f ( T ) T ln g
.
(3.16b)
2
1
2
The first of these relations exactly coincides with Eq. (1.45), which was derived in Sec. 1.4 from the equation of state PV = NT, using thermodynamic identities. At that stage, this equation of state was just postulated, but now we can derive it by calculating the pressure from the second of Eqs. (1.35), and Eq. (16a):
F
NT
P
.
(3.17)
V
V
T
So, the equation of state of the ideal classical gas, with density n N/ V, is indeed given by Eq. (1.44): NT
P
nT .
(3.18)
V
Hence we may use Eqs.(1.46)-(1.51), derived from this equation of state, to calculate all other thermodynamic variables of the gas. For example, using Eq. (1.47) with f( T) given by Eq. (16b), for the internal energy and the specific heat of the gas we immediately get
6 As a reminder, we have already used this rule (twice) in Sec. 2.6, with particular values of g.
7 For the opposite limit when N = g = 1, Eq. (15) yields the results obtained, by two alternative methods, in the solutions of Problems 2.8 and 2.9. Indeed, for N = 1, the “correct Boltzmann counting” factor N! equals 1, so that the particle distinguishability effects vanish – naturally.
Chapter 3
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df T 3
C
E
V
1
3
E N f T
T
NT ,
c
,
(3.19)
V
dT 2
N
N T
2
V
in full agreement with Eq. (8) and hence with the equipartition theorem.
Much less trivial is the result for entropy, which may be obtained by combining Eqs. (1.46) and (16a):
F
V
df T
( )
S
N ln
.
(3.20)
T V
N
dT
This formula,8 in particular, provides the means to resolve the following gas mixing paradox (sometimes called the “Gibbs paradox”). Consider two volumes, V 1 and V 2, separated by a partition, each filled with the same gas, with the same density n, at the same temperature T, and hence with the same pressure P.
Now let us remove the partition and let the gas portions mix; would the total entropy change? According to Eq. (20), it would not, because the ratio V/ N = n, and hence the expression in the square brackets is the same in the initial and the final state, so that the entropy is additive, as any extensive variable should be. This makes full sense if the gas particles in both parts of the volume are truly identical, i.e. the partition’s removal does not change our information about the system. However, let us assume that all particles are distinguishable; then the entropy should clearly increase because the mixing would decrease our information about the system, i.e. increase its disorder. A quantitative description of this effect may be obtained using Eq. (11). Repeating for Z dist the calculations made above for Z, we readily get a different formula for entropy:
df ( T )
3 / 2
mT
dist
S
N ln V
,
f ( T ) T ln g
.
(3.21)
dist
dist
dT
2
2
Please notice that in contrast to the S given by Eq. (20), this entropy includes the term ln V
instead of ln( V/ N), so that S dir is not proportional to N (at fixed temperature T and density N/ V). While for distinguishable particles this fact does not present any conceptual problem, for indistinguishable particles it would mean that entropy was not an extensive variable, i.e. would contradict the basic assumptions of thermodynamics. This fact emphasizes again the necessity of the correct Boltzmann counting in the latter case.
Using Eq. (21), we can calculate the change of entropy due to mixing two gas portions, with N 1
and N 2 distinguishable particles, at a fixed temperature T (and hence at unchanged function f dist): V V
V V
S
N N
V V N
V N
V N
N
.
(3.22)
dist
1
2 ln(
)
1
2
ln
ln
1
1
2
2
ln 1
2
ln 1
2
0
1
2
V
V
1
2
Note that for a particular case, V 1 = V 2 = V/2, Eq. (22) reduces to the simple result, S dist = ( N 1 + N 2) ln2, which may be readily understood in terms of the information theory. Indeed, allowing each particle of the total number N = N 1 + N 2 to spread to a twice larger volume, we lose one bit of information per particle, i.e. I = ( N 1 + N 2) bits for the whole system. Let me leave it for the reader to show that Eq. (22) is also valid if particles in each sub-volume are indistinguishable from each other, but different from 8 The result represented by Eq. (20), with the function f given by Eq. (16b), was obtained independently by O.
Sackur and H. Tetrode as early as in 1911, i.e. well before the final formulation of quantum mechanics in the late 1920s.
Chapter 3
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those in another sub-volume, i.e. for mixing of two different gases.9 However, it is certainly not applicable to the system where all particles are identical, stressing again that the correct Boltzmann counting (12) does indeed affect the gas entropy, even though it may be not as consequential as the Maxwell distribution (5), the equation of state (18), and the average energy (19).
In this context, one may wonder whether the change (22) (called the mixing entropy) is experimentally observable. The answer is yes. For example, after free mixing of two different gases, and hence boosting their total entropy by S dist, one can use a thin movable membrane that is semipermeable, i.e. whose pores are penetrable for particles of one type only, to separate them again, thus reducing the entropy back to the initial value, and measure either the necessary mechanical work
W = T S dist or the corresponding heat discharge into the heat bath. Practically, measurements of this type are easier in weak solutions 10 – systems with a small concentration c << 1 of particles of one sort ( solute) within much more abundant particles of another sort ( solvent). The mixing entropy also affects the thermodynamics of chemical reactions in gases and liquids.11 Note that besides purely thermal-mechanical measurements, the mixing entropy in some conducting solutions ( electrolytes) is also measurable by a purely electrical method, called cyclic voltammetry, in which a low-frequency ac voltage, applied between two solid-state electrodes embedded in the solution, is used to periodically separate different ions, and then mix them again.12
Now let us briefly discuss two generalizations of our results for ideal classical gases. First, let us consider such gas in an external field of potential forces. It may be described by replacing Eq. (3) with 2
p
k
U (r ) ,
(3.23)
k
2
k
m
where r k is the position of the k th particular particle, and U(r) is the potential energy of the particle. If the potential U(r) is changing in space sufficiently slowly,13 Eq. (4) is still applicable, but only to small volumes, V → dV = d 3 r whose linear size is much smaller than the spatial scale of substantial variations of the function U(r). Hence, instead of Eq. (5), we may only write the probability dW of finding the particle in a small volume d 3 rd 3 p of the 6-dimensional phase space: 9 By the way, if an ideal classical gas consists of particles of several different sorts, its full pressure is a sum of independent partial pressures exerted by each component – the so-called Dalton law. While this fact was an important experimental discovery in the early 1800s, for statistical physics this is just a straightforward corollary of Eq. (18), because in an ideal gas, the component particles do not interact.
