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Grand-canonical ensemble
Gabriel T. Landi
- November 21, University of S˜ao Paulo
- 1 The Grand-canonical ensemble Contents
- 1.1 Equilibrium minimizes the thermodynamic potential
- 1.2 Non-interacting Hamiltonians
- 1.3 Fluctuations
- 1.4 The “classical” limit
- 2 Quantum gases
- 2.1 The density of states
- 2.2 Density of states and Green’s functions (optional)
- 2.3 First quantum correction to the classical limit
- 3 Fermi gases
- 3.1 Ground-state
- 3.2 Finite temperatures
- 4 Bose gases and Bose-Einstein condensation
- 4.1 Solving for the chemical potential
- 4.2 Physics below Tc
1 The Grand-canonical ensemble
The key behind second quantization is to remove the restriction that the number of particles is fixed. Instead, the theory is built around the idea of Fock space, where the number of particles is not fixed. This is highly advantageous when dealing with many- body systems. This same idea, when extended to finite temperatures, is what we call the Grand canonical ensemble. What we want is to consider some finite temperature density matrix ρ ∼ e−βH^ where the number of particles is not fixed, but can fluctuate. However, we cannot let it fluctuate arbitrarily since that would make no physical sense. Instead, the basic idea of the grand-canonical ensemble is to impose that the num- ber of particles is only fixed on average. That is, we impose that
〈 N〉ˆ = N, (1.1)
where ˆN is the number operator and N is the number of particles in the system. In some systems, the number of particles does indeed fluctuate. This happens, for instance, in chemical solutions: if we look at a certain region of a liquid, the number of molecules there is constantly fluctuating due to molecules moving around from other parts. Of course, in many other systems on the other hand, the number of particles is fixed. However, it turns out that even in these cases, pretending it can may still give good answers, provided N is large (the thermodynamic limit). The reason is because, as we will show below, the variance of ˆN scales as var( ˆN) ∼ N so that √ var( ˆN) 〈 N〉ˆ
N
which is thus small when N is large. Hence, when N is large, the grand-canonical ensemble will give accurate answers, even if the number of particles is not actually allowed to fluctuate. This is the idea behind ensemble equivalence: we are allowed to use an ensemble where the number of particles fluctuate, even though it actually doesn’t, because in the thermodynamic limit the fluctuations are small. Our focus here will be on a system described within the language of second quan- tization, with a Hamiltonian H and a number operator ˆN. We assume that
[H, Nˆ] = 0 , (1.3)
meaning that the number of particles is a conserved quantity. This means that H and Nˆ can be simultaneously diagonalized. The eigenvalues of ˆN are all possible numbers of particles N. Thus, H is divided in sectors with well defined N; in other words, H is block diagonal, so there are no terms connecting sectors with different N. The eigenvalues E are thus labeled by two indices E(N, j), where j labels the quantum states within each sector. Suppose now that the system is in thermal equilibrium with exactly N particles. The corresponding canonical partition function will be
Z(N) =
j
e−βE(N,^ j). (1.4)
This is a constrained sum since we are only summing over that sector that has ex- actly N particles. This constraint makes it notoriously difficult to compute the sum in practice.
which occurs when ρ = ρcan. The state with the smallest free energy is thus the thermal state (1.9). Alternatively, we could recast the problem in terms of entropy: we write
Feq = −T ln Z = Ueq − T S (^) eq, (1.12)
where Ueq = 〈H〉th and S (^) eq = S (ρcan) are the equilibrium energies and entropies. Using this together with Eq. (1.8), we can rearrange Eq. (1.10) to read
S (ρ) = S (^) eq + β
[
〈H〉ρ − Ueq
]
− S (ρ||ρcan). (1.13)
This is an identity comparing the entropy of any quantum state ρ with the entropy of the equilibrium state (1.9). Consider now the following question: out of all states ρ having 〈H〉ρ = Ueq, which has the highest entropy? Eq. (1.13) makes the answer to this question transparent: If 〈H〉ρ = Ueq then second term vanishes. Since the first term is independent of ρ, to make S (ρ) as large as possible we must minimize the second term. The minimum is thus achieved when ρ = ρcan.
Max Ent principle for the grand-canonical ensemble
The logic we just used can now be extended to the grand-canonical ensemble. After all, if the principle works for the canonical ensemble (and we know it does because it matches experiment), then it must also work for the grand-canonical ensemble. To impose the condition (1.1), we ask
Out of all states ρ with fixed 〈H〉ρ = U and 〈 N〉ˆ ρ = N, which has the highest entropy?
