Low Temperature Properties of the Fermi-Dirac,
Boltzman and Bose-Einstein Equations
William C. Troy
Department of Mathematics
University of Pittsburgh, Pittsburgh PA 15260
Abstract
We investigate low temperature (T ) properties of three classical quan-
tum statistics models: (I) the Fermi-Dirac equation, (II) the Boltzman
equation, and (III) the Bose-Einstein equation. It is widely assumed
that each of these equations is valid for all T > 0. For each equa-
tion we prove that this assumption leads to erroneous predictions as
T → 0+. Our approach to correct these errors gives new low tempera-
ture predictions which contradict previous theory. We examine a two
state paramagnetism system and show how our new low temperature
prediction compares favorably with experimental data.
Keywords temperature, Gamma function, paramagnetism
We study low temperature properties of three quantum statistics models:
(I) The Fermi-Dirac equation for identical fermions.
(II) The Boltzman equation for distinguishable particles.
(III) The Bose-Einstein equation for identical bosons.
Before stating these equations we follow Griffiths ([1], Ch. 5) and describe the
general setting: assume that N particles are put in an arbitrary potential,
whose energies are E1 < E2 < · · ·, with corresponding degeneracies d1, d2, ...
Low Temperature Properties of Quantum Physics Models 2
For each i ≥ 1, let Ni denote the most probable number of particles having
energy Ei.
(I) The Fermi-Dirac equation ([1], Ch. 6 and [2, 3]) for identical fermions is
Ni =di
e(Ei−µ)
kT + 1, 0 < T < ∞, (0.1)
where k is Boltzman’s constant. The chemical potential, µ(T ), satisfies
µ(0) = EF (the Fermi level). A well known property (e.g. [1], Ch. 5) is
limT→0+
Ni =
di if Ei < µ(0)
0 if Ei > µ(0).(0.2)
Equivalently, it follows from (0.1) that the fundamental ratio Ei−µkT satisfies
Ei − µ
kT= ln
(
di
Ni− 1
)
, 0 < Ni < di, (0.3)
and therefore
limNi→0+
Ei − µ
kT= ∞ and lim
Ni→d−i
Ei − µ
kT= −∞. (0.4)
(II) The Boltzman equation ([1], p. 237) for distinguishable particles is
Ni = die−
Ei−µ
kT , 0 < T < ∞. (0.5)
(IIII) The Bose-Einstein equation ([1], p. 239 and [4, 5]) for identical bosons is
Ni =di − 1
e(Ei−µ)
kT − 1, 0 < T < ∞, i ≥ 1. (0.6)
A well known property of µ(T ) ([1], Ch. 5) for both (0.5) and (0.6) is that
µ(0) = 0. (0.7)
It follows from (0.5)-(0.6)-(0.7) that both (0.5) and (0.6) satisfy
limT→0+
Ni = 0 and limNi→0+
Ei − µ
kT= ∞, i ≥ 1. (0.8)
Low Temperature Properties of Quantum Physics Models 3
Goals. In this section we focus on low temperature predictions of the Fermi-
Dirac equation (0.1). We have two goals: (i) we use a microcanonical ap-
proach to prove that predictions (0.2)-(0.4) are erroneous; (ii) to correct
these errors we develop formulas which replace (0.1) and (0.3), and which
give new predictions. In the Appendix we address these issues for a param-
agnetism model, and the Boltzman and Bose-Einstein equations.
The Fermi-Dirac Model. To prove that (0.2) and (0.4) are erroneous,
we examine the derivation of (0.1). We follow Griffiths ([1], pp. 233-241).
Assume that the N particles are identical fermions. Let
(N1,N2,N3, ...) (0.9)
denote a configuration such that, for each i ≥ 1, Ni is the number of particles
having energy Ei. The total number of particles, N, and energy, E, satisfy
N =
∞∑
n=1
Nn and E =
∞∑
n=1
NnEn. (0.10)
Let Q (N1, N2, N3, ...) denote the number of distinct states corresponding to
configuration (0.9). Then Q =∏
∞
n=1dn!