10 Interestingly, the statistical mechanics of weak solutions is very similar to that of ideal gases, with Eq. (18) recast into the following formula (derived in 1885 by J. van ’t Hoff), PV = cNT, for the partial pressure of the solute. One of its corollaries is that the net force (called the osmotic pressure) exerted on a semipermeable membrane is proportional to the difference of the solute concentrations it is supporting.
11 Unfortunately, I do not have time for even a brief introduction into this important field, and have to refer the interested reader to specialized textbooks – for example, P. A. Rock, Chemical Thermodynamics, University Science Books, 1983; or P. Atkins, Physical Chemistry, 5th ed., Freeman, 1994; or G. M. Barrow, Physical Chemistry, 6th ed., McGraw-Hill, 1996.
12 See, e.g., either Chapter 6 in A. Bard and L. Falkner, Electrochemical Methods, 2nd ed., Wiley, 2000 (which is a good introduction to electrochemistry as the whole); or Sec. II.8.3.1 in F. Scholz (ed.), Electroanalytical Methods, 2nd ed., Springer, 2010.
13 Quantitatively, the effective distance of substantial variations of the potential, T/ U(r), has to be much larger than the mean free path l of the gas particles, i.e. the average distance a particle passes its successive collisions with its counterparts. (For more on this notion, see Chapter 6 below.)
Chapter 3
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3
p 2
3
U (r)
dW w(r, p) d rd p,
w(r, p) const exp
.
(3.24)
2 mT
T
Hence, the Maxwell distribution of particle velocities is still valid at each point r, so that the equation of state (18) is also valid locally. A new issue here is the spatial distribution of the total density, n(r) N w(r, p d 3
)
p ,
(3.25)
of all gas particles, regardless of their momentum/velocity. For this variable, Eq. (24) yields14
U (r)
n(r) n( )
0 exp
,
(3.26)
T
where the potential energy at the origin (r = 0) is used as the reference of U, and the local gas pressure may be still calculated from the local form of Eq. (18):
U (r)
P(r) n(r T
) P( )
0 exp
.
(3.27)
T
A simple example of numerous applications of Eq. (27) is an approximate description of the Earth’s atmosphere. At all heights h << R E ~ 6106 m above the Earth’s surface (say, above the sea level), we may describe the Earth gravity effect by the potential U = mgh, and Eq. (27) yields the so-called barometric formula
h
T
k T
P( h) P( )
0 exp
,
h
B K
with
.
(3.28)
h
0
mg
mg
0
For the same N2, the main component of the atmosphere, at T K = 300 K, h 0 ≈ 7 km. This gives the correct order of magnitude of the atmosphere’s thickness, though the exact law of the pressure change differs somewhat from Eq. (28), because the flow of radiation from Sun and Earth cause a relatively small deviation of the atmospheric air from the thermal equilibrium: a drop of its temperature T with height, with the so-called lapse rate of about 2% (~6.5 K) per km.
The second generalization I need to discuss is to particles with internal degrees of freedom. Now ignoring the potential energy U(r), we may describe them by replacing Eq. (3) with p 2
' ,
(3.29)
k
2 m
k
where k’ describes the internal energy spectrum of the k th particle. If the particles are similar, we may repeat all the above calculations, and see that all their results (including the Maxwell distribution, and the equation of state) are still valid, with the only exception of Eq. (16), which now becomes
mT 3/ 2
'k
f T
( ) T ln g
1 ln
.
(3.30)
2
exp
2
'
T
k
14 In some textbooks, Eq. (26) is also called the Boltzmann distribution, though it certainly should be distinguished from Eq. (2.111).
Chapter 3
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As we already know from Eqs. (1.50)-(1.51), this change may affect both specific heats of the ideal gas – though not their difference, cV – cP = 1. They may be readily calculated for usual atoms and molecules, at not very high temperatures (say the room temperature of ~25 meV), because in these conditions, k’ >> T for most their internal degrees of freedom, including the electronic and vibrational ones. (The typical energy of the lowest electronic excitations is of the order of a few eV, and that of the lowest vibrational excitations is only an order of magnitude lower.) As a result, these degrees of freedom are “frozen out”: they are in their ground states, so that their contributions exp{- k’/ T} to the sum in Eq. (30), and hence to the heat capacity, are negligible. In monoatomic gases, this is true for all degrees of freedom besides those of the translational motion, already taken into account by the first term in Eq. (30), i.e. by Eq. (16b), so that their specific heat is typically well described by Eq. (19).
The most important exception is the rotational degrees of freedom of diatomic and polyatomic molecules. As quantum mechanics shows,15 the excitation energy of these degrees of freedom scales as
2/2 I, where I is the molecule’s relevant moment of inertia. In the most important molecules, this energy is rather low (e.g. for N2, it is close to 0.25 meV, i.e. ~1% of the room temperature), so that at usual conditions they are well excited and, moreover, behave virtually as classical degrees of freedom, each giving a quadratic contribution to the molecule’s energy, and hence obeying the equipartition theorem, i.e. giving an extra contribution of T/2 to the energy, i.e. ½ to the specific heat.16 In polyatomic molecules, there are three such classical degrees of freedom (corresponding to their rotations about three principal axes17), but in diatomic molecules, only two.18 Hence, these contributions may be described by the following generalization of Eq. (19):
,
3/2
gases,
monoatomic
for
c
(3.31)
V
5/2,
of
gases
for
molecules,
diatomic
3,
of
gases
for
molecules.
polyatomic
Please keep in mind, however, that as the above discussion shows, this simple result is invalid at very low and very high temperatures; its most notable violation is that the thermal activation of vibrational degrees of freedom for many important molecules at temperatures of a few thousand K.
3.2. Calculating
Now let us discuss properties of ideal gases of free, indistinguishable particles in more detail, paying special attention to the chemical potential – which, for some readers, may still be a somewhat mysterious aspect of the Fermi and Bose distributions. Note again that particle indistinguishability requires the absence of thermal excitations of their internal degrees of freedom, so that in the balance of this chapter such excitations will be ignored, and the particle’s energy k will be associated with its
“external” energy alone: for a free particle in an ideal gas, with its kinetic energy (3).
15 See, e.g., either the model solution of Problem 2.12 (and references therein), or QM Secs. 3.6 and 5.6.
16 This result may be readily obtained again from the last term of Eq. (30) by treating it exactly like the first one was and then applying the general Eq. (1.50).
17 See, e.g., CM Sec. 4.1.
18 This conclusion of the quantum theory may be interpreted as the indistinguishability of the rotations about the molecule’s symmetry axis.