It is easy to address this question using the result from the canonical ensemble. For, as we saw in (1.6), all we need to do is replace H with H − μ Nˆ. Eq. (1.13) then becomes
S (ρ) = S (^) th + β
〈H〉ρ − U
〈 N〉 −ˆ N
− S (ρ||ρeq), (1.14)
where the relative entropy is now between the state ρ and the grand-canonical state (1.5). With this formula, we are then able to conclude that the state (1.5) is the state with the highest possible entropy, given that 〈H〉ρ = U and 〈 N〉ˆ ρ = N.
The thermodynamic potential
Just like we went back and forth between free energy and entropy in the case of the canonical ensemble, in the case of the grand-canonical ensemble we can move between entropy and a new thermodynamic quantity, called thermodynamic potential (I guess at this point people ran out of creativity for naming it). It is defined as
Φ(ρ) = 〈H〉ρ − μ〈 N〉ˆ ρ − T S (ρ). (1.15)
Rearranging (1.14) we get
Φ(ρ) = Φeq + T S (ρ||ρeq), (1.16)
where Φeq = Ueq − μNeq − T S (^) eq.
Eq. (1.16) allows us to conclude that the grand-canonical state (1.5) is the state mini- mizes the thermodynamic potential.
From now on, since we will be mostly interested in equilibrium states, I will simplify the notation and write only
Φ = U − μN − T S , (1.17)
where it is now implied that all quantities refer to equilibrium at the state (1.5). We can also express Eq. (1.17) more simply as
Φeq = −T ln Q. (1.18)
This is the grand-canonical analog of Feq = −T ln Z. In fact, if we set μ = 0 we recover exactly the canonical ensemble.
If we write the eigenvalues of the Hamiltonian as E(N, j), as in Eq. (1.4), then the grand-canonical partition function Q in Eq. (1.5) can be written more explicitly as
Q =
N, j
e−β(E(N,^ j)−μN).
We can now factor out this sum as
Q =
N
e−βμN^
j
e−βE(N,^ j).
The sum in j is now exactly the canonical partition function Z(N) in Eq. (1.4). This therefore provides us with a link between the canonical and grand-canonical partition functions:
Q =
N
zN^ Z(N), z = e−βμ, (1.19)
where z is called the fugacity (a name due to historical reasons). We therefore see that Z(N) is obtained from a series expansion of Q as a function of z. Mathematically, this transformation between Z(N) and Q is called the Z transform; it is the discrete version of the Laplace transform.
1.2 Non-interacting Hamiltonians
We saw when we studied second quantization, that non-interacting Hamiltonians could always be written as
H =
α
εαa† αaα, (1.20)
where α is a set of single-particle states and the operators aα can be either bosonic or fermionic. The number operator may be similarly written as
Nˆ =
α
a† αaα. (1.21)
Thus, the grand-canonical state (1.5) becomes
ρeq =
Q
exp
− β
α
(εα − μ)a† αaα
Conversely, for Bosons we get
tr
ˆne−λnˆ
∑^ ∞
n= 0
ne−λn^ =
e−λ (e−λ^ − 1)^2
Combining this with Eq. (1.26) yields the Bose-Einstein distribution
〈ˆnα〉 =
eβ(εα−μ)^ − 1
, (Bosons).
To summarize, in the case of non-interacting Fermions the thermody- namic potential Φ = −T ln Q reads
Φ = −T
α
ln
1 + e−β(εα−μ)
and the average occupation number of each state α is given by the Fermi-Dirac distribution 〈ˆnα〉 =
eβ(εα−μ)^ + 1
Conversely, in the case of non-interacting Bosons we get
Φ = T
α
ln
1 − e−β(εα−μ)
and the Bose-Einstein distribution
〈ˆnα〉 =
eβ(εα−μ)^ − 1
In the case of Fermions, the value of μ can in principle be arbitrary. But for Bosons, we could have 〈nˆα〉 < 0, which is of course unphysical. The condition for this not to happen is to have μ < εα for all α; or,
μ < min(εα), for Bosons. (1.33)
This restriction on μ in the case of Bosons has dramatic consequences, being at the core of Bose-Einstein condensation. We will study this in detail later on.
