pα =

where ZG is the normalization constant:

Eα − µNα

kB T

exp −

.

ZG =

α


Grand Partition Function ZG

ZG is more than just a normalization constant.

From the expression for the entropy

µN

T

E

T

S = −kB

Pα log Pα =

+ kB log ZG

α

−kB T log ZG = E − TS − µN = G

the Gibbs free energy!


Thermodynamics with Grand Canonical Ensemble

Partition function ZG

G (T , µ, V , · · · ) = −kB T log ZG (T , µ, V , · · · )

dG = −SdT − Nd µ − PdV + · · ·

Thermodynamic quantities

∂ G

∂ T

∂ G

∂µ

∂ G

∂ V

S = −

,

N = −

,

P = −

, ···


More about the Entropy

S

kB

= −

Pα log Pα = (E − µN ) + log ZG

α

∂β

E − µN = −β

log ZG

∂β

S

kB

1 − β

=

log ZG


More about Pressure

The elementary consideration leads to

∂ G

∂ V

P = −

Since the Gibbs free energy is an extensive quantity, it should

be directly proportional to V . It means that

∂ G

∂ V

= constant.

The constant should be the pressure P :

G = −kB T log ZG = PV .


Summary of Ensembles

Microcanonical ensemble

Γ(E , N , V , · · · ) =

δ(E − Eα )

α

S (E , N , V , · · · ) = kB log Γ(E , N , V , · · · )

Canonical ensemble

kB T

Z (T , N , V , · · · ) =

exp −

α

A(E , N , V , · · · ) = −kB T log Z (T , N , V , · · · )

Grand canonical ensemble

Eα − µNα

kB T

ZG (T , N , V , · · · ) =

exp −

α

G (E , µ, V , · · · ) = −kB T log FG (T , µ, V , · · · )


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