Chapter 3
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Let us start from the classical gas, and recall the conclusion of thermodynamics that is just the Gibbs potential per unit particle – see Eq. (1.56). Hence we can calculate = G/ N from Eqs. (1.49) and (16b). The result,
3 / 2
2
V
N 2
T ln
f T T T ln
,
(3.32a)
N
gV mT
which may be rewritten as
3 / 2
2
N 2
exp
,
(3.32b)
T gV mT
gives us some information about not only for a classical gas but for quantum (Fermi and Bose) gases as well. Indeed, we already know that for indistinguishable particles, the Boltzmann distribution (2.111) is valid only if Nk << 1. Comparing this condition with the quantum statistics (2.115) and (2.118), we see again that the condition of the gas behaving classically may be expressed as
k
exp
1
(3.33)
T
for all k. Since the lowest value of k given by Eq. (3) is zero, Eq. (33) may be satisfied only if exp{/ T} << 1. This means that the chemical potential of a classical gas has to be not just negative, but also “strongly negative” in the sense
T.
(3.34a)
According to Eq. (32), this important condition may be represented as
T T ,
(3.34b)
0
with T 0 defined as
2 / 3
2 / 3
2
2
2
Quantum
N
n
scale of
T
,
(3.35)
0
2 / 3
2
temperature
m gV
m g
g
mr ave
where r ave is the average distance between the gas particles:
1/ 3
1
V
r
.
(3.36)
ave
1/ 3
n
N
In this form, the condition (34) is very transparent physically: disregarding the factor g 2/3 (which is typically of the order of 1), it means that the average thermal energy of a particle, which is always of the order of T, has to be much larger than the energy of quantization of particle’s motion at the length r ave. An alternative form of the same condition is19
1
/ 3
r
g
r ,
where r
.
(3.37)
ave
c
c
1/ 2
( mT )
For a typical gas (say, N2, with m 14 m p 2.310-26 kg) at the standard room temperature ( T =
k B300K 4.110-21 J), the correlation length r c is close to 10-11 m, i.e. is significantly smaller than the 19 In quantum mechanics, the parameter r c so defined is frequently called the correlation length – see, e.g., QM
Sec. 7.2 and in particular Eq. (7.37).
Chapter 3
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SM: Statistical Mechanics
physical size a ~ 310-10 m of the molecule. This estimate shows that at room temperature, as soon as any practical gas is rare enough to be ideal ( r ave >> a), it is classical, i.e. the only way to observe quantum effects in the translational motion of molecules is very deep refrigeration. According to Eq.
(37), for the same nitrogen molecule, taking r ave ~ 102 a ~ 10-8 m (to ensure that direct interaction effects are negligible), the temperature should be well below 1 mK.
In order to analyze quantitatively what happens with gases when T is reduced to such low values, we need to calculate for an arbitrary ideal gas of indistinguishable particles. Let us use the lucky fact that the Fermi-Dirac and the Bose-Einstein statistics may be represented with one formula: 1
N
,
(3.38)
/ T
e
1
where (and everywhere in the balance of this section) the top sign stands for fermions and the lower one for bosons, to discuss fermionic and bosonic ideal gases in one shot.
If we deal with a member of the grand canonical ensemble (Fig. 2.13), in which not only T but also is externally fixed, we may use Eq. (38) to calculate the average number N of particles in volume V. If the volume is so large that N >> 1, we may use the general state counting rule (13) to get 3
2
gV
gV
d p
gV
4 p
dp
N
N 3
d k
.
(3.39)
2 3
23
[ ( p)]/ T
e
1 23 [ ( p)]/ T
0 e
1
In most practical cases, however, the number N of gas particles is fixed by particle confinement (i.e. the gas portion under study is a member of a canonical ensemble – see Fig. 2.6), and hence rather than N
should be calculated. Let us use the trick already mentioned in Sec. 2.8: if N is very large, the relative fluctuation of the particle number, at fixed , is negligibly small ( N/ N ~ 1/ N << 1), and the relation between the average values of N and should not depend on which of these variables is exactly fixed.
Hence, Eq. (39), with having the sense of the average chemical potential, should be valid even if N is exactly fixed, so that the small fluctuations of N are replaced with (equally small) fluctuations of .
Physically, in this case the role of the -fixing environment for any sub-portion of the gas is played by the rest of it, and Eq. (39) expresses the condition of self-consistency of such chemical equilibrium.
So, at N >> 1, Eq. (39) may be used for calculating the average as a function of two independent parameters: N (i.e. the gas density n = N/ V) and temperature T. For carrying out this calculation, it is convenient to convert the right-hand side of Eq. (39) to an integral over the particle’s energy ( p) = p 2/2 m, so that p = (2 m)1/2, and dp = ( m/2)1/2 d, getting 3 / 2
1/ 2
gVm
d
Basic
N
.
(3.40) equation
2
3 (
2
)/ T
0 e
1
for
This key result may be represented in two other, more convenient forms. First, Eq. (40), derived for our current (3D, isotropic and parabolic-dispersion) approximation (3), is just a particular case of the following self-evident state-counting relation
N g( ) N
d ,
(3.41)
0
where
Chapter 3
Page 10 of 34
SM: Statistical Mechanics
g dN
d
(3.42)
states
is the temperature-independent density of all quantum states of a particle – regardless of whether they are occupied or not. Indeed, according to the general Eq. (4), for our simple model (3), 3 / 2
dN
d gV 4
gVm
g
states
3
1/ 2
g ( )
p
,
(3.43)
3
d
d 2
3
2
3
3
2
so that we return to Eq. (39).
On the other hand, for some calculations, it is convenient to introduce the following dimensionless energy variable: / T, to express Eq. (40) via a dimensionless integral: 3 / 2
1/ 2
gV ( mT )
d
N
.
(3.44)
2
3
2
/ T
0 e
1
As a sanity check, in the classical limit (34), the exponent in the denominator of the fraction under the integral is much larger than 1, and Eq. (44) reduces to
gV ( mT 3/ 2
)
1/ 2
d
gV ( mT )3/ 2
N
.
(3.45)
2
3
exp 1/2 e d ,
-
at μ T
/
2
3
2
0 e
T
2
T 0
By the definition of the gamma function (),20 the last integral is just (3/2) = 1/2/2, and we get 3 / 2
2
3
2
2
T
0
exp N
2
,
(3.46)
3 / 2
T
gV ( mT )
T
which is exactly the same result as given by Eq. (32), obtained earlier in a rather different way – from the Boltzmann distribution and thermodynamic identities.