1.3 Fluctuations
Consider the thermodynamic potential Φ = −T ln tr e−β(H−μ^ Nˆ)
. Differentiating with respect to μ, we get
−
∂μ
tr
Ne−β(H−μ^
Nˆ))
tr
e−β(H−μ^ Nˆ)
).^ (1.34)
Thus we conclude that
〈 N〉ˆ = −
∂μ
This result is general, holding even in the presence of interactions. As a sanity check, differentiating Eqs. (1.29) or (1.31) with respect to μ yields
〈 N〉ˆ =
α
eβ(εα−μ)^ ± 1
α
〈nˆα〉, (1.36)
as expected. Here the + sign is for Fermions and the minus for Bosons. Next differentiate (1.34) again with respect to μ. This leads to
∂^2 Φ
∂μ^2
= β
tr
N^2 e−β(H−μ^ Nˆ)
tr
e−β(H−μ^ Nˆ) ) −^ β
[
tr
Ne−β(H−μ^ Nˆ)
tr
e−β(H−μ^ Nˆ)
] 2
We recognize in this the variance of ˆN, var( ˆN) = 〈 Nˆ^2 〉 − 〈 N〉ˆ^2. Whence,
var( ˆN) =
β
∂〈 N〉ˆ
∂μ
β
∂^2 Φ
∂μ^2
This result is similar to what we found before in the canonical ensemble for the heat capacity or the susceptibility. It shows that the fluctuations are proportional to the derivative of the average with respect to μ. As a consequence, we see that since 〈 N〉ˆ = N is the number of particles, var( ˆN) will be similarly extensive in N. This means we can write ∂〈 N〉ˆ ∂μ
:= Nκ. (1.38)
The constant κ is intensive and can be shown in thermodynamics to be related to the compressibility of the system. This therefore implies that √ var( ˆN) 〈 N〉ˆ
κ Nβ
The relative fluctuations therefore scale proportionally to 1/
N, which becomes neg- ligible in the thermodynamic limit. This, as already discussed before, is the reason why we can use the grand-canonical ensemble even in those situations where the number of particles does not actually fluctuate.
1.4 The “classical” limit
The above results have to distinguish between Fermions and Bosons. When is this really necessary? Is it possible to have limiting cases where it does not matter if the particles are Fermions and Bosons? This is normally called the “classical” limit be- cause it is the limit where quantum indistinguishibility no longer matters. I personally don’t like this name because there are so many “classical limits” these days, you never which one people mean. In any case, if we look at Eqs. (1.30) and (1.32), we see that what distinguishes them is the ±1 term. If this term was negligible, the results for Fermions and Bosons would be the same. The distinction between Fermions and Bosons therefore becomes irrelevant whenever
eβ(εα−μ)^ 1. (1.40)
where
εk =
ℏ^2 k^2 2 m
k^21 +... + k^2 d 2 m
where, here, I reintroduced ℏ just for completeness. Notice how the energies depend only on k = |k|. The condition (1.33) for the chemical potential in the case of Bosons becomes, in this case, μ < 0. (2.4)
Thus, while for Fermions the chemical potential is arbitrary, for Bosons it is always negative.
2.1 The density of states
The average occupation number is given by Eqs. (1.30) or (1.32):
〈 N〉ˆ =
k,s
eβ(εk^ −μ)^ ± 1
To carry out this sum, we transform it into an integral. This is justified when the box size L is large, so that the discreteness of (2.1) becomes very fine. It is actually easier to do this in a slightly more general context. Consider an arbitrary sum of the form ∑
k,s
f (k),
where f (k) is an arbitrary function which depends only on the absolute value of k. To convert it into an integral we introduce the convenient 1:
( L
2 π
)d ∆k 1... ∆kd (2.6)
We then get (^) ∑
k,s
f (k) =
k
∆k 1... ∆kd f (k),
where we also introduced the factor (2S +1) which comes from the sum over s. Written in this way, the expression has the form of a Riemann sum. When L is large, the ∆ki become infinitesimal, so that we are allowed to convert the sum to an integral:
k,s
f (k) = (2S + 1)
( L
2 π
)d ∫ ddk f (k). (2.7)
This result is already nice. It provides a recipe to go from a k sum to a k integral. But we can also go a step further and exploit the fact that f (k) depends only on k = |k|. Introduce the d-dimensional solid angle ddk = kd−^1 dk dΩd. We can then carry out the integral over dΩd. The result is the area of a d-dimensional sphere:
∫ dΩd =
dπd/^2 Γ
( (^) d 2 +^1
) ,^ (2.8)
where Γ(x) is the Gamma function. For instance, if d = 3 the above expression sim- plifies to the familiar 4π, whereas if d = 2 we get 2π. With this change, Eq. (2.7) becomes
∑
k,s
f (k) = (2S + 1)
( L
2 π
)d (^) dπd/ 2
Γ
( (^) d 2 +^1
∫^ ∞
0
dk kd−^1 f (k). (2.9)
I know this seem a bit messy. But if you think about it, everything here is just a bunch of silly numbers. These numbers represent the coefficient that you have to multiply when you want to go from a sum to an integral. Finally, we can also go one step further and convert the k integral into an integral over ε. To do this we change variables using Eq. (2.3):
ε =
ℏ^2 k^2 2 m
→ dε =
ℏ^2 k m
dk.
