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Grand-canonical ensembles
As we know, we are at the point where we can deal with almost any classical problem (see below), but for quantum systems we still cannot deal with problems where the translational degrees of freedom are described quantum mechanically and particles can interchange their locations – in such cases we can write the expression for the canonical partition function, but because of the restriction on the occupation numbers we simply cannot calculate it! (see end of previous write-up). Even for classical systems, we do not know how to deal with problems where the number of particles is not fixed (open systems). For example, suppose we have a surface on which certain types of atoms can be adsorbed (trapped). The surface is in contact with a gas containing these atoms, and depending on conditions some will stick to the surface while other become free and go into the gas. Suppose we are interested only in the properties of the surface (for instance, the average number of trapped atoms as a function of temperature). Since the numbers of atoms on the surface varies, this is an open system and we still do not know how to solve this problem. So for these reasons we need to introduce grand-canonical ensembles. This will finally allow us to study quantum ideal gases (our main goal for this course). As we expect, the results we’ll obtain at high temperatures will agree with the classical predictions we already have, however, as we will see, the low-temperature quantum behavior is really interesting and worth the effort! Like we did for canonical ensembles, I’ll introduce the formalism for classical problems first, and then we’ll generalize to quantum systems. So consider an open system in contact with a thermal and particle reservoir.
reservoir:
system:
R R R R R
E,V,N,T,p, μ
E , V , N ,T , p , μ (^) R
Figure 1: Model of an open thermodynamic system. The system is in contact with a thermal and particle reservoir (i.e., both energy and particles get exchanged between the two, but E + ER = ET = const, N + NR = NT = const). The reservoir is assumed to be much bigger than the system, E ≪ ER, N ≪ NR. In equilibrium T = TR, μ = μR.
As we know (see thermo review), the macrostate of such a system will be characterized by its temperature T (equal to that of the reservoir), its volume V and its chemical potential μ (which will also be equal to that of the reservoir) – again, I am using a classical gas as a typical example. We call an ensemble of very many copies of our open system, all prepared in the same equilibrium macrostate T, V, μ, a grandcanonical ensemble. As always, our goal is to find out the relation- ships that hold at equilibrium between the macroscopic variables, i.e. in this case to find out how U, S, p, 〈N 〉, ... depend on T, V, μ. Note that because the number of particles is no longer fixed, we can only speak about the ensemble average 〈N 〉 (average number of particles in the container for
given values of T, V, μ) from now on.
1 Classical grand-canonical ensemble
As was the case for the canonical ensemble, our goal is to find the density of probability ρg.c.(N, q, p) to find the system in a given microstate – once we know this, we can compute any ensemble average and answer any question about the properties of the system. Note that since the number of microsystems (atoms or whatever may be the case) that are inside the system varies, we will specify N explicitly from now on: a microstate is characterized by how many microsystems are in the system in that microstate, N , and for each for these microsystems we need f generalized coordinates and f generalized momenta to describe its behavior, so we have a total of 2N f microscopic variables q, p. So we need to figure out ρg.c.(N, q, p). We will follow the reasoning we used for canonical ensembles
- have a quick look over that and you’ll see the parallels. In fact, I’m going to copy and paste text from there, making only the appropriate changes here and there. First, we reduce the problem to one that we already know how to solve, by noting that the total system = system + reservoir is isolated, and so we can use microcanonical statistics for it. The microstate of the total system is characterized by the generalized coordinates N, q, p; NR, qR, pR, where the latter describe the microsystems that are inside the reservoir. Clearly, N + NR = NT is the total number of microsystems in the total system, which is a constant. So in fact the total system’s microstate is characterized by: N, q, p; NT − N, qR, pR. Its Hamiltonian is HT (N, q, p; NR, qR, pR) = H(q, p) + HR(qR, pR) – here I won’t write the dependence of the Hamiltonian on N explicitly, but we know it’s there: for example, in the kinetic energy we sum over the energies of the microsystems inside the system, and the number of terms in the sum is N , i.e. how many microsystems are contributing to the energy in that particular microstate. Based on the microcanonical ensemble results, we know that the density of probability to find the total system in the microstate N, q, p; NT − N, qR, pR^ is:
ρmc(N, q, p; NT −N, qR, pR) =
{ (^1) ΩT (ET ,δET ,V,VR,NT ) if^ ET^ ≤ HT^ (N,^ q,^ p;^ NT^ −^ N,^ q
R, pR) ≤ ET + δET 0 otherwise
where ΩT (ET , δET , V, VR, NT ) is the multiplicity for the total system. This is the same as saying that the total probability to find the system in a microstate with N microsystems that are located between q, q + dq and p, p + dp AND the reservoir to have the remaining NT − N microsystems located between qR, qR^ + dqR^ and pR, pR^ + dpR^ is:
dqdpdqRdpR GN GNT −N hN f^ +(NT^ −N^ )fR
ρmc(N, q, p; NT − N, qR, pR).