Unfortunately, in the general case of arbitrary , the integral in Eq. (44) cannot be worked out analytically.21 The best we can do is to use T 0, defined by Eq. (35), to rewrite Eq. (44) in the following convenient, fully dimensionless form:
2
/ 3
1/ 2
T
1
d
,
(3.47)
2
/ T
T
0
2
0 e
1
and then use this relation to calculate the ratios T/ T 0 and / T 0 (/ T)( T/ T 0), as functions of / T
numerically. After that, we may plot the results versus each other, now considering the first ratio as the argument. Figure 1 below shows the resulting plots, for both particle types. They show that at high temperatures, T >> T 0, the chemical potential is negative and approaches the classical behavior given by Eq. (46) for both fermions and bosons – just as we could expect. However, at temperatures T ~ T 0 the type of statistics becomes crucial. For fermions, the reduction of temperature leads to changing its 20 See, e.g., MA Eq. (6.7a).
21 For the reader’s reference only: for the upper sign, the integral in Eq. (40) is a particular form (for s = ½) of a special function called the complete Fermi-Dirac integral Fs, while for the lower sign, it is a particular case (for s
= 3/2) of another special function called the polylogarithm Li s. (In what follows, I will not use these notations.) Chapter 3
Page 11 of 34
SM: Statistical Mechanics
sign from negative to positive, and then approaching a constant positive value called the Fermi energy,
F 7.595 T 0 at T 0. On the contrary, the chemical potential of a bosonic gas stays negative, and then turns into zero at a certain critical temperature T c 3.313 T 0. Both these limits, which are very important for applications, may (and will be :-) explored analytically, separately for each statistics.
10
F
8
fermions
T 0
6
4
2
0
Fig. 3.1. The chemical potential of an ideal
T
0
2
gas of N >> 1 indistinguishable quantum
particles, as a function of temperature at a
4
bosons
T
fixed gas density n N/ V (i.e. fixed T
6
c
0 n 2/3),
for two different particle types. The dashed
8
line shows the classical approximation (46),
100 1 2 3 4 5 6 7 8 9
valid only at T >> T 0.
10
T / T 0
Before carrying out such studies (in the next two sections), let me show that, rather surprisingly, for any non-relativistic, ideal quantum gas, the relation between the product PV and the energy, 2
PV E ,
(3.48) Ideal gas:
3
PV vs. E
is exactly the same as follows from Eqs. (18) and (19) for the classical gas, and hence does not depend on the particle statistics. To prove this, it is sufficient to use Eqs. (2.114) and (2.117) for the grand thermodynamic potential of each quantum state, which may be conveniently represented by a single formula,
(
T ln 1
) / T
k
e
,
(3.49)
k
and sum them over all states k, using the general summation formula (13). The result for the total grand potential of a 3D gas with the dispersion law (3) is
3 / 2
gV
2
p
m T
gVm
T
ln1
(
/ 2 ) /
e
4
2
p dp T
ln
(
)/
e
T d
(3.50)
3
1
1/2
2
.
0
2 2 3
0
Working out this integral by parts, exactly as we did it with the one in Eq. (2.90), we get 2
3 / 2
3 / 2
gVm
d
2
g ( )
N d
(3.51)
T
3
.
3 2 2 3
(
) /
0 e
1
3 0
But the last integral is just the total energy E of the gas:
2
gV
p
4 2
3 / 2
3 / 2
p dp
gVm
d
Ideal
E
g ( )
N d
gas:
(3.52)
p T
T
3
3
2
,
2
[ ( )
]/
(
) /
m e
energy
0
1
2 2 3
0 e
1 0
Chapter 3
Page 12 of 34
SM: Statistical Mechanics
so that for any temperature and any particle type, = –(2/3) E. But since, from thermodynamics, = –
PV, we have Eq. (48) proved. This universal relation22 will be repeatedly used below.
3.3. Degenerate Fermi gas
Analysis of low-temperature properties of a Fermi gas is very simple in the limit T = 0. Indeed, in this limit, the Fermi-Dirac distribution (2.115) is just the step function:
,
1 for ,
N
(3.53)
,
0
for ,
- see by the bold line in Fig. 2a. Since = p 2/2 m is isotropic in the momentum space, in that space the particles, at T = 0, fully occupy all possible quantum states inside a sphere (frequently called either the Fermi sphere or the Fermi sea) with some radius p F (Fig. 2b), while all states above the sea surface are empty. Such degenerate Fermi gas is a striking manifestation of the Pauli principle: though in thermodynamic equilibrium at T = 0 all particles try to lower their energies as much as possible, only g of them may occupy each translational (“orbital”) quantum state. As a result, the sphere’s volume is proportional to the particle number N, or rather to their density n = N/ V.
N
(a)
pz
(b)
1
T 0
Fig. 3.2. Representations of the
0
p
Fermi sea: (a) on the Fermi
p
y
F
T
p
F
x
distribution plot, and (b) in the
momentum space.
0
F
Indeed, the radius p F may be readily related to the number of particles N using Eq. (39), with the upper sign, whose integral in this limit is just the Fermi sphere’s volume:
p F
gV
gV
4
2
3
N
p dp
p
.
(3.54)
2
4
3
2
3
0
3
F
Now we can use Eq. (3) to express via N the chemical potential (which, in the limit T = 0, it bears the special name of the Fermi energy F)23:
2 / 3
1/ 3
2
2
4
Fermi
p
N
9
energy
F
2
6
T 595
.
7
T ,
(3.55a)
F
T 0
0
0
2 m
2 m
gV
2
where T 0 is the quantum temperature scale defined by Eq. (35). This formula quantifies the low-temperature trend of the function ( T), clearly visible in Fig. 1, and in particular, explains the ratio F/ T 0
mentioned in Sec. 2. Note also a simple and very useful relation,
22 For gases of diatomic and polyatomic molecules at relatively high temperatures, when some of their internal degrees of freedom are thermally excited, Eq. (48) is valid only for the translational-motion energy.
23 Note that in the electronic engineering literature, is usually called the Fermi level, for any temperature.
Chapter 3
Page 13 of 34
SM: Statistical Mechanics
3
N
3
N
,
i.e. g ( )
,
(3.55b)
F
2 g ( )
3
F
2
3
F
F
that may be obtained immediately from the comparison of Eqs. (43) and (54).
The total energy of the degenerate Fermi gas may be (equally easily) calculated from Eq. (52): p
gV
F p 2
4
3
2
gV
p 5
E
F
,
(3.56)
2
4 p dp
3
2 m
2
2
5
5
0
N
3
m
F
showing that the average energy, E/ N, of a particle inside the Fermi sea is equal to 3/5 of that (F) of the particles in the most energetic occupied states, on the Fermi surface. Since, according to the formulas of Chapter 1, at zero temperature H = G = N, and F = E, the only thermodynamic variable still to be calculated is the gas pressure P. For it, we could use any of the thermodynamic relations P =
( H – E)/ V or P = –( F/ V) T, but it is even easier to use our recent result (48). Together with Eq. (56), it yields
1/ 3
2 E
2
N
36 4
2
5 / 3
n
P
P 3.035 P ,
where P nT
.
