Download Chapter 10 Grand canonical ensemble and more Study notes Statistical mechanics in PDF only on Docsity!
Chapter 10
Grand canonical ensemble
10.1 Grand canonical partition function
The grand canonical ensemble is a generalization of the canonical ensemble where the restriction to a definite number of particles is removed. This is a realistic representation when then the total number of particles in a macroscopic system cannot be fixed.
Heat and particle reservoir. Consider a sys- tem A 1 in a heat and particle reservoir A 2. The two systems are in equilibrium with the thermal equilibrium.
- Thermal equilibrium results form the ex- change of heat. The two temperature are then equal: T = T 1 = T (^2)
- The equilibrium with respect to particle exchange leads to identical chemical poten- tials: μ = μ 1 = μ 2.
Energy and particle conservation. We assume that the system A 2 is much larger than the system A 1 , i.e., that
E 2 � E 1 , N 2 � N 1 ,
with N 1 + N 2 = N = const. E 1 + E 2 = E = const.
where N and E are the particle number and the energy of the total system A = A 1 + A 2.
Hamilton function. The overall Hamilton function is defined as the sum of the Hamilton functions of A 1 and A 2 :
H(q, p) = H 1 (q(1), p(1), N 1 ) + H 2 (q(2), p(2), N 2 ).
For the above assumption to be valid, we neglect interactions among particles in A 1 and A 2 : H 12 = 0.
118 CHAPTER 10. GRAND CANONICAL ENSEMBLE
This is a valid assuming for most macroscopic systems.
Microcanonical ensemble. Since the total system A is isolated, its distribution function is given in the microcanonical ensemble as
ρ(q, p) =
Ω(E, N )
δ(E − H 1 − H 2 ) ,
as in (9.1), with t
Ω(E, N ) =
d 3 N^ q d 3 N^ p δ(E − H 1 − H 2 )
being the density of states.
Sub-macroscopic particle exchange. The total entropy of the combined system is given by the microcanonical expression
S = k (^) B ln
Ω(E, N )Δ
Γ 0 (N )
, Γ 0 (N ) = h 3 N^ N 1 !N 2! , (10.1)
where Δ is the width of the energy shell. Compare (9.4) and Sect. 9.4.
The reason that Γ 0 (N ) ∼ N 1 !N 2! in (10.1), and not ∼ N !, stems from the assumption that there is no particles exchange on the macroscopic level. This is in line with the observation made in Sect. 9.6 that energy fluctuations, i.e. the exchange of energy between a system and the heat reservoir, scale like 1/
N relatively to the internal energy. We come back to this issue in Sect. 10.3.2.
Integrating out the reservoir. We are now interested in the system A 1. As we did for the canonical ensemble, we integrate the total probability density ρ(q, p) over the phase space of the reservoir A 2. We obtain
ρ 1 (q(1), p(1), N 1 ) ≡
dq(2) dp(2) ρ(q, p)
dq(2) dp(2) δ(E − H 1 − H 2 , N ) Ω(E, N )
≡
Ω 2 (E − H 1 , N − N 1 )
Ω(E, N )
in analogy to (9.2).
Expanding in E 1 and N 1. With E 1 � E and N 1 � N , we can approximate the slowly varying logarithm of Ω 2 (E 2 , N 2 ) around E 2 = E and N 2 = N and thus obtain:
S 2 (E − E 1 , N − N 1 ) = k (^) B ln
Ω 2 (E − E 1 , N − N 1 )Δ
Γ 0 (N )
= S 2 (E, N ) − E 1
∂S 2
∂E 2
N (^2)
� E 2 = E
N 2 = N
− N 1
∂S 2
∂N 2
E (^2)
� E 2 = E
N 2 = N
120 CHAPTER 10. GRAND CANONICAL ENSEMBLE
10.2 Grand canonical potential
In Sect. 5.4 we defined the grand canonical potential ∗^ as
Ω(T, V, μ) = F (T, V, N ) − μN , Ω = −P V ,
and showed with Eq. (5.21) that
U − μN =
∂β
βΩ
∂β
ln Z(T, V, μ) , (10.8)
where we have used in the last step the representation (10.7) of U − μN = ∂ ln Z/∂β.