Note that even though the microsystems are the same, they may have different number of degrees of freedom in the system and the reservoir – think about the previous example with atoms adsorbed on a surface. It is very likely that we need different degrees of freedom to describe the state of an atom when on the surface (inside the system) than when it’s in the gas (the reservoir). However, the total number of atoms on the surface plus inside the gas is fixed, and that’s all that matters. We do not care/need to know what the reservoir is doing, all we want to know is the probability that the system is in a microstate which has N microsystems between q, q + dq and p, p + dp. To
Note that here “sum over all microstates” means to sum over all possible numbers N of microsystems in the system, and for each N to “sum” over all possible locations/momenta of the microsystems. Of course, for classical systems we know that the “sum” over locations/momenta is really an integral, because these are continuous variables. You might argue that we should stop the sum over N at NT , but since NT is by definition much much bigger than the average number of microsystems in the system, it will turn out that we can use the upper limit to be infinity – it makes calculations easier and the “error” can be made arbitrarily small by making the reservoir bigger, i.e. NT → ∞. Note also that Z is a function of T (through the β in exponent), of V (the integrals over positions are restricted to the volume V ), and μ (again through the exponent). Now that we know the grandcanonical density of probability, we can calculate the internal energy
U = 〈H(q, p)〉 =
∑^ ∞
N =
∫ (^) dqdp
GN hN f^
ρg.c.(N, q, p)H(q, p) =
Z
∑^ ∞
N =
∫ (^) dqdp
GN hN f^
H(q, p)e−β[H(q,p)−μN^ ]
Here we have to be a bit careful. We can’t simply use the trick with the derivative with respect to β, since this will bring down both H (which we want), but also μN (which we don’t want):
∂β
e−β[H(q,p)−μN^ ]^ = [H(q, p) − μN ] e−β[H(q,p)−μN^ ]^6 = H(q, p)e−β[H(q,p)−μN^ ]
So here’s what we do. Let me define: α = βμ
and use this instead of μ as a variable, so that Z = Z(β, V, α) and
ρg.c.(N, q, p) =
Z
e−βH(q,p)+αN
Now, we have to pretend that α and β are independent variables, i.e. we “forget” for a bit what is definition of α, we pretend that it’s just some quantity totally unrelated to β. If this was true, then we could use:
−
∂β
e−βH(q,p)+αN^ = H(q, p)e−βH(q,p)+αN
and we could then write:
U =
Z
[ −
∂β
] (^) ∞ ∑
N =
∫ (^) dqdp
GN hN f^
e−βH(q,p)+αN^ →
U = −
Z(β, V, α)
∂β
Z(β, V, α) = −
∂β
ln Z(β, V, α)
So the point is to treat α as a variable independent of β while we take the derivative, and only after we’re done with taking the derivative to remember that α = βμ. This “forgetfulness” is very useful since it allows us to calculate another ensemble average very simply, namely:
〈N 〉 =
∑^ ∞
N =
∫ (^) dqdp
GN hN f^
ρg.c.(N, q, p)N =
Z(β, V, α)
∑^ ∞
N =
∫ (^) dqdp
GN hN f^
N e−βH(q,p)+αN
Again, treating α and β as independent variables while we take derivatives, we have:
∂ ∂α
e−βH(q,p)+αN^ = N e−βH(q,p)+αN
so that we avoid doing the integrals and we find:
〈N 〉 =
Z(β, V, α)
∂α
Z(β, V, α) =
∂α
ln Z(β, V, α).