Nn!(dn−Nn)! , and the most probable
configuration is the one such that ln (Q) is maximized subject to (0.10).
The method of Lagrange multipliers is the standard technique to obtain
this maximum: define
G = ln
(
∞∏
n=1
dn!
Nn! (dn − Nn)!
)
+ α
[
N −
∞∑
n=1
Nn
]
+ β
[
E −
∞∑
n=1
NnEn
]
.
(0.11)
The standard statistical mechanics approach to simplify (0.11) is to apply
Stirling’s approximation ln(M !) = M ln(M) − M, M ≫ 1. This gives
G =∑
∞
n=1 [dn ln (dn) − Nn ln (Nn) − (dn − Nn) ln (dn − Nn)]
+ α [N −∑
∞
n=1 Nn] + β [E −∑
∞
n=1 NnEn] .(0.12)
It follows from (0.12) that, for each i ≥ 1,
∂G
∂Ni= − ln (Ni) + ln (di − Ni) − α − βEi, 0 < Ni < di. (0.13)
Low Temperature Properties of Quantum Physics Models 4
Solve ∂G∂Ni
= 0 for Ni, and set α = − µkT , β = 1
kT ([1], pp. 239-241). Then Ni,
the most probable number of particles with energy Ei, satisfies the Fermi-
Dirac equation (0.1).
The Mathematical Error. The derivation of (0.1) uses the Stirling ap-
proximations ln(Ni!) = Ni ln(Ni)−Ni and ln((di −Ni)!) = (di −Ni) ln(di −
Ni) − (di − Ni), where
di ≫ 1, Ni ≫ 1 and di − Ni ≫ 1. (0.14)
However, Ni → d−i or Ni → 0+ as T → 0+ in (0.2). Thus, (0.14) is violated
when T → 0+. Therefore, it is erroneous to let T → 0+ in (0.2). Finally,
Ni → 0+ and Ni → d−i in (0.4), which also violates (0.14). Thus, (0.4) is
also erroneous.
Remark 1. Because of these errors, we cannot conclude from the Fermi-
Dirac model (0.1) that lowest possible temperature is zero.
To correct the errors described above, we make use of the Gamma function,
the unique analytic continuation of N !, which satisfies
Γ(x) > 0 ∀x > 0 and Γ′(x) > 0 ∀x ≥ 2, (0.15)
and
N ! = Γ(N + 1) ∀N ≥ 0. (0.16)
Thus, we replace each term in (0.11) of the form ln(M !) with the exact value
ln(M !) = ln(Γ(M + 1)), and (0.11) becomes
G = ln
(
∞∏
n=1
Γ(dn + 1)
Γ(Nn + 1)Γ(dn − Nn + 1)
)
+α
[
N −
∞∑
n=1
Nn
]
+β
[
E −
∞∑
n=1
NnEn
]
.
(0.17)
It follows from (0.17 ) that, for each i ≥ 1,
∂G
∂Ni=
Γ′ (di − Ni + 1)
Γ (di − Ni + 1)−
Γ′ (Ni + 1)
Γ (Ni + 1)− α − βEi 0 < Ni < di. (0.18)
Low Temperature Properties of Quantum Physics Models 5
Set ∂G∂Ni
= 0, α = − µkT , β = 1
kT , and obtain a new formula for the dimen-
sionless ratio Ei−µkT which replaces (0.3):
Ei − µ
kT=
Γ′ (di − Ni + 1)
Γ (di − Ni + 1)−
Γ′ (Ni + 1)
Γ (Ni + 1), 0 < Ni < di. (0.19)
We use (0.19) to address the following important question:
(Q1) How does the ratio Ei−µkT behave as Ni → 0+ or as Ni → d−i ?