Grand canonical potential. From (10.8) we may determine (up-to a constant) the grand canonical potential with
Ω(T, V, μ) = −k (^) B T ln Z(T, V, μ) , Z = e −βΩ^ (10.9)
as the logarithm of the grand canonical potential. Note the analog to the relation F = −k (^) B T ln Z (^) N valid within the canonical ensemble.
Calculating with the grand canonical ensemble. To summarize, in order to obtain the thermodynamic properties of our system in contact with a heat and particle reservoir, we do the following:
1 - calculate the grand canonical partition function:
Z =
�^ ∞
N =
d 3 N^ q d 3 N^ p h 3 N^ N!
e −β(H−μN^ )^ ;
2 - calculate the grand canonical potential:
Ω = −k (^) B T ln Z = −P V, dΩ = −SdT − P dV − N dμ ;
3 - calculate the remaining thermodynamic properties through the equations:
S = −
∂T
V,μ
P = −
∂V
T,μ
V
N = −
∂μ
T,V
and the remaining thermodynamic potentials through Legendre transformations. ∗ (^) Don’t confuse the grand canonical potential Ω(T, V, μ) with the density of microstates Ω(E)!
10.3. FUGACITY 121
10.3 Fugacity
By using the definition of the partition function in the canonical ensemble,
Z (^) N =
h 3 N^ N!
d 3 N^ q d 3 N^ p e −βH^ ,
we can rewrite the partition function in the grand canonical ensemble as
Z(T, V, μ) =
�^ ∞
N =
e μN^ β^ Z (^) N (T ) =
�^ ∞
N =
z N^ Z (^) N (T ) , (10.10)
where z = exp(μ/k (^) B T ) is denoted as the fugacity. Equation (10.10) shows that Z(T, μ) is the discrete Laplace transform of ZN (T ).
10.3.1 Particle distribution function
In the canonical ensemble the particle number N was fixed, whereas it is a variable in the in the grand canonical ensemble. We define with
w (^) N = e βμN^
Z N (T )
Z(T, μ)
the probability that the system at temperature T and with chemical potential μ contains N particles. It is normalized:
�^ ∞
N =
w (^) n =
�^ ∞
N =
e βμN^ Z (^) N (T ) Z(T, μ)
Average particle number. That mean particle number �N � is given by
�N � =
�^ ∞
N =
N w (^) N (T, V ) =
N =0 N z^
N Z N (T, V )
N =0 z^
N Z N (T, V ) ,^ (10.12)
which can be rewritten as
�N � =
β
∂μ
ln Z(T, V, μ)
T,V
or, alternatively, as
�N � = z
∂z
ln Z(T, V, μ)
T,V
when making use of the definition of fugacity z.
Chemical potential. Conversely, the chemical potential for a given average particle number N = �N �, μ = μ(T, V, N ) ,
is obtained by by inverting (10.13).
10.3. FUGACITY 123
In fact, this can be proven within the grand canonical ensemble in the same way as it was proven in Sect. 9.6 that the energy fluctuations in the canonical ensemble fulfill the thermal stability criterion
C (^) V = k (^) B β 2
H − �H�
Intensive variables. We start be defining the free energy per particle, a = F/N , as
a(T, v) ≡
F (T, V, N )
N
v = V /N , (10.20)
where the v is the volume per particle (the specific volume).
- a = F/N is intensive and a function of the intensive variable T. It can therefore not be a independently a function of the volume V or the particle number N (which are extensive), only of V /N (which is intensive).
- The circumstance that a is a function of V /N only allows to transform derivative with respect to V and to N into each other, the reason of using this representation.
Chemical potential. With (10.20) we find
μ =
∂F
∂N
N F/N
∂N
= a + N
∂a ∂N
= a + N
∂a ∂(V /N )
∂(V /N )
� ∂��N �
−V /N 2
when using (5.12), namely that dF = −SdT − P dV + μdN. This relation yields
μ = a − v
∂a ∂v
∂μ ∂v
= −v
∂ 2 a ∂v 2
Note, that μ is intensive.