So we can easily also calculate the average number of microsystems in the system. We will look at some examples soon and you’ll see that doing this is simple in practice, it just requires a bit of attention when taking the derivatives. This approach can be extended easily to find (check!) that:
〈H^2 〉 =
Z(β, V, α)
∂^2
∂β^2
Z(β, V, α)
and
〈N 2 〉 =
Z(β, V, α)
∂^2
∂α^2
Z(β, V, α).
Of course, we would need these quantities to calculate standard deviations. Now, this trick I described above, with using α and β, is what people usually do, and what is given in textbooks etc. All is needed is that while you take the derivatives, you treat α and β as independent variable. For reasons which escape me, some students think that this is too fishy and refuse to use this trick. So here is an alternate trick, which is almost as good (takes just a bit more work) and gives the precise same answers at the end of the day. Let’s look again at:
U = 〈H(q, p)〉 =
∑^ ∞
N =
∫ (^) dqdp
GN hN f^
ρg.c.(N, q, p)H(q, p) =
Z
∑^ ∞
N =
∫ (^) dqdp
GN hN f^
H(q, p)e−β[H(q,p)−μN^ ]
Clearly we’d like to not have to do the integrals explicitly, so we have to get rid of the H somehow. If you do not like the trick with introducing α, then we can do this. First, introduce an x in front of the Hamiltonian, in the exponent, and ask that x be set to 1 at the end of the calculation, since clearly:
U =
Z
∑^ ∞
N =
∫ (^) dqdp
GN hN f^
H(q, p)e−β[xH(q,p)−μN^ ]
∣∣ ∣∣ ∣ x= Now we can take the derivative with respect to x, so that we have:
U =
Z
[ −
β
∂x
Z(x)
]∣∣ ∣∣ ∣x=
where
Z(x) =
∑^ ∞
N =
∫ (^) dqdp
GN hN f^
e−β[xH(q,p)−μN^ ]
can be quickly calculated (this is where the extra work comes in), just like you calculated Z = Z(x =
- (it’s just a matter of tracking where the extra x goes in some gaussians). So, putting these together, and since we set x = 1 at the end, we have:
U = −
β
∂x
ln Z(x)
∣∣ ∣∣ ∣ x=
and similarly
〈H^2 〉 =
β^2
∂^2
∂x^2
Z(x)
∣∣ ∣∣ ∣ x=
We can also calculate the internal energy and 〈N 〉 from
U = −
∂β
ln Z(β, V, α); 〈N 〉 =
∂α
ln Z(β, V, α);
where α, β are treated as independent variables while we take the derivatives, after which we can set α = βμ. Similarly, we can calculate averages of 〈H^2 〉, 〈N 2 〉, 〈HN 〉, ..., but using the proper number of derivatives with respect to α and β. If you do not like this, use the trick with introducing the x, and then setting it to be 1 after all derivatives were taken. Any other ensemble averages are calculated starting from the definition of an ensemble average and the known density of probability for the grandcanonical ensemble, and by doing all integrals over positions/momenta and sum over N. One last thing. Remember that for the canonical partition function and non-interacting systems, we could use the factorization theorem to simplify the calculation. It turns out we can do a similar thing here. First, start with the general definition of the grand-canonical partition function:
Z(T, V, μ) =
∑^ ∞
N =
∫ (^) dqdp
GN hN f^
e−β[H(q,p)−μN^ ]^ =
∑^ ∞
N =
eβμN
∫ (^) dqdp
GN hN f^
e−βH(q,p)
Now we recognize that the integrals are simply the canonical partition function for a system with N particles, so:
Z(T, V, μ) =
∑^ ∞
N =
eβμN^ Z(T, V, N )
So in fact, if we know how to calculate Z (which we do) there isn’t much left of the calculation. Let’s simplify even further. For non-interacting systems where particles can move and exchange positions (such as gases), we know from the factorization theorem that:
Z(T, V, N ) =
N!