The right side of (0.19) is a decreasing function of Ni. Thus,
limNi→0+
Ei − µ
kT=
Γ′ (di + 1)
Γ (di + 1)− Γ′ (1) > 0, (0.20)
limNi→d−i
Ei − µ
kT= −
(
Γ′ (di + 1)
Γ (di + 1)− Γ′ (1)
)
< 0, (0.21)
−
(
Γ′ (di + 1)
Γ (di + 1)− Γ′ (1)
)
<Ei − µ
kT<
Γ′ (di + 1)
Γ (di + 1)− Γ′ (1) ∀Ni ∈ (0, di) .
(0.22)
Properties (0.20)-(0.21)-(0.22) answer (Q1).
Remark 2. The differences between predictions of (0.3) and formula (0.19)
become clear when Ni → 0+ or Ni → d−i . Properties (0.4) and (0.20)-
(0.21)-(0.22), which are derived from (0.3) and (0.19), demonstrate these
differences. First, (0.4) shows that the ratio Ei−µkT changes sign and becomes
unbounded as Ni → 0+ or Ni → d−i . This prediction is fundamentally
flawed because it is derived from (0.3), which is equivalent to the Fermi-
Dirac model (0.1) whose derivation requires Ni ≫ 1 and di − Ni ≫ 1. In
contrast, properties (0.20)-(0.21)-(0.22), which are derived from (0.19), show
that Ei−µkT remains bounded over the entire range 0 ≤ Ni ≤ di.
Our next goal is to show how to predict lowest temperature values. For
i ≥ 1 we let Ti denote the lowest temperature of particles with energy Ei,
and address the following questions:
(Q2) Is there an exact formula for Ti?
(Q3) Is Ti > 0, or is Ti = 0?
Low Temperature Properties of Quantum Physics Models 6
(Q4) How does T behave as Ni → 0+ or as Ni → d−i ? How does Ni behave
as T decreases to Ti?
The first step towards answering (Q2)-(Q4) is to make reasonable assump-
tions on the chemical potential, µ(T ), the energy levels, Ei, and the degrees
of freedom, di.
(H1) µ(T ) ≡ µ = constant.
(H2) There is an integer L ≥ 1 such that Ej < µ if j ≤ L, Ej > µ if j > L,
(H3) di ≫ 1, i ≥ 1, and N =∑L
j=1 dj.
Remark 3 In the Appendix we investigate low temperature properties of
a two state paramagnetism model. There we show how (H1)-(H3) are
satisfied (i. e. µ = 0, L = 1, E1 < 0 < E2 and d1 = d2 = N ≫ 1).
These properties allow us to completely analyze the model and make new
low temperature predictions which contradict previous theory, and compare
favorably with experimental data for the organic free radical DPPH [6, 7].
We now continue with the analysis and answer (Q2)-(Q4). It is hoped
that our results will provide insights for rigorous studies of more general
settings (e.g. for more general forms of µ(T )). Assumptions (H1)-(H3)
allow us to reduce our investigation to two cases, Ei < µ and Ei > µ.
Case (i) Ei < µ. We solve (0.19) for T, and conclude that
T =Ei − µ
k(
Γ′(di−Ni+1)Γ(di−Ni+1) − Γ′(Ni+1)
Γ(Ni+1)
) > 0 ⇐⇒di
2< Ni < di. (0.23)
It follows from (0.23) and (H1)-(H2) that T is a decreasing function of
Ni ∈(
di
2 , di
)
, and that
Ti ≡ limNi→d−i
T =µ − Ei
k(
Γ′(di+1)Γ(di+1) − Γ′(1)
) > 0. (0.24)
Property (0.24) answers (Q2), and the first part of (Q4) when Ni → d−i .