(3.57)
3 V
5 F V
125 0
0
0
0
2 / 3
mg
From here, it is straightforward to calculate the bulk modulus (reciprocal compressibility),24
P
2
N
K V
,
(3.58)
V
F
3
V
T
which may be simpler to measure experimentally than P.
Perhaps the most important example25 of the degenerate Fermi gas is the conduction electrons in metals – the electrons that belong to outer shells of the isolated atoms but become shared in solid metals, and as a result, can move through the crystal lattice almost freely. Though the electrons (which are fermions with spin s = ½ and hence with the spin degeneracy g = 2 s + 1 = 2) are negatively charged, the Coulomb interaction of the conduction electrons with each other is substantially compensated by the positively charged ions of the atomic lattice, so that they follow the simple model discussed above, in which the interaction is disregarded, reasonably well. This is especially true for alkali metals (forming Group 1 of the periodic table of elements), whose experimentally measured Fermi surfaces are spherical within 1% – even within 0.1% for Na.
Table 1 lists, in particular, the experimental values of the bulk modulus for such metals, together with the values given by Eq. (58) using the F calculated from Eq. (55) with the experimental density of the conduction electrons. The agreement is pretty impressive, taking into account that the simple theory 24 For a general discussion of this notion, see, e.g., CM Eqs. (7.32) and (7.36).
25 Recently, nearly degenerate gases (with F ~ 5 T) have been formed of weakly interacting Fermi atoms as well –
see, e.g., K. Aikawa et al., Phys. Rev. Lett. 112, 010404 (2014), and references therein. Another interesting example of the system that may be approximately treated as a degenerate Fermi gas is the set of Z >> 1 electrons in a heavy atom. However, in this system the account of electron interaction via the electrostatic field they create is important. Since for this Thomas-Fermi model of atoms, the thermal effects are unimportant, it was discussed already in the quantum-mechanical part of this series (see QM Chapter 8). However, its analysis may be streamlined using the notion of the chemical potential, introduced only in this course – the problem left for the reader’s exercise.
Chapter 3
Page 14 of 34
SM: Statistical Mechanics
described above completely ignores the Coulomb and exchange interactions of the electrons. This agreement implies that, surprisingly, the experimentally observed rigidity of solids (or at least metals) is predominantly due to the kinetic energy (3) of the conduction electrons, rather than any electrostatic interactions – though, to be fair, these interactions are the crucial factor defining the equilibrium value of n. Numerical calculations using more accurate approximations (e.g., the Density Functional Theory26), which agree with experiment with a few-percent accuracy, confirm this conclusion.27
Table 3.1. Experimental and theoretical parameters of electrons’ Fermi sea in some alkali metals28
Metal
F (eV)
K (GPa)
K (GPa)
(mcal/moleK2) (mcal/moleK2)
Eq. (55)
Eq. (58)
experiment
Eq. (69)
experiment
Na
3.24
923
642
0.26
0.35
K
2.12
319
281
0.40
0.47
Rb
1.85
230
192
0.46
0.58
Cs
1.59
154
143
0.53
0.77
Looking at the values of F listed in this table, note that room temperatures ( T K ~ 300 K) correspond to T ~ 25 meV. As a result, virtually all experiments with metals, at least in their solid or liquid form, are performed in the limit T << F. According to Eq. (39), at such temperatures, the occupancy step described by the Fermi-Dirac distribution has a non-zero but relatively small width of the order of T – see the dashed line in Fig. 2a. Calculations for this case are much facilitated by the so-called Sommerfeld expansion formula 29 for the integrals like those in Eqs. (41) and (52):
2
d
2
( )
Sommerfeld
I ( T ) ( ) N d ( ) d
T
, for T ,
expansion
(3.59)
6
d
0
0
where () is an arbitrary function that is sufficiently smooth at = and integrable at = 0. To prove this formula, let us introduce another function,
df
f
( )
( ' ) d ' ,
that
so
,
(3.60)
d
0
and work out the integral I( T) by parts:
df
I ( T )
N
d
N
df
d
0
0
26 See, e.g., QM Sec. 8.4.
27 Note also a huge difference between the very high bulk modulus of metals ( K ~ 1011 Pa) and its very low values in usual, atomic gases (for them, at ambient conditions, K ~105 Pa). About four orders of magnitude of this difference is due to that in the particle density N/ V, but the balance is due to the electron gas’ degeneracy. Indeed, in an ideal classical gas, K = P = T( N/ V), so that the factor (2/3) F in Eq. (58), of the order of a few eV in metals, should be compared with the factor T 25 meV in the classical gas at room temperature.
28 Data from N. Ashcroft and N. D. Mermin, Solid State Physics, W. B. Saunders, 1976.
29 Named after Arnold Sommerfeld, who was the first (in 1927) to apply quantum mechanics to degenerate Fermi gases, in particular to electrons in metals, and may be credited for most of the results discussed in this section.
Chapter 3
Page 15 of 34
SM: Statistical Mechanics
N
N f f d N f
d .
(3.61)
0
0
0
As evident from Eq. (2.115) and/or Fig. 2a, at T << the function – N()/ is close to zero for all energies, besides a narrow peak of the unit area, at . Hence, if we expand the function f() in the Taylor series near this point, just a few leading terms of the expansion should give us a good approximation:
2
df
d f
N
I ( T )
1
f ()
2
d
2
d
2
d
0
N
N
( ' ) d '
d ()
d
0
0
0
(3.62)
1 d()
N
2
d .
2 d
0
In the last form of this relation, the first integral over equals N( = 0) – N( = = 1, the second one vanishes (because the function under it is antisymmetric with respect to the point = ), and only the last one needs to be dealt with explicitly, by working it out by parts and then using a table integral:30
N
d
d
2
2
2
1
2
d T
d 4 2
T
4 2
T
(3.63)
d
0
e 1
0 e
1
12
Being plugged into Eq. (62), this result proves the Sommerfeld formula (59).
The last preparatory step we need to make is to account for a possible small difference (as we will see below, also proportional to T 2) between the temperature-dependent chemical potential ( T) and the Fermi energy defined as F (0), in the largest (first) term on the right-hand side of Eq. (59), to write
F
2 2 d()
2 2 d()
I T
( )
( ) d
. (3.64)
F ( )
T
I (0) F ()
T
6
d
6
d
0
Now, applying this formula to Eq. (41) and the last form of Eq. (52), we get the following results (which are valid for any dispersion law (p) and even any dimensionality of the gas): 2
dg
2
( )
N ( T ) N (0) g
T
(3.65)
F (
)
,
6
d
2
d
E( T ) E(0) g
T
g .