[z(T, V )]N
Using this in the above sum, we find:
Z(T, V, μ) =
∑^ ∞
N =
N!
[ z(T, V )eβμ
]N = exp
( eβμz(T, V )
)
since the sum is just the expansion of the exponential function. Those of you with good memory will be delighted to learn that eβμ^ has its own special name, namely fugacity. For problems where the microsystems are distinguishable because they are located at different spatial location (crystal-type problems), Gibbs’ factor is 1 and:
Z(T, N ) = [z(T )]N
and therefore:
Z(T, V, μ) =
∑^ ∞
N =
[ eβμz(T )
]N
1 − eβμz(T )
if |eβμz(T )| < 1 (otherwise the geometric series is not convergent). Of course, we’ll find that this condition is generally satisfied. So the conclusion is that once we calculate z (just as we did it for canonical ensembles), we immediately have Z, i.e. dealing with a grandcanonical classical system really does not involve any more work/math that dealing with a canonical system – at least so far as classical problems are concerned.
1.1 Classical ideal gas
Let’s check how this works for a classical ideal gas – our favorite classical model. Assume a system of volume V in contact with a thermal and simple atom reservoir with temperature and chemical potential T, μ. Let’s calculate the average number of particles in the container, 〈N 〉, their internal energy U and their pressure p. Because these are simple, non-interacting atoms, they have f = 3 degrees of freedom, and we can use factorization theorem, and GN = N !. Of course, we know that:
z(T, V ) =
h^3
∫ d~r
∫ d~pe−β^ 2 ~pm^2 = V
( 2 πmkB T h^2
) (^32) → Z(T, V, N ) =
zN N!
(three identical gaussians plus one integral over the volume). Therefore (see formula above):
Z =
∑^ ∞
N =
eβμN^ Z(T, V, N ) =
∑^ ∞
N =
N!
[ eβμz
]N = exp
eβμV
( 2 πmkB T h^2
) 32
and so:
φ(T, V, μ) = −kB T ln Z = −kB T eβμV
( 2 πmkB T h^2
) (^32)
This is indeed an extensive quantity. It is maybe not so obvious that it has the right (energy) units, but you should be able to convince yourselves that that is true (remember that z is a number, and μ is an energy). To calculate 〈N 〉 we have two alternatives, either as a partial derivative of φ with respect to μ (I’ll let you do this) or using the trick with α and β. Let’s do it by the second method. First, we replace βμ → α everywhere where they appear together. We then find:
Z(β, V, α) = exp
eαV
( 2 πm βh^2
) 32 (^) → ln Z(β, V, α) = eαV
( 2 πm βh^2
) (^32)
Now, assuming α and β to be independent variables, we have:
〈N 〉 =
∂α
ln Z(β, V, α) = eαV
( 2 πm βh^2
) (^32)
This looks a bit strange, but let’s not loose heart – this tells us how 〈N 〉 depends upon T, V, μ (the variables which characterize the macrostate for the open system) and it is not something we’ve looked at before. Notice that you can extract how μ depends on T, V, 〈N 〉 from this – you should do that and see that the result agrees with what we obtained for canonical ensembles, if we replace N → 〈N 〉. The internal energy is:
U = −
∂β
ln Z(β, V, α) =
2 β
eαV
( 2 πm βh^2
) (^32) → U =
〈N 〉kB T
So yeeeii! It works!