Property (0.24) shows that Ti is strictly positive, and suggests that the tem-
perature of particles with energy Ei < µ cannot be lowered below Ti. Thus,
Low Temperature Properties of Quantum Physics Models 7
property (0.24) answers (Q3) when Ei < µ. Further physical interpretation
of Ti depends on the specific physical setting being analyzed (e.g. see the
analysis of the two point paramagnetism system in the Appendix). Finally,
since equation (0.23) gives T as a decreasing function of Ni ∈(
di
2 , di
)
, the
implicit function theorem guarantees that it can be inverted, and although
it is difficult to obtain an explicit, simple formula for the inversion, we can
conclude two important qualitative properties:
(i) Ni is a decreasing function of T, and
(ii) the converse of (0.24) holds, namely
limT→T+
i
Ni = di . (0.25)
Property (0.25) answers the second part of (Q4), and predicts that all states
with energy Ei < µ become occupied when T reaches the positive value Ti. It
remains a challenging problem to derive an explicit formula for Ni in terms
of T when Ei < µ.
Case (ii) Ei > µ. We solve (0.19) for T, and conclude that
T =Ei − µ
k(
Γ′(di−Ni+1)Γ(di−Ni+1) − Γ′(Ni+1)
Γ(Ni+1)
) > 0 ⇐⇒ 0 < Ni <di
2. (0.26)
It follows from (0.26) that T is an increasing function of Ni ∈(
0, di
2
)
, and
Ti ≡ limNi→0+
T =Ei − µ
k(
Γ′(di+1)Γ(di+1) − Γ′(1)
) > 0. (0.27)
Property (0.27) answers (Q2), and the first part of (Q4) when Ni → 0+.
Property (0.27) shows that Ti is strictly positive, and suggests that the
temperature of particles with energy Ei > µ cannot be lowered below Ti.
Thus, (0.27) answers the second part of question (Q3) when Ei > µ. Again,
we point out that further interpretation of Ti depends on the particular
physical system being studied. Finally, since equation (0.26) gives T as an
Low Temperature Properties of Quantum Physics Models 8
increasing function of Ni ∈(
0, di
2
)
, then as above, the implicit function the-
orem guarantees that it can be inverted to give Ni as an increasing function
of T ∈ (Ti,∞) , and that the converse of (0.27) holds, namely
limT→T+
i
Ni = 0. (0.28)
Property (0.28) answers the second part of (Q4) , and predicts that no state
with energy Ei > µ is occupied when T reaches Ti. It remains a challenging
problem to derive an explicit formula for Ni in terms of T when Ei > µ.
Remark 4. We address a slight technical issue. It follows from (0.15) that
Γ(x) is increasing when x ≥ 2. However, this monotonicity property does
not hold on the entire interval [1, 2] since Γ′(1) ≈ −.577 is negative. It
is possible that analysis of a particular physical system may require that
our continuation of N ! be modified so that it is monotone increasing for all
x ∈ [1, 2]. An example of such a modification (suggested by the referee) is
Γnew(x) =
1 if 1 ≤ x ≤ 2
Γ(x) if x > 2.(0.29)
We now show that modification (0.29) has a negligible effect on the predic-
tions of lowest temperature. First, it follows from (0.29) that Γ′
new(1+) = 0.