(3.66)
F
( )
2
( )
6
d
If the number of particles does not change with temperature, N( T) = N(0), as in most experiments, Eq.
(65) gives the following formula for finding the temperature-induced change of :
30 See, e.g., MA Eqs. (6.8c) and (2.12b), with n = 1.
Chapter 3
Page 16 of 34
SM: Statistical Mechanics
2 2 1 dg()
T
.
(3.67)
F
6
g() d
Note that the change is quadratic in T and negative, in agreement with the numerical results shown with the red line in Fig. 1. Plugging this expression (which is only valid when the magnitude of the change is much smaller than F) into Eq. (66), we get the following temperature correction to the energy: 2
2
E( T ) E( )
0
g() T ,
(3.68)
6
where within the accuracy of our approximation, may be replaced with F. (Due to the universal relation (48), this result also gives the temperature correction to the Fermi gas’ pressure.) Now we may use Eq. (68) to calculate the heat capacity of the degenerate Fermi gas:
Low-T
2
E
heat
C
T,
with
g
.
(3.69)
V
F
capacity
T
3
V
According to Eq. (55b), in the particular case of a 3D gas with the isotropic and parabolic dispersion law (3), Eq. (69) reduces to
2
2
N
CV
T
,
i.e. c
1.
(3.70)
2
V
N
2
F
F
This important result deserves a discussion. First, note that within the range of validity of the Sommerfeld approximation ( T << F), the specific heat of the degenerate gas is much smaller than that of the classical gas, even without internal degrees of freedom: cV = 3/2 – see Eq. (19). The physical reason for such a low heat capacity is that the particles deep inside the Fermi sea cannot pick up thermal excitations with available energies of the order of T << F, because the states immediately above them are already occupied. The only particles (or rather quantum states, due to the particle indistinguishability) that may be excited with such small energies are those at the Fermi surface, more exactly within a surface layer of thickness ~ T << F, and Eq. (70) presents a very vivid manifestation of this fact.
The second important feature of Eqs. (69)-(70) is the linear dependence of the heat capacity on temperature, which decreases with a reduction of T much slower than that of crystal vibrations – see Eq.
(2.99). This means that in metals the specific heat at temperatures T << T D is dominated by the conduction electrons. Indeed, experiments confirm not only the linear dependence (70) of the specific heat,31 but also the values of the proportionality coefficient CV/ T for cases when F can be calculated independently, for example for alkali metals – see the two rightmost columns of Table 1 above. More typically, Eq. (69) is used for the experimental measurement of the density of states on the Fermi surface, g(F) – the factor which participates in many theoretical results, in particular in transport properties of degenerate Fermi gases (see Chapter 6 below).
31 Solids, with their low thermal expansion coefficients, provide a virtually-fixed-volume confinement for the electron gas, so that the specific heat measured at ambient conditions may be legitimately compared with the calculated cV.
Chapter 3
Page 17 of 34
SM: Statistical Mechanics
3.4. Bose-Einstein condensation
Now let us explore what happens at the cooling of an ideal gas of bosons. Figure 3a shows the same plot as Fig. 1b, i.e. the result of a numerical solution of Eq. (47) with the appropriate (lower) sign in the denominator, on a more appropriate, log-log scale. One can see that the chemical potential
indeed tends to zero at some finite “critical temperature” T c. This temperature may be found by taking
= 0 in Eq. (47), reducing it to a table integral:32
2
/ 3
2
/ 3
1/ 2
1
d
1
3 3
BEC:
T T
T
313
.
3
T ,
(3.71) critical
c
0
0
0
2
2
2
e 1
2
2 2
temperature
0
the result explaining the T c/ T 0 ratio mentioned in Sec. 2 and indicated in Fig. 1.
(a)
(b)
100
10
8
PV NT
10
6
PV
3.313
)
NT
0
4
)
1
T 0
2
1.701
0
0.1
0
2
4
6
8
10
T / T 0
Fig. 3.3. The Bose-Einstein condensation:
(a) the chemical potential of the gas and (b)
0.01
1
10
100
its pressure, as functions of temperature. The
T / T 0
dashed line corresponds to the classical gas.
3.313
Let us have a good look at the temperature interval 0 < T < T c, which cannot be directly described by Eq. (40) (with the appropriate negative sign in the denominator), and hence may look rather mysterious. Indeed, within this range, the chemical potential , cannot either be negative or equal zero, because according to Eq. (71), in this case, Eq. (40) would give a value of N smaller than the number of particles we actually have. On the other hand, cannot be positive either, because the integral (40) would diverge at due to the divergence of N() – see, e.g., Fig. 2.15. The only possible resolution of the paradox, suggested by A. Einstein in 1925, is as follows: at T < T c, the chemical potential of each particle of the system still equals exactly zero, but a certain number ( N 0 of N) of them are in the ground state (with p 2/2 m = 0), forming the so-called Bose-Einstein condensate, usually referred to as the BEC. Since the condensate particles do not contribute to Eq. (40) (because of 32 See, e.g., MA Eq. (6.8b) with s = 3/2, and then Eqs. (2.7b) and (6.7e).
Chapter 3
Page 18 of 34
SM: Statistical Mechanics
the factor 1/2 = 0), their number N 0 may be calculated by using that formula (or, equivalently, Eq. (44)), with = 0, to find the number ( N – N 0) of particles still remaining in the gas, i.e. having energy > 0: 3 / 2
1/ 2
gV ( mT )
d
N N
.
(3.72)
0
2
3
2
0 e 1
This result is even simpler than it may look. Indeed, let us write it for the case T = T c, when N 0 = 0:33
3 / 2
1/ 2
gV ( mT )
d
N
c
.
(3.73)
2
3
2
0 e
1
Dividing both sides of Eqs. (72) and (73), we get an extremely simple and elegant result: 3 / 2
3 / 2
N N
T
T
0
,
that
so
N N 1
, for T T .