system). α are all the needed quantum numbers to describe these levels – degeneracies are very important! These energies (and the single-particle wavefunctions associated with them) are usually called single-particle energies or single-particle orbitals. To make things more clear, let’s look at some examples as we go along. First, a simple “crystal”- like example. Assume we have a surface with a total number NT of sites where simple atoms could be trapped. If a trap is empty, its energy is zero. If it catches an atom, its energy is lowered to −ǫ 0 < 0. This surface is in contact with a gas of atoms (the reservoir), with known T and μ. The question could be, for instance, what is the average number 〈N 〉 of atoms trapped on the surface. What are the single-particle orbitals, in this case? Well, if we have a single atom in the system= surface with traps, it must be trapped in some site or other, and the energy will be −ǫ 0. We could use as “quantum number” an integer 1 ≤ n ≤ NT which tells us at which site is the atom trapped – so here we have a NT degenerate spectrum of single-particle states, all with the same energy −ǫ 0. As a second example, let’s consider a quantum gas problem. Assume simple atoms with quantum dynamics in a cubic box of volume V = L^3. Of course, we’ll generally want to know what are the properties of the quantum gas when the system is in contact with a thermal and particle reservoir with known T, μ. However, right now all we want to know, is what are the single particle levels. For this, we must find the spectrum (the eigenstates) when there is just one atom in the system. This we know how to do. In this case the Hamiltonian is:
ˆh = − ¯h
2 2 m
( d^2 dx^2
d^2 dy^2
d^2 dz^2
)
and the eigenstates are:
enx,ny ,nz =
h^2 8 mL^2
(n^2 x + n^2 y + n^2 z )
where nx, ny, nz = 1, 2 , ... are strictly positive integers. So here three quantum number α = nx, ny, nz characterize the single-particle orbitals (if you don’t remember where this formula comes from, it is a simple generalization of the 1d case we solved when we looked at multiplicities, when we discussed microcanonical ensembles). Strictly speaking, atoms also have some spin S, so we should actually include a 4th quantum number α = nx, ny, nz , m where m = −S, −S + 1, ...S − 1 , S is the spin projection. For any other problem we can figure out the single-particle orbitals (or states) similarly. Now let’s go back to our general description, where eα are the energies of all possible single-particle orbitals. What happens if there are more microsystems in the system? Well, let’s start with two. Because these are non-interacting microsystems, they occupy the same set of single-particle orbitals. So to characterize the state, we now need two pairs of quantum numbers, say α and α′, to tell us which two states are occupied. The energy is simply eα + eα′^. For example, if there is a second atom on the surface, it must also be trapped in some trap or other, just like the first one, so I could specify the state by saying atom 1 is trapped at site n while atom 2 is trapped at site n′. Of course, the energy is − 2 ǫ 0. Similarly, a second atom in the box will be in some eigenstates en′ x,n′ y ,n′ z and the energy will be the sum of the two. The wavefunction for the total system is now:
Ψ(~r 1 , ~r 2 ) = φα(~r 1 )φβ (~r 2 )
where φα(~r) is the single-particle wavefunction associated with the single-particle state eα. Right? WRONG! What quantum mechanics tells us is that if the particles are identical, their wavefunction must be either symmetric (for so-called bosonic particles, i.e. whose spin S is an integer) or antisymmetric (for so-called fermionic particles, i.e. whose spin S is half-integer ) to exchanges of the two, i.e.
