PHY304: Statistical Mechanics · 2021. 5. 11. · Ideal Gas in the Grand Canonical Ensemble That...

PHY304: StatisticalMechanicsLecture 24,Monday, March 8, 2021

Dr. Anosh JosephIISER Mohali

The Grand Canonical Ensemble [Cont’d]

In the last lecture we encountered the grand canonicalpartition function

Z(T , V ,µ) =∞∑

N=0

1h3N

∫d3Nq

∫d3Np e−β[H(qν,pν)−µN]. (1)

The above expression is for distinguishable particles.

PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali


For indistinguishable particles we need to add theGibbs’ correction factor.

Microstates differing only by a different enumerationof the N particles should not be counted as differentmicrostates.

This correction factor ensures that.



Z(T , V ,µ) =∞∑

N=0

1N !h3N

∫d3Nq

∫d3Np e−β[H(qν,pν)−µN].

(2)

From Eq. (2) we find

Z(T , V ,µ) =∞∑

N=0

(eβµ

)NZ(T , V , N). (3)



That is, the grand canonical partition function is theweighted sum of all canonical partition functions.

The weighting factor

z = eβµ (4)

is called fugacity.



In Eq. (3) we can identify the principle which connectsthe microcanonical, canonical, and grand canonicalensembles.

The canonical partition function Z was formed as thesum of all microcanonical “partition functions” g ...

... at energy E, volume V , and particle number N ,weighted by the Boltzmann factor e−βE.



We have

Z(β, N , V ) =∑

E

e−βEg(E, V , N). (5)

Here the energy E is now non longer a fixed quantity,but only its mean value 〈E〉 = U is fixed.



However, the temperature T = (kBβ)−1 has fixed value

given by the heat bath.

The grand canonical partition function Z is formed asthe sum of all canonical partition functions Z ...

... at temperature T , particle number N , and volumeV , weighted by eβµN .

In general, for non-interacting systems (withindistinguishable particles), we have

Z(T , V , N) =1N !

[Z(T , V , 1)]N . (6)



In we insert this into Eq. (3) we get

Z(T , V ,µ) =

∞∑N=0

1N !

[eβµZ(T , V , 1)

]N= exp

[eβµZ(T , V , 1)

]. (7)



This shows that we can directly write down Z(T , V ,µ)for many problems which we have already treated inthe canonical formalism.


Ideal Gas in the Grand Canonical Ensemble

We can make use of Eq. (7) to compute Z(T , V ,µ) inthis case.

For an ideal gas we have

Z(T , V , 1) =Vλ3 , (8)

with

λ =

(h2

2πmkBT

)1/2

. (9)



Thus

Z(T , V ,µ) = exp[eβµZ(T , V , 1)

]= exp

[eβµV

(2πmkBT

h2

)3/2]

. (10)



The grand canonical potential φ has the form

φ(T , V ,µ) = −kBT lnZ(T , V ,µ),

= −kBTeβµV(

2πmkBTh2

)3/2

. (11)



From this we can get the equations of state

−∂φ

∂T

∣∣∣∣∣V ,µ

= S(T , V ,µ) = eβµV(

2πmkBTh2

)3/2

kB

[52− βµ

],

(12)

−∂φ

∂V

∣∣∣∣∣T ,µ

= p(T , V ,µ) = kBTeβµ(

2πmkBTh2

)3/2

, (13)

−∂φ

∂µ

∣∣∣∣∣T ,V

= N(T , V ,µ) = eβµV(

2πmkBTh2

)3/2

. (14)



If we insert Eq. (14) into Eq. (13) we get the ideal gasequation.

If we insert Eq. (14) into Eq. (12) we get the wellknown expression for entropy S(T , V , N) of the idealgas.

From Eq. (11) we have

−kBT lnZ(T , V ,µ) = −kBTeβµV(

2πmkBTh2

)3/2

= −kBTN . (15)



That is, for the case of the ideal gas

lnZ = N . (16)

In some cases we may need to consider systems withother thermodynamic variables, instead of (E, V , N),(T , V , N), or (T , V ,µ).



For example, we can consider (T , p, N).

We can obtain the partition function involving thesevariables another Laplace transformation.