Thus
dk kd−^1 =
m ℏ^2
2 mε ℏ^2
) d 2 − 1 dε.
Eq. (2.9) then finally becomes
∑
k,s
f (k) = (2S + 1)
( L
2 π
)d (^) dπd/ 2
Γ
( (^) d 2 +^1
2 m ℏ^2
)d/ 2 ∫∞
0
dε ε
d 2 − 1 f (ε).
This still looks somewhat messy. But what we do now is to throw every- thing that is ugly under the carpet by defining the density of states (DOS)
D(ε) = (2S + 1)
( L
2 π
)d (^) dπd/ 2
Γ
( (^) d 2 +^1
2 m ℏ^2
)d/ 2 ε
d 2 − 1
. (2.10)
so that our recipe finally becomes
∑
k,s
f (k) =
∫^ ∞
0
dε D(ε) f (ε). (2.11)
The DOS therefore quantifies the weight that you get from going from a k sum to an integral in ε. With this expression, we can now write down any thermodynamic quantity we wish. For instance, the average number of particles will be 〈 N〉ˆ =
dε D(ε)¯n(ε), (2.12)
where n¯(ε) :=
eβ(ε−μ)±^1
��(ϵ)
��(ϵ)
Figure 1: The poles of tr G() [Eq. (2.20)] occur at = εk − is.
Here 1/A is just a fancy way of writing the matrix inverse A−^1. This is interpreted as a continuous function of a parameter . Moreover, s is a tiny number which is left there to ensure that G never blows up when touches one of the eigenvalues εk. Sometimes we do not write the lim s→ 0 explicitly. But this limit is always there in principle. The point I want to stress here is that the Green’s function (2.20) actually contains the DOS in it. To see that, take the trace of G() using the |k〉 basis:
tr G() =
k
〈k|
+ is − H
|k〉
k
+ is − εk
k
( − εk) − is ( − εk)^2 + s^2
We see that tr G() will have poles in the complex plane whenever = εk − is (see Fig. 1). Thus, by knowing the pole structure of tr G() one can infer the eigenvalues of H. Next let us focus on the imaginary part of tr G():
Im
[
tr G()
]
= − lim s→ 0
k
s ( − εk)^2 + s^2
We now use the identity
lim s→ 0
s x^2 + s^2
= πδ(x), (2.22)
which leads to Im
[
tr G()
]
= −π
k
δ( − εk). (2.23)
The quantity on the right-hand side is nothing but the DOS (2.18). Thus, we conclude that
D() = −
π
Im
[
tr G()
]
π
lim s→ 0
Im
k
+ is − εk
This kind of relation is used to evaluate the DOS numerically in systems for which the energy levels εk are too complicated. Finally, we mention that the trace in Eq. (2.24) can be taken with respect to any basis we want. This allows us to define density of states for specific sectors. For
instance, suppose we have a model where the energy eigenvalues depend on the spin projection; something like εk,σ. We can then use the basis |k, σ〉 to take the trace in (2.24), which allows us to decompose
D() =
σ
Dσ(), Dσ() = −
π
Im
k
〈kσ|G()|kσ〉. (2.25)
The quantities Dσ() can be interpreted as the density of states within the sector of spin σ. This kind of idea therefore allows us to address how many states are available within a given subspace. As another example, suppose we actually have a tight-binding model where we label the position states as |n〉, with n = 1 , 2 ,... , L. In this case we can write (2.24) as
D() =
∑^ L
n= 1
Dn(), Dn() = −
π
Im〈n|G()|n〉. (2.26)
Each Dn() therefore quantifies the density of states available at position n. It would be interesting to apply this to the Aubry-Andr´e model. I don’t think anyone ever did this...