When we replace Γ′(1) with Γ′
new(1+) = 0 in (0.24) and (0.27) we obtain
the new predictions
T newi =
µ − Ei
kΓ′(di+1)Γ(di+1)
when µ > Ei, (0.30)
T newi =
Ei − µ
kΓ′(di+1)Γ(di+1)
when µ < Ei. (0.31)
Comparing (0.24) and (0.30), and also (0.27), and (0.31), we find that
T newi > Ti, i ≥ 1, and that the relative change between Ti and T new
i satisfies
|Relative Change| =|Ti − T new
i |
Ti=
|Γ′(1)|Γ′(di+1)Γ(di+1)
≈.577
Γ′(di+1)Γ(di+1)
, i ≥ 1. (0.32)
Low Temperature Properties of Quantum Physics Models 9
It follows from (H3) that di ≫ 1. Also, the digamma function Γ′(x)Γ(x) satisfies
Γ′(di+1)Γ(di+1) ≈ ln(di + 1) when di ≫ 1. Thus, (0.32) reduces to
|Relative Change| ≈.577
ln(di + 1)≪ 1, i ≥ 1. (0.33)
We conclude from (0.32)-(0.33) that replacing Γ′(1) with Γ′
new(1+) = 0
in (0.24) and (0.27) has a negligible effect on the prediction of lowest tem-
perature. Next, it is also important to determine how the limiting behavior
of the ratio Ei−µkT given in (0.20) and (0.21) changes when Γ′(1) is replaced
with zero. This replacement leads to the new limiting results
limNi→0+
Ei − µ
kT=
Γ′ (di + 1)
Γ (di + 1)> 0, (0.34)
limNi→d−i
Ei − µ
kT= −
Γ′ (di + 1)
Γ (di + 1)< 0. (0.35)
The magnitude of the relative change between predictions (0.20) and (0.34),
and also between (0.21) and (0.35), satisfies
|Relative Change| =|Γ′(1)|
Γ′(di+1)Γ(di+1) − Γ′(1)
<|Γ′(1)|Γ′(di+1)Γ(di+1)
≪ 1, i ≥ 1, (0.36)
since |Γ′(1)| ≈ .577 and di ≫ 1, i ≥ 1. We conclude from (0.34)-(0.35)-
(0.36) that replacing Γ′(1) with zero has a negligible effect on the limiting
behavior of Ei−µkT . In the Appendix we consider four instances where Γ′(1)
appears, and show (Remarks 6, 7, 8 and 10) that replacing Γ′(1) with zero
also produces negligible effects on predictions. Finally, we note that other
modifications of the continuation of N ! may be appropriate when N is small.
The construction of such modifications should be guided by experimental
data for specific physical systems.
Conclusions and future research.
Classical theory of the Fermi-Dirac equation (0.1) predicts that lowest tem-
perature iz zero for each i ≥ 1. Our most important theoretical advance,
which contradicts classical theory, is the proof that lowest temperature, Ti,
Low Temperature Properties of Quantum Physics Models 10
is strictly positive for each i ≥ 1. It is a challenging problem to determine if
this property demonstrates a mathematical shortcoming of the theoretical
underpinnings of quantum statistics, or if it holds up to further scrutiny,
both experimental and theoretical. Towards this end, a comprehensive
Schrodinger equation based study may give insights into lowest attainable
temperatures. In the Appendix we take further steps in this direction:
(i) As we pointed out above, we show how our methods apply to a two-
state paramagnetism model of magnetic properties of the organic free radical
DPPH [6, 7]. For specific experimental data given in [6, 7], we show that
the lowest theoretical predicted value of temperature is strictly positive, and
that the predicted value satisfies the physical requirement that it lies below
the lowest experimental temperature value.
(ii) We examine the accuracy of low temperature predictions of the Boltz-
man equation (0.5). Boltzman equations form core components of the par-
tition function method (Prathia [8], Ch. 6) for deriving the mean value
Fermi-Dirac equation
〈Ni〉 =di
e(Ei−µ)
kT + 1, 0 < T < ∞, (0.37)
and the mean value Bose-Einstein equation
〈Ni〉 =di − 1
e(Ei−µ)
kT − 1, 0 < T < ∞. (0.38)
Boltzman equations are also core components of widely diverse models rang-
ing from Planck’s black body radiation to Bose-Einstein condensation. As
above, we show how errors arise in predictions of the Boltzman equation (0.5)
at low T, and derive a new formula (see (A.25)) which replaces (0.5). We
also describe the importance and technical challenge of combining (A.25)
with the partition function approach to derive new mean value formulas
which replace (0.37) and (0.38) at low T.