We have

Ξ(T , p, N) =∑V

e−γVpZ(T , V , N), (17)

with a Lagrange multiplier γ.



This partition function is very convenient when wehave systems with a given temperature, particlenumber, and pressure.

Here, the volume is no longer fixed, but at a constantpressure a mean value of the volume 〈V 〉, will beestablished.

The logarithm of all partition functions treated up tonow could be related to thermodynamic potentials.



That is,

g ↔ S = kB ln g, (18)

Z ↔ F = −kBT ln Z , (19)

Z ↔ φ = −kBT lnZ. (20)

We can also find a potential associated with Ξ.



Let ρΞ denote the phase space density related to Ξ,

ρΞ =e−βH−γpV∑

V∫

d3Nq∫

d3Np 1h3N exp[−βH − γpV ]

. (21)



Then we have

S = 〈−kB ln ρΞ〉

=∑V

∫d3Nqd3Np

h3N ρΞ [kB lnΞ(T , p, N) + kBβH + kBγpV ]

= kB lnΞ+ kBβ〈H〉+ kBγp〈V 〉. (22)



That is,−kBT lnΞ = U − TS + kBγTp〈V 〉. (23)

By a procedure analogues to the one performed in thecase of Z, we can identify γ with β.



Thus we getG = −kBT lnΞ. (24)

Thus, the Gibbs’ free enthalpy is the thermodynamicpotential associated with the partition function Ξ.


Fluctuations in the Grand Canonical Ensemble

Earlier we calculated the probability pi,N of finding asystem of the grand canonical ensemble at a particlenumber N and in the phase space point i.

We got the expression

pi,N =1Z

e−β(Ei−µN). (25)



Here Ei is the energy corresponding to the phase spacecell i, and Z is the grand canonical partition function

Z =∑i,N

e−β(Ei−µN). (26)

From Eq. (25) we can, in analogy to the canonicalensemble, calculate the probability density p(E, N) ...

... of finding a system of the ensemble at energy E (nomatter which micro state i) and at particle number N .



If gN (E) is the number of microstates i in the energyinterval (E, E + dE) at particle number N , then

p(E, N) =1Z

gN (E)e−β(E−µN), (27)

and the grand canonical partition function is given by

Z =

∞∑N=1

∫∞0

dE gN (E) e−β(E−µN). (28)



That is, when the particle number N is fixed, thedistribution of the energies in the grand canonicalensemble is the same as in the canonical ensemble.

In addition, however, there is still a distribution in theparticle number N .

We can calculate the most probable values for theenergy and the particle number.



From Eq. (27)

∂p(E, N)

∂E

∣∣∣E=E∗

= 0 =⇒ ∂gN (E)

∂E

∣∣∣∣∣E=E∗

− βgN (E∗) = 0.

(29)

That is,∂gN (E)

∂E

∣∣∣∣∣E=E∗

= βgN (E∗). (30)



We have1

gN (E)

∂gN (E)

∂E

∣∣∣∣∣E=E∗

= β. (31)

We also have

gN (E)∆E ≈ Ω(E, V , N). (32)



On the left hand side of Eq. (31) multiplying bothnumerator and denominator by ∆E and using ∂∆E

∂E = 0(the thickness ∆E of the energy shell is independent ofthe energy E) we get

∂ lnΩ∂E

∣∣∣∣∣E=E∗

=1

kBT. (33)



Thus the most probable energy of the grand canonicalensemble, just as in the canonical case, is given by

∂S∂E

∣∣∣∣∣E=E∗

=1T

, (34)

and is thus identical to the fixed energy of themicrocanonical case.



The most probable particle number N∗ must obey

∂p(E, N)

∂N

∣∣∣∣∣N=N∗

= 0 =⇒ ∂gN (E)

∂N

∣∣∣∣∣N=n∗

+ βµgN (E) = 0

(35)

We have∂gN (E)

∂N

∣∣∣∣∣N=N∗

= −βµgN (E). (36)

or∂S∂N

∣∣∣∣∣N=N∗

= −µ

T. (37)



That is, N∗ is also identical to the given particlenumber N of the microcanonical case.