2.3 First quantum correction to the classical limit
We saw in Sec. 1.4 that the classical limit corresponds to eβ(εk^ −μ)^ 1. Since this must be true for all energies, this implies that e−βμ^ 1. Since β > 0 we therefore see that in the classical limit μ < 0. Let us now expand on this and include also the first quantum correction to this classical limit. For convenience, introduce a symbol
ζ =
This way we can write the occupation number as
n¯(ε) =
eβ(ε−μ)^ + ζ
We now write this as
n¯(ε) = e−β(ε−μ)^
1 + ζe−β(ε−μ)^
When e−β(ε−μ)^ 1 we can then expand this using (1 + x)−^1 ' 1 − x. This then yields
n¯(ε) ' e−β(ε−μ)
[
1 − ζe−β(ε−μ)
]
For concreteness, let us now focus on the case of a quantum gas in 3D. To be a bit more concise, let us also write the density of states, Eq. (2.17), as
D(ε) = αV
ε, α = (2S + 1)
2 π^2
( (^) m
ℏ^2
�
μ
� � �
Figure 2: First quantum correction to the chemical potential, comparing the classical case (C) with that of Bosons (B) and Fermions (F). The curves were made using Eq. (2.37).
done using the following inverse series: 1
y = ax + bx^2 + O(x)^3 → x =
y a
by^2 a^3
We then get
z = λ^3 N/V +
ζ 22 /^3
λ^3 N/V
Or, in terms of the chemical potential,
βμ = ln
λ^3 N/V +
ζ 22 /^3
λ^3 N/V
The situation is depicted in Fig. 2. For very high temperatures the curves for Bosons and Fermions coincide with the classical limit [Eq. (2.33)]. As the system is cooled down, however, the quantities V/N and λ^3 start to become of the same magnitude and the indistinguishability of the particles begins to matter. As a consequence, the curves for Bosons and Fermions deviate from the classical behavior. In particular, as can be seen in Eq. (2.37), the chemical potential for Bosons (ζ = −1) is always smaller than that for Fermions (ζ = +1). Next let us look at the average energy (2.14). Using again the expansion of ¯n(ε) in Eq. (2.29) and carrying out the integrals, as before, we find
U =
3 T
V
λ^3
z − ζ 25 /^2
z^2
This result is correct, but is expressed in terms of the chemical potential (in z). This is not very useful. It would be better to have the result expressed in terms of N. To do that, we can use Eq. (2.36) to get rid of z. We also need
z^2 =
λ^3 N/V + ζ 22 /^3
λ^3 N/V
' (λ^3 N/V)^2.
(^1) This can be understood as follows. Start with y = ax + bx (^2). We now assume that the inverse can also be expressed in a power series, as x = cy + dy^2 , for some coefficients c and d. Plugging this ansatz in y = ax + bx^2 yields the equation y = acy + (ad + bc^2 )y^2 + O(y^3 ). In order to satisfy this (up to second order) we must then have c = 1 /a and d = −bc^2 /a, which leads to Eq. (2.35).
We then get
z − ζ 25 /^2
z^2 = λ^3 N/V + ζ 22 /^3
λ^3 N/V
ζ 25 /^2
(λ^3 N/V)^2
= λ^3 N/V +
ζ 25 /^2
(λ^3 N/V)^2.
Substituting this in Eq. (2.38) and simplifying a bit, we get
U =
NT
ζ 25 /^2
λ^3 N/V
This result is pretty cool. The value of 3NT/2 is simply the classical internal energy of an ideal gas. When quantum corrections become important, however, we see that for Fermions the Pauli exclusion principle increases the energy of the gas, whereas for Bosons the energy is reduced instead. What is surprising about this is that there are no interactions involved; the gas is ideal and the energy is purely kinetic. A set of Bosons or Fermions at the same temperature will thus have different energies, even though they do not interact. Thus, for instance, if we have a mixture of He-3 (which are fermions) and He-4 (which are bosons), each species will have a different average energy, solely due to quantum effects.
3 Fermi gases
Let us now focus on the case of Fermions. The most important example of a Fermi gas are the electrons in a metal. The conduction electrons are only weakly bound to the atomic nuclei, so that they can pretty much more around freely. Their only constraint is that they cannot leave the metal; there is a potential barrier to do so (which is the work function for those who studied the photoelectric effect). Thus, electrons in a metal are naturally trapped in a box of length L, so that the momentum quantization rules of Sec. applies.