Pressure. For the pressure P , an intensive variable, we find likewise that
P = −
∂F
∂V
N F/N
∂V
= −N
∂a ∂(V /N )
∂(V /N )
� ∂��V �
1 /N
which results in
P = −
∂a ∂v
∂P
∂v
∂ 2 a ∂v 2
∂μ ∂v
= v
∂P
∂v
where we have used in the last step the comparison with (10.21).
124 CHAPTER 10. GRAND CANONICAL ENSEMBLE
Particle fluctuations in intensive variables. Using intensive variables for (10.17),
β V
�N 2 � − �N � 2
∂ 2 P (T, v) ∂μ 2
∂v
∂P
∂μ
∂v ∂μ
the last relation of (10.22) and (10.15), namely ∂P/∂μ = 1/v, we obtain
∂ 2 P (T, v) ∂μ 2
∂v
v
− 1 /v 2
v
∂v ∂P
−κ (^) T
, κ (^) T = −
V
∂V
∂P
v
∂v ∂P
Compressibility. Taking the results together we obtain
κ (^) T =
βv 2 V
�N 2 � − �N � 2
= β
V
N
�N 2 � − �N � 2
N
for the compressibility κT.
- The compressibility is strictly positive, κT ≥ 0, which implies that the system constricts when the pressure increases. The basic stability condition.
- The compressibility is finite if the size of the fluctuations
�N 2 � − �N � 2 vanishes in the thermodynamic limit relative to the number of particles N present in the system.
Response vs. fluctuations. The specific heat CV and the compressibility κT are mea- sure the response of the system to a change of variables.
- A temperature gradient ΔT to the heat reservoir leads to a transfer of heat,
ΔQ = C (^) V ΔT,
which is proportional to CV. Compare Sect. 3.2.
- An increase in temperature by ΔP leads to relative decrease of the volume by ΔV V
= κ (^) T ΔP,
which is proportional to κT.
It is not a coincidence, that both response functions, CV and κ (^) T , as given respectively by (10.19) and (10.24), are proportional to the fluctuations of the involved variables.
A system in which a certain variable A is not fluctuating (and hence fixed) cannot respond to perturbations trying to change A. The response always involves the fluctuation �� A − �A�
= �A 2 � − �A� 2.
126 CHAPTER 10. GRAND CANONICAL ENSEMBLE
with (10.28) we find immediately the equation of state
V P = k (^) B T N
of the ideal gas
Entropy. Evaluating the entropy
S = −
∂T
V,μ
, Ω(T, V, μ) = −k (^) B T e μ/k^ B^ T^
V
λ (^3) T
λ (^) T = h √ 2 π m k (^) B T
we find
S = −
T
−μ k (^) B T 2
T
μ k (^) B T
which leads with (10.29), Ω = −k (^) B T N , to
S(T, V, μ) = kB N
μ k (^) B T
, N = e μ/k^ B^ T^
V
λ (^3) T
Sackur-Tetrode equation. Inverting N for μ/(k (^) B T ) we obtain for the entropy with
S = k (^) B N
ln
V
N λ (^3) T
the Sackur-Tetrode equation, which coincides with the both the microcanonical and the canonical expressions, (9.20) and (8.25). The three ensembles yield identical results in the thermodynamic limit.
10.4. THE IDEAL GAS IN THE GRAND CANONICAL ENSEMBLE 127
10.4.1 Summary ensembles
Phase space integration Thermodynamic potential
Microcanonical Γ(E, V, N ) =
E<H<E+Δ
d 3 N^ q d 3 N^ p S(E, V, N ) = kB ln
h 3 N^ N!
ensemble
Ω(E, V, N ) =
∂E
d 3 N^ q d 3 N^ p δ(E − H)
Canonical Z (^) N (T, V ) =
d 3 N^ q d 3 N^ p h 3 N^ N!
e −βH^ F (T, V, N ) = −k (^) B T ln Z (^) N (T, V ) ensemble =
h 3 N^ N!
0
dE Ω(E) e −βH^ Z (^) N = e −βF
Grand canonical Z(T, μ) =
N =0 e^
μN β (^) Z (^) N (T ) Ω(T, V, μ) = −k (^) B T ln Z(T, V, μ) ensemble Z = e −βΩ^ = e
P V kB T