Ψ(~r 1 , ~r 2 ) = ±Ψ(~r 2 , ~r 1 ) → |Ψ(~r 1 , ~r 2 )|^2 = |Ψ(~r 2 , ~r 1 )|^2
This is what indistinguishability really means. If the particles are truly identical, then there is no measurement whatsoever that we can perform to tell us which of the two particles is at ~r 1 and which at ~r 2 , so if we exchanged their positions we should see no difference (same probability to find them at those locations). So going back, it follows that for two fermions, the two-particle wavefunction must be: ΨF (~r 1 , ~r 2 ) = φα(~r 1 )φβ (~r 2 ) − φα(~r 2 )φβ (~r 1 )
while for bosons, we must have:
ΨB (~r 1 , ~r 2 ) = φα(~r 1 )φβ (~r 2 ) + φα(~r 2 )φβ (~r 1 )
(there is actually an overall normalization factor, but that is just a constant that has no relevance for our discussion). This immediately tells us that we cannot have two fermions occupy the same single-electron or- bitals, i.e. α 6 = β always. If α = β we find ΨF (~r 1 , ~r 2 ) = 0, which is not allowed (wavefunctions must normalize to 1). This is known as Pauli’s exclusion principle. There is no such restriction for bosons, there we can have any number of bosons occupying the same single-particle state. If we now look at these two-particle wavefunctions, we see that it’s just as likely that particle 1 is in state α and 2 in β, as it is to have 1 in state β and 2 in state α. Therefore, it makes much more sense to characterize this state by saying that there is a particle in state α and a particle in state β, and not attempt anymore to say which is which – they’re identical and either one could be in either state with equal probability. We can rephrase this by saying that of all single-particle orbitals, only α and β are occupied, while all other ones are empty. In fact, we can define an occupation number which is an integer nα associated to each single-particle level. For empty levels nα = 0, while for occupied levels nα counts how many particles are in that particular orbital. For bosons, nα = 0, 1 , 2 , 3 , ... – could be any number between 0 and infinity. For fermions, nα = 0 or 1! Because of the exclusion principle, we cannot have more than 1 fermion occupying a state. This is the difference between fermions and bosons. It might not look like much, but as we will see, it will lead to extraordinarily different behavior of fermions vs. bosons at low temperatures, i.e. where quantum behavior comes into play. We can now generalize. For any number of microsystems (particles) present in the system, we can specify a possible microstate by giving the occupation numbers for all the single-particle orbitals (if there is an infinite number of orbitals, such as for a particle in a box, lots and lots of occupations numbers will be zero; but we still have to list all – infinite number – of them). So the microstate is now specified through the values {nα} of all occupation numbers in that microstate. Allowing all numbers {nα} to take all their possible values will generate all possible microstates for all possible numbers of microsystems. Of course, the total number of microsystems (particles) in the microstate must be
N{nα} =
∑ α
nα
i.e. we go through all levels and sum how many particles are occupying each of them – clearly this sum gives the total number of particles in that microstate. The energy of this microstate is:
E{nα} =
∑ α
nαeα
again we go through all levels, for each one which is occupied (nα 6 = 0) we add how many particles are in that level, nα, times the energy of each one of them, eα. Again, I think this should be quite
a finite number of single-particle levels indexed by the integer α → n, 1 ≤ n ≤ NT which tells us at which site is the single atom trapped. Let’s assume that the atoms are fermionic, i.e. there can be at most 1 atom in any trap (you might want to ask me some questions here ....). The microstate is now described by the occupation numbers n 1 , ..., nNT where ni is zero if trap i is empty and 1 if trap 1 is occupied by an atom. The number of atoms in the microstate is N =