3.1 Ground-state
The density of states (DOS) for a 3D electron gas is given by Eq. (2.17) with S = 1 /2; viz,
D(ε) = αV
ε, α =
π^2
( (^) m ℏ^2
As our first step, let us compute the Fermi level. Recall that the word “Fermi” is associated with “highest filled”. The Fermi level is a zero temperature property; it refers to the ground-state. We pile up the electrons, state by state, until we reach a total of Ne particles. The Fermi energy will thus be the energy of the highest filled state. This is one of those cases where the DOS comes quite in handy. The DOS already contains all factors that come from changing from a k-sum to an integral in energy.
Figure 3: Grid of available states in k-space. The Fermi energy is determined by specifying how many points lie inside a sphere of radius kF.
The average energy per electron is therefore 3/5 of the Fermi energy. It is not exactly 1 /2 because the density of states (3.1) is not uniform in ε: the higher the energy, the higher is the number of available states. From the Fermi energy and momentum we can also compute other “Fermi” quan- tities. For instance, the Fermi velocity is
vF = ℏkF m
m
(3π^2 ne)^1 /^3. (3.8)
This yields the typical velocity of electrons around the Fermi level. The reason why this is important is because the electrons around the Fermi level are the ones which can be excited to empty states. And it is through these excitations that the physics happens. For instance, when you apply an electric potential, the electrons will begin to move around. This “moving around” means that you are exciting electrons to states above the Fermi level. The typical velocities of these electrons will therefore be of the order of vF. Similarly, we can also define the Fermi temperature from the relation
kBTF = εF , (3.9)
where, just for now, I reinstitute Boltzmann’s constant. The Fermi temperature is sim- ply a measure of the typical energies involved in an electron gas, but measured in Kelvin instead of eV. For typical metals the Fermi energy is εF ∼ 1 − 10 eV. This yields a Fermi temperature of the order of 10^4 K, a remarkably high value. This result is extremely important. It means that as far as the electron gas is concerned, room temperature (300 K) is actually a very very low temperature. Thus, the regime which matters for electron gases is actually the far opposite as that studied in the first quan- tum correction, Sec. 2.3. The interesting regime here is actually in the deep quantum regime, where quantum effects are very strong.
3.2 Finite temperatures
Now let us turn to the Fermi gas at finite temperatures. The Fermi-Dirac distribution
n¯(ε) =
eβ(ε−μ)^ + 1
μ ϵ
� �
Figure 4: Fermi-Dirac distribution (3.10) as a function of ε for low temperatures.
looks somewhat like Fig. 4 at low temperatures. It is essentially 1 when ε < μ and 0 otherwise. Indeed, in the limit where T → 0 (or β → ∞) the Fermi-Dirac function becomes the Heaviside theta function:
lim T → 0 n¯(ε) = θ(μ − ε), (3.11)
where
θ(x) =
1 x > 0 ,
0 x < 0.
In the limit of zero temperature, all states below μ are occupied, whereas all states above μ are empty. If we think about it for a second, we therefore conclude that at zero temperatures the chemical potential becomes the Fermi energy:
lim T → 0 μ = εF. (3.13)
The chemical potential can therefore be viewed as a sort of generalization of the notion of Fermi energy to finite temperatures. In this case the Fermionic occupations are not sharp, but instead are a bit blurred like in Fig. 4. In the vicinity of μ the different states have some finite probabilities of being occupied, which are neither 0 nor 1. The average number of particles and average energy are computed using Eqs. (2.12) and (2.14):
Ne = 〈 N〉ˆ =
∫^ ∞
0
dε D(ε)¯n(ε), (3.14)
U = 〈H〉 =
∫^ ∞
0
dε D(ε) ε n¯(ε). (3.15)
As discussed in the previous section, all that matters for fermionic systems like this are very low temperatures. The integral in Eq. (3.14), for instance, will look a bit like that in Fig. 5. If we had T = 0 we would get exactly the integral in Eq. (3.2). Since T , 0 the integral is distorted. But since all that matters are low temperatures, the integral is not distorted too much. Only slightly. And only in the vicinity of μ. The integrals in (3.14) and (3.15) are from 0 to ∞. However, the Fermi-Dirac distribution n¯(ε) essentially cuts this off above μ. The cut-off is not 100% sharp, but it is pretty abrupt.
