PHY304: Statistical Mechanics · 2021. 5. 11. · Ideal Gas in the Grand Canonical Ensemble That...
Transcript of 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
The Grand Canonical Ensemble [Cont’d]
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.
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
The Grand Canonical Ensemble [Cont’d]
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)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
The Grand Canonical Ensemble [Cont’d]
That is, the grand canonical partition function is theweighted sum of all canonical partition functions.
The weighting factor
z = eβµ (4)
is called fugacity.
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
The Grand Canonical Ensemble [Cont’d]
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.
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
The Grand Canonical Ensemble [Cont’d]
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.
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
The Grand Canonical Ensemble [Cont’d]
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)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
The Grand Canonical Ensemble [Cont’d]
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)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
The Grand Canonical Ensemble [Cont’d]
This shows that we can directly write down Z(T , V ,µ)for many problems which we have already treated inthe canonical formalism.
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
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)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Ideal Gas in the Grand Canonical Ensemble
Thus
Z(T , V ,µ) = exp[eβµZ(T , V , 1)
]= exp
[eβµV
(2πmkBT
h2
)3/2]
. (10)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Ideal Gas in the Grand Canonical Ensemble
The grand canonical potential φ has the form
φ(T , V ,µ) = −kBT lnZ(T , V ,µ),
= −kBTeβµV(
2πmkBTh2
)3/2
. (11)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Ideal Gas in the Grand Canonical Ensemble
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)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Ideal Gas in the Grand Canonical Ensemble
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)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Ideal Gas in the Grand Canonical Ensemble
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).
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Ideal Gas in the Grand Canonical Ensemble
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.
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Ideal Gas in the Grand Canonical Ensemble
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.
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Ideal Gas in the Grand Canonical Ensemble
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.
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Ideal Gas in the Grand Canonical Ensemble
Let ρΞ denote the phase space density related to Ξ,
ρΞ =e−βH−γpV∑
V∫
d3Nq∫
d3Np 1h3N exp[−βH − γpV ]
. (21)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Ideal Gas in the Grand Canonical Ensemble
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)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Ideal Gas in the Grand Canonical Ensemble
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.
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Ideal Gas in the Grand Canonical Ensemble
Thus we getG = −kBT lnΞ. (24)
Thus, the Gibbs’ free enthalpy is the thermodynamicpotential associated with the partition function Ξ.
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
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)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Fluctuations in the Grand Canonical Ensemble
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 .
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Fluctuations in the Grand Canonical Ensemble
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)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Fluctuations in the Grand Canonical Ensemble
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.
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Fluctuations in the Grand Canonical Ensemble
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)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Fluctuations in the Grand Canonical Ensemble
We have1
gN (E)
∂gN (E)
∂E
∣∣∣∣∣E=E∗
= β. (31)
We also have
gN (E)∆E ≈ Ω(E, V , N). (32)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Fluctuations in the Grand Canonical Ensemble
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)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Fluctuations in the Grand Canonical Ensemble
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.
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Fluctuations in the Grand Canonical Ensemble
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)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Fluctuations in the Grand Canonical Ensemble
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)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Fluctuations in the Grand Canonical Ensemble
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)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Fluctuations in the Grand Canonical Ensemble
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.
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Fluctuations in the Grand Canonical 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)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Fluctuations in the Grand Canonical Ensemble
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)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Fluctuations in the Grand Canonical Ensemble
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 .
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Fluctuations in the Grand Canonical Ensemble
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)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Fluctuations in the Grand Canonical Ensemble
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)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Fluctuations in the Grand Canonical Ensemble
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.
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Fluctuations in the Grand Canonical Ensemble
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)
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Fluctuations in the Grand Canonical Ensemble
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.
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Fluctuations in the Grand Canonical Ensemble
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.
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
Fluctuations in the Grand Canonical Ensemble
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.
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
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).
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali
End
PHY304: Statistical Mechanics Dr. Anosh Joseph, IISER Mohali