∑NT i=1 ni, and the energy is^ E^ =^
∑NT i=1(−ǫ^0 )ni^ = −ǫ 0 N , and E − μN = (−ǫ 0 − μ)
∑NT i=1 ni. In this case:
ZF =
∑^1 n 1 =
∑^1 n 2 =
∑^1 nNT =
e−β(E−μN^ )^ =
∑^1 n 1 =
∑^1 n 2 =
∑^1 nT =
e−β(−ǫ^0 −μ)(n^1 +···+nNT^ )
∑^1 n 1 =
e−β(−ǫ^0 −μ)n^1
∑^1 n 2 =
e−β(−ǫ^0 −μ)n^2
(^) · · ·
∑^1 nNT =
e−β(−ǫ^0 −μ)nNT
(^) =
[ 1 + e−β(−ǫ^0 −μ)
]NT
In this case, there is a finite number of single particle orbitals, and each contributes the same since they all have the same energy −ǫ 0. In the general case, each orbital contributes 1 + e−β(eα−μ), and we must multiply over all the orbitals. That’s precisely what the general formula for ZF means. All the other formulae we have derived for classical grand-canonical system hold unchanged, in particular: φF = −kB T ln Z = −kB T
∑ α
ln
[ 1 + e−β(eα−μ)
]
For different fermionic systems we’ll have different energies ǫα and number of levels α, but this formula will always hold. Let us do the same for a bosonic system. In that case:
ZB =
∏ α
∑^ ∞
nα=
(^) e−β^
∑ α nα(eα−μ)
since occupation numbers can now be anything. Again we can factorize the product:
ZB =
∏ α
∑^ ∞
nα=
e−βnα(eα−μ)
Each sum is an infinite geometric series. Note that we must have:
e−β(ǫα−μ)^ ≤ 1
in order for each of these series to be convergent. Since β > 0, it follows that for a bosonic system, we must always have μ ≤ eα for all single particle levels and therefore
for bosons, we must have: μ ≤ eGS
where eGS is the energy of the single-particle ground-state. This restriction will turn out to have important consequences. For fermions we have no restrictions for the chemical potential. If the restriction μ ≤ eGS is satisfied, then each geometric series ∑∞ n=0 x n (^) = 1/(1−x) is convergent,
and we find:
ZB =
∏ α
1 − e−β(eα−μ)
)
and φB = −kB T ln ZB = +kB T
∑ α
ln
[ 1 − e−β(eα−μ)
]
In fact, because of the similarities of the formulae, we can group together the results for both fermions and bosons and write: Z =
∏ α
( 1 ± e−β(eα−μ)
)± 1
and φ = −kB T ln Z = ∓kB T
∑ α
ln
( 1 ± e−β(eα−μ)
)
where the upper sign is for fermions, and the lower sign is for boson systems. From partial derivatives of φ we can calculate S, p, 〈N 〉, as usual, since dφ = −SdT − pdV − 〈N 〉dμ. We can also use the tricks with α and β to find U and 〈N 〉. Let’s remember them, and check that they still hold. First, we replace βμ → α everywhere this product appear. In terms of α and β, we have:
pμstate =
Z
e−βEμstate+αNμstate
where from normalization,
Z(α, β, ...) =
∑
μstates
e−βEμstate+αNμstate
By definition,
U =
∑ μstates
pμstateEμstate =
Z
∑ μstates
Eμstatee−βEμstate+αNμstate^ = −
Z
∂β
Z(α, β, ...) = −
∂β
ln Z(α, β, ...)
if, while taking the derivative, we pretend that α and β are independent variables. Similarly,
〈N 〉 =
∑ μstates
pμstateNμstate =
Z
∑ μstates
Nμstatee−βEμstate+αNμstate^ =
Z
∂α
Z(α, β, ...) =
∂α
ln Z(α, β, ...)
So indeed, we have precisely the same formulae as before. This is because the only difference is what is meant by
∑ μstates. For classical systems that implies a sum over^ N^ and many integrals over all classical degrees of freedom; for quantum systems, this is a sum over all possible occupation numbers. But nothing in the derivation depended on such details. For our quantum system, after replacing βμ → α, we have:
ln Z(α, β, ...) = ±
∑ γ
ln
( 1 ± e−βeγ^ +α
)
(again, upper sign for fermions, lower sign for bosons). I prefer to call the quantum numbers γ this time, since α = βμ is now taken. If we take the derivatives, we find that:
U = −
∂β
ln Z(α, β, ...) = −(±)
∑ γ
∓eγ e−βeγ^ +α 1 ± e−βeγ^ +α^
∑ γ
eγ
eβ(eγ^ −μ)^ ± 1
and
〈N 〉 =
∂α
ln Z(α, β, ...) = (±)
∑ γ
e−βeγ^ +α 1 ± e−βeγ^ +α^
∑ γ
eβ(eγ^ −μ)^ ± 1
(we can go back to α → βμ after we took the derivatives. Results are generally in terms of μ.