(3.74a)
0
c
N
T
T
c
c
Please note that this result is only valid for the particles whose motion, within the volume V, is free – in other words, for a system of free particles confined within a rigid-wall box of volume V. In most experiments with the Bose-Einstein condensation of dilute gases of neutral (and hence very weakly interacting) atoms, they are held not in such a box, but at the bottom of a “soft” potential well, which may be well approximated by a 3D quadratic parabola: U(r) = m2 r 2/2. It is straightforward (and hence left for the reader’s exercise) to show that in this case, the dependence of N 0( T) is somewhat different: 3
T
N N 1
, for
*
T T ,
(3.74b)
0
*
c
T
c
where T *
c is a different critical temperature, which now depends on , i.e. on the confining potential’s
“steepness”. (In this case, V is not exactly fixed; however, the effective volume occupied by the particles at T = T *
c is related to this temperature by a formula close to Eq. (71), so that all estimates given above are still valid.) Figure 4 shows one of the first sets of experimental data for the Bose-Einstein condensation of a dilute gas of neutral atoms. Taking into account the finite number of particles in the experiment, the agreement with the simple theory is surprisingly good.
Returning to the spatially-uniform Bose system, let us explore what happens below the critical temperature with its other parameters. Formula (52) with the appropriate (lower) sign shows that approaching T c from higher temperatures, the gas energy and hence its pressure do not vanish – see the red line in Fig. 3b. Indeed, at T = T c (where = 0), that formula yields34
3 / 2
5/2
3 / 2
3 / 2
5/2
m T
d
m T
5 5
c
c
E( T ) gV
gV
0.7701 NT ,
(3.75)
c
c
2
3
2
3
2
e 1
2
2 2
0
so that using the universal relation (48), we get the pressure value,
P T 2 E( T )
(5 / 2) N
N
c
T 0.5134
T 1.701 P ,
(3.76)
c
c
c
0
3 V
(3 / 2) V
V
33 This is, of course, just another form of Eq. (71).
34 For the involved dimensionless integral see, e.g., MA Eqs. (6.8b) with s = 5/2, and then (2.7b) and (6.7c).
Chapter 3
Page 19 of 34
Essential Graduate Physics
SM: Statistical Mechanics
which is somewhat lower than, but comparable to P(0) for the fermions – cf. Eq. (57).
Fig. 3.4. The total number N of trapped 87Rb
atoms (inset) and their ground-state fraction
N
0 /N, as functions of the ratio T/ T c, as measured
N
in one of the pioneering experiments – see J.
0
Ensher et al., Phys. Rev. Lett. 77, 4984 (1996). In
N
this experiment, T *
c was as low as 0.2810-6 K.
The solid line shows the simple theoretical
dependence N( T) given by Eq. (74b), while other
lines correspond to more detailed theories taking
into account the finite number N of trapped
atoms. © 1996 APS, reproduced with permission.
*
T / T c
Now we can use the same Eq. (52), also with = 0, to calculate the energy of the gas at T < T c, 3 / 2
5 / 2
3 / 2
m T
d
E T gV
.
(3.77)
2
3
2
0 e 1
Comparing this relation with the first form of Eq. (75), which features the same integral, we immediately get one more simple temperature dependence:
5 / 2
T
E( T ) E T
, for T T .
(3.78) BEC:
c
c
T
energy
c
From the universal relation (48), we immediately see that the gas pressure follows the same dependence: 5 / 2
T
P( T ) P T
, for T T .
(3.79) BEC:
c
c
T
pressure
c
This temperature dependence of pressure is shown with the blue line in Fig. 3b. The plot shows that for all temperatures (both below and above T c) the pressure is lower than that of the classical gas of the same density. Now note also that since, according to Eqs. (57) and (76), P( T c) P 0 V-5/3, while according to Eqs. (35) and (71), T c T 0 V-2/3, the pressure (79) is proportional to V-5/3/( V-2/3)5/2 = V 0, i.e. does not depend on the volume at all! The physics of this result (which is valid at T < T c only) is that as we decrease the volume at a fixed total number N of particles, more and more of them go to the condensate, decreasing the number ( N – N 0) of particles in the gas phase, but not changing its spatial density pressure. Such behavior is very typical for the coexistence of two different phases of the same matter – see, in particular, the next chapter.
The last thermodynamic variable of major interest is heat capacity, because it may be most readily measured. For temperatures T T c, it may be easily calculated from Eq. (78):
E
5 3/ 2
T
C ( T )
E( T )
,
(3.80)
V
c
T
2 5/2
T
N , V
c
Chapter 3
Page 20 of 34
SM: Statistical Mechanics
so that below T c, the capacity increases with temperature, at the critical temperature reaching the value 5 E( T )
C ( T )
c
1.925 N,
(3.81)
V
c
2 T c
which is approximately 28% above that (3 N/2) of the classical gas. (As a reminder, in both cases we ignore possible contributions from the internal degrees of freedom.) The analysis for T T c is a little bit more cumbersome because differentiating E over temperature – say, using Eq. (52) – one should also take into account the temperature dependence of that follows from Eq. (40) – see also Fig. 1.
However, the most important feature of the result may be predicted without the calculation (which is being left for the reader’s exercise). Namely, since at T >> T c the heat capacity has to approach the classical value 1.5 N, starting from the value (81), it must decrease with temperature at T > T c, thus forming a sharp maximum (a “cusp”) at the critical point T = T c – see Fig. 5.
3
3.313
2.5
2 1.925
CV
1.5
N
1
Fig. 3.5. Temperature dependences of the heat
0.5
capacity of an ideal Bose-Einstein gas,
numerically calculated from Eqs. (52) and (40)
0
0
2
4
6
8
10
for T T c, and given by Eq. (80) for T T c.
T / T
0
Such a cusp is a good indication of the Bose-Einstein condensation in virtually any experimental system, especially because inter-particle interactions (unaccounted for in our simple discussion) typically make this feature even more substantial, frequently turning it into a weak (logarithmic) singularity. Historically, such a singularity was the first noticed, though not immediately understood sign of the Bose-Einstein condensation, observed in 1931 by W. Keesom and K. Clusius in liquid 4He at its - point (called so exactly because of the characteristic shape of the CV( T) dependence) T = T c 2.17
K. Other milestones of the Bose-Einstein condensation studies include:
- the experimental discovery of superconductivity (which was later explained as the result of the Bose-Einstein condensation of electron pairs) by H. Kamerlingh-Onnes in 1911;
- the development of the Bose-Einstein statistics, and predicting the condensation, by S. Bose and A. Einstein, in 1924-1925;
- the discovery of superfluidity in liquid 4He by P. Kapitza and (independently) by J. Allen and D. Misener in 1937, and its explanation as a result of the Bose-Einstein condensation by F. and H.
Londons and L. Titza, with further significant elaborations by L. Landau – all in 1938;
- the explanation of superconductivity as a result of electron binding into Cooper pairs, with a simultaneous Bose-Einstein condensation of the resulting bosons, by J. Bardeen, L. Cooper, and J.