Here also, in analogy to the canonical case, we have

N∗ = 〈N〉 = Nmc, (38)

andE∗ = 〈E〉 = Emc. (39)



The mean energy coincides with the thermodynamicinternal energy U , and thus also with the fixed energyE given in the microcanonical case.

For the mean particle number

〈N〉 =∑i,N

Npi,N

=1Z

∑i,N

Ne−β(Ei−µN)

=∂

∂µ(kBT lnZ)

∣∣∣∣∣T ,V

= −∂φ

∂µ

∣∣∣∣∣T ,V

. (40)



The mean particle number 〈N〉 is identical to thethermodynamic particle number

N = −∂φ

∂µ

∣∣∣∣∣T ,V

, (41)

which was equal to the fixed given particle number ofthe microcanonical ensemble.



The deviations of the mean values in the grandcanonical ensemble are given by the standarddeviations of the distributions

σ2N = 〈N2〉− 〈N〉2. (42)

We have

〈N2〉 =∑i,N

N2pi,N

=1Z

∑i,N

N2e−β(Ei−µN)

=(kBT )2

Z

∂2

∂µ2Z

∣∣∣∣∣T ,V

. (43)



Since

kBT∂Z

∂µ

∣∣∣∣∣T ,V

= Z · 〈N〉 (44)

we have

〈N2〉 = kBTZ

∂

∂µ(Z · 〈N〉)

∣∣∣∣∣T ,V

= 〈N〉2 + kBT∂〈N〉∂µ

∣∣∣∣∣T ,V

. (45)



That is,

σ2N = kBT

∂〈N〉∂µ

∣∣∣∣∣T ,V

= kBT∂N∂µ

∣∣∣∣∣T ,V

. (46)

In the last equation, 〈N〉 has been replaced by thethermodynamic particle number N .



From the above equation we get the relative meansquare fluctuation in the particle density n(= N/V ).

σ2N

N2 =kBTN2

∂N∂µ

∣∣∣∣∣T ,V

. (47)

In terms of the variable

v =VN

, (48)

we may write

σ2N

N2 =kBTv2

V 2∂V/v∂µ

∣∣∣∣∣T ,V

= −kBTV

(∂v∂µ

) ∣∣∣∣∣T

. (49)



To put this relation into a more practical form, werecall the thermodynamic relation (Gibbs-Duhemrelation)

dµ = vdp − sdT , (50)

according to which

dµ(at constant T ) = vdp. (51)



The above equation then takes the form

σ2N

N2 = −kBTV

1v

(∂v∂p

) ∣∣∣∣∣T

=kBTVκT , (52)

where κT is the isothermal compressibility of thesystem.

Thus, the relative root-mean-square fluctuation in theparticle density of the given system goes like O(N−1/2)and, hence, negligible.



However, there are exceptions: situationsaccompanying phase transitions.

For instance, at a critical point the compressibilitydiverges, so it is no longer intensive.

For the case of experimental liquid-vapor criticalpoints,

κT (Tc) ∼ N0.63. (53)



Accordingly, the root mean square density fluctuationsgrow faster than N0.5 – in this case, like N0.82.

Thus, in the region of phase transitions, especially atthe critical points, we encounter unusually largefluctuations in the particle density of the system.

Such fluctuations indeed exist and account forphenomena like critical opalescence.



Critical opalescence is caused by the occurrence ofdensity fluctuations in the fluid with a correlationlength comparable to the wavelength of light.

These cause certain wavelengths of light to scatterwhich gives rise to the colored or cloudy appearance.



It is clear that under these circumstances theformalism of the grand canonical ensemble could, inprinciple, ...

... lead to results that are not necessarily identical tothe ones following from the corresponding canonicalensemble.

In such cases, it is the formalism of the grandcanonical ensemble that will have to be preferredbecause only this one will provide a correct picture ofthe actual physical situation.


References

I W. Greiner, L. Neise, H. Stocker, and D. Rischke,Thermodynamics and Statistical Mechanics,Springer (2001).

I R. K. Pathria and Paul D. Beale, StatisticalMechanics, Elsevier; Third edition (2011).


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PHY304: Statistical Mechanics · 2021. 5. 11. · Ideal Gas in the Grand Canonical Ensemble That...

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