For bosons, the average occupation number of a level with energy ǫγ is:
〈nγ 〉 =
eβ(eγ^ −μ)^ − 1
This is called the Bose-Einstein distribution. Let’s analyze it a bit. First, remember that for bosons we must always have μ ≤ eGS → eγ − μ ≥ 0. Now we see that this is very necessary, since with this restriction eβ(eγ^ −μ)^ ≥ 1 and the average occupation numbers are positive! They must be positive – the average of any number whose only allowed values are 0, 1 , 2 , ... cannot be negative. Unlike for fermions, however, we see that an average occupation number could be anything between 0 and infinity. In fact, let us consider T = 0 behavior. Here we have two cases: (1) μ < eGS. In this case eβ(eγ^ −μ)^ → ∞ as β → ∞, T → 0, so all average occupation numbers become vanishingly small. Since 〈N 〉 =
∑ γ 〈nγ^ 〉, if all^ 〈nγ^ 〉 →^ 0 then^ 〈N^ 〉 →^ 0.^ This can happen if conditions are such that particles would rather not stay in the system (they prefer to be in the bath at low-temperatures). But this is a rather boring case. The more interesting case is when (ii) μ = eGS. In this case, the average occupation numbers are still zero for all higher energy levels, but we see that the ground-state itself has an infinite occupation number! Clearly, that can’t be quite right – indeed, we’ll have to do this analysis more carefully when we study the so-called Bose-Einstein condensation. What this result tells us, though, is that for bosons, at T = 0 all particles that are in the system (however many they may be) occupy the ground-state orbital. This does make sense! We know that at T = 0 we expect the system to go into its ground-state. Of course, the lowest total energy is obtained when we place the particles in the orbitals with the lowest possible energies. For bosons, we are allowed to place all of them in the single-particle ground-state orbital, and that is indeed the lowest total energy possible. For fermions, we cannot put more that 1 particle in a state, therefore to get the lowest total energy possible, we occupy all the lowest energy states available with one particle in each – which is what the Fermi-Dirac distribution predicts at T = 0. So you see how the change of sign in the denominator leads to extremely different results! You might now wonder how is it possible that a gas of bosonic atoms and a gas of fermionic atoms will behave the same at high-temperatures, given how differently they behave at low-T. After all, we know that at high-T we should have agreement with the classical predictions, for e.g. find that pV = N kB T , etc. Since there is only one set of relationships for classical gases, it follows that both bosons and fermions should behave the same at high temperatures. For this to happen, clearly these average occupation numbers should also be equal at high-temperatures, otherwise we should be able to tell the difference somehow. Interestingly enough, this indeed happens. What we will show a bit later on (you’ll have to just believe me for now) is that at high-temperature we have to set μ to be an extremely large negative number, μ ≪ −kB T , if we want to have large numbers of particles in the box, 〈N 〉 ∼ 1023. All one-particle levels eγ = enx,ny ,nz in a box have positive energies, and therefore now β(eγ − μ) ≫ 1 → eβ(eγ^ −μ)^ ≫ 1 → eβ(eγ^ −μ)^ ± 1 ≈ eβ(eγ^ −μ). So the sign doesn’t make any difference anymore, and at high temperatures we find both for fermions and bosons that:
〈nγ 〉 ≈ e−β(eγ^ −μ)^ ≪ 1
As we will show soon, in this limit we indeed find agreement with the expected classical results.