Schrieffer in 1957;
Chapter 3
Page 21 of 34
SM: Statistical Mechanics
- the discovery of superfluidity of two different phases of 3He, due to the similar Bose-Einstein condensation of pairs of its fermion atoms, by D. Lee, D. Osheroff, and R. Richardson in 1972;
- the first observation of the Bose-Einstein condensation in dilute gases (87Ru by E. Cornell, C.
Wieman, et al. , and 23Na by W. Ketterle et al.) in 1995.
The importance of the last achievement stems from the fact that in contrast to other Bose-Einstein condensates, in dilute gases (with the typical density n as low as ~ 1014 cm-3) the particles interact very weakly, and hence many experimental results are very close to the simple theory described above and its straightforward elaborations – see, e.g., Fig. 4.35 On the other hand, the importance of other Bose-Einstein condensates, which involve more complex and challenging physics, should not be underestimated – as it sometimes is.
Perhaps the most important feature of any Bose-Einstein condensate is that all N 0 condensed particles are in the same quantum state, and hence are described by exactly the same wavefunction. This wavefunction is substantially less “feeble” than that of a single particle – in the following sense. In the second quantization language,36 the well-known Heisenberg’s uncertainty relation may be rewritten for the creation/annihilation operators; in particular, for bosons,
ˆ a ˆ†
a 1
(3.82)
Since a ând †
ˆ a are the quantum-mechanical operators of the complex amplitude a = A exp{ i} and its complex conjugate a* = A exp{– i}, where A and are real amplitude and phase of the wavefunction, Eq. (82) yields the following approximate uncertainty relation (strict in the limit << 1) between the number of particles N = AA* and the phase :
N
½ .
(3.83)
This means that a condensate of N >> 1 bosons may be in a state with both phase and amplitude of the wavefunction behaving virtually as c-numbers, with very small relative uncertainties: N << N,
<< 1. Moreover, such states are much less susceptible to perturbations by experimental instruments.
For example, the electric current carried along a superconducting wire by a coherent Bose-Einstein condensate of Cooper pairs may be as high as hundreds of amperes. As a result, the “strange” behaviors predicted by the quantum mechanics are not averaged out as in the usual particle ensembles (see, e.g., the discussion of the density matrix in Sec. 2.1), but may be directly revealed in macroscopic, measurable dynamics of the condensate.
For example, the density j of the electric “supercurrent” of the Cooper pairs may be described by the same formula as the well-known usual probability current density of a single quantum particle,37 just multiplied by the electric charge q = –2 e of a single pair, and the pair density n: 35 Such controllability of theoretical description has motivated the use of dilute-gas BECs for modeling of renowned problems of many-body physics – see, e.g. the review by I. Bloch et al., Rev. Mod. Phys. 80, 885
(2008). These efforts are assisted by the development of better techniques for reaching the necessary sub-K
temperatures – see, e.g., the recent work by J. Hu et al., Science 358, 1078 (2017). For a more general, detailed discussion see, e.g., C. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, 2nd ed., Cambridge U.
Press, 2008.
36 See, e.g., QM Sec. 8.3.
37 See, e.g., QM Eq. (3.28).
Chapter 3
Page 22 of 34
SM: Statistical Mechanics
q
j qn
A ,
(3.84)
m
where A is the vector potential of the (electro)magnetic field. If a superconducting wire is not extremely thin, the supercurrent does not penetrate into its interior.38 As a result, the integral of Eq. (84), taken along a closed superconducting loop, inside its interior (where j = 0), yields
q
A d Δ 2 M
,
r
(3.85)
C
where M is an integer. But, according to the basic electrodynamics, the integral on the left-hand side of this relation is nothing more than the flux of the magnetic field B piercing the wire loop area A. Thus we immediately arrive at the famous magnetic flux quantization effect:
2
2
Φ
d r Φ
M
,
Φ
where
B
2.07 10 15
Wb
,
(3.86)
n
0
0
q
A
which was theoretically predicted in 1950 and experimentally observed in 1961. Amazingly, this effect holds even “over miles of dirty lead wire”, citing H. Casimir’s famous expression, sustained by the coherence of the Bose-Einstein condensate of Cooper pairs.
Other prominent examples of such macroscopic quantum effects in Bose-Einstein condensates include not only the superfluidity and superconductivity as such, but also the Josephson effect, quantized Abrikosov vortices, etc. Some of these effects are briefly discussed in other parts of this series.39
3.5. Gases of weakly interacting particles
Now let us discuss the effects of weak particle interaction effects on properties of their gas.
(Unfortunately, I will have time to do that only very briefly, and only for classical gases.40) In most cases of interest, particle interaction may be well described by a certain potential energy U, so that in the simplest model, the total energy is
N
p 2
E k Ur ,.., r ,..., r ,
(3.87)
1
j
N
k 1 2 m
where r k is the radius-vector of the k th particle’s center.41 First, let us see how far would the statistical physics allow us to proceed for an arbitrary potential U. For N >> 1, at the calculation of the Gibbs statistical sum (2.59), we may perform the usual transfer from the summation over all quantum states of the system to the integration over the 6 N-dimensional space, with the correct Boltzmann counting: 38 This is the Meissner-Ochsenfeld (or just “Meissner”) effect which may be also readily explained using Eq. (84) combined with the Maxwell equations – see, e.g., EM Sec. 6.4.
39 See EM Secs. 6.4-6.5, and QM Secs. 1.6 and 3.1.
40 A concise discussion of the effects of weak interactions on the properties of quantum gases may be found, for example, in Chapter 10 of the textbook by K. Huang, Statistical Mechanics, 2nd ed., Wiley, 2003.
41 One of the most significant effects neglected by Eq. (87) is the influence of atomic/molecular angular orientations on their interactions.
Chapter 3
Page 23 of 34
SM: Statistical Mechanics
E / T
1
2
N
g
N p j
U r
r
m
N
3
3
( ,... )
1
3
3
Z e
d p d p
d r d r
N
N
m
N! 2
exp
...
exp
...
3
N
k 1
2
1
1
mT
T
(3.88)
1
2
N
N
g V
N p
j
U r
r N
3
3
1
( ,... )
1
3
3
d p d p
d r d r
N
N
N
N! 2
exp
...
exp
...
.
3
N
mT
k 1 2
1
1
V
T
But according to Eq. (14), the first operand in the last product is just the statistical sum of an ideal gas (with the same g, N, V, and T), so that we may use Eq. (2.63) to write
1
3
3
U / T
F F
T ln
d r ... d r e
F
T ln 1 1
3
3
U / T
d r d r e
, (3.89)
ideal
N
1
N
ideal
N
...
1