(iii) We investigate the accuracy of low temperature predictions of the Bose-
Einstein model (0.6). Again, we show how errors arise in predictions of (0.6)
Low Temperature Properties of Quantum Physics Models 11
at low T, and derive a new formula (see (A.37)) which replaces (0.6). Finally,
we describe the results of our recent study [9] of the case µ(T ) ≡ 0. This
setting has important applications to quantum computing devices [10, 11, 12,
13, 14]. Our low temperature prediction (see (A.44)) improves the previous
Bose-Einstein equation based prediction [11].
A Appendix
Here we examine low temperature predictions of (I) an ideal two state para-
magnetism system in which the Fermi-Dirac model plays a central role, (II)
the Boltzman equation (0.5), and (III) the Bose-Einstein equation (0.6).
It is widely assumed that each model is valid for all T > 0. We show how
this assumption leads to erroneous predictions, and describe our approach
to correct these errors. Although there is some repetition, it is necessary to
include all details in order to obtain maximum new insight.
Part (I). We follow Schroeder ([6], pp.98-108). Consider N ≫ 1 electrons
in a uniform magnetic field B̄. Let N = N1 + N2, where N1 is the number
of electrons in the up state, N2 = N − N1 is the number in the down state.
Let µ̄ denote the magnetic dipole moment of each electron. Assume that
µz = µB = 9.27 × 10−24 (JoulesTesla ), and that B̄ points +z direction, i.e. in the
up state direction of the electrons. Electrons in the up state have energy
U1 = −µBB, and electrons in the down state have energy U2 = µBB. The
total energy and magnetization of the system are U = µBB(N2 − N1) and
M = µB(N1 − N2). Setting N2 = N − N1 gives
U = µBB(N − 2N1) and M = µB(2N1 − N), 0 < N1 < N, (A.1)
Next, express U and M in terms of T. The number of ways to distribute N1
dipoles in the up state, and N2 in the down state, over N total dipoles is
W = N !N1!N2!
. Thus, entropy S = k ln(
N !N1!N2!
)
. Setting N2 = N − N1 gives
S = k ln
(
N !
N1!(N − N1)!
)
. (A.2)
Low Temperature Properties of Quantum Physics Models 12
Applying Stirling’s approximation, when N1 ≫ 1 and N − N1 ≫ 1, re-
duces (A.2) to
S = k (N ln(N) − N1 ln(N1) − (N − N1)(ln(N − N1)) , 0 < N1 < N.
(A.3)
Combine the relationship 1T = ∂S
∂N1= dS/dN1
dU/dN1with (A.1) and (A.3), and ob-
tain the Fermi-Dirac equation for N1, the most probable number of electrons
in the up state:
N1 =N
exp(
−2µBBkT
)
+ 1, 0 < T < ∞. (A.4)
Inversion of (A.4) shows that the corresponding temperature satsifies
T = −2µBB
k ln(
NN1
− 1) > 0 ⇐⇒
N
2< N1 < N. (A.5)
We conclude from (A.5) that T is a decreasing function of Ni ∈(
N2 ,N
)
,
and that
limN1→N−
T = 0. (A.6)
Thus, the lowest temperature of particles in the upstate is zero.
Likewise, N2, the most probable number of electrons in the down state,
satisfies the Fermi-Dirac equation
N2 =N
exp(
2µBBkT
)
+ 1, 0 < T < ∞. (A.7)
Inversion of (A.7) shows that the corresponding temperature satisfies
T =2µBB
k ln(
NN2
− 1) > 0 ⇐⇒ 0 < N2 <
N
2. (A.8)
It follows from (A.8) that
limN2→0+
T = 0. (A.9)
Thus, the lowest temperature of particles in the downstate is also zero.
Low Temperature Properties of Quantum Physics Models 13
Next, we show that assumptions (H1)-(H3) are satisfied. A comparison
of (A.4) and (A.6) with the Fermi-Dirac model (0.1) shows that the chemical
potential µ(T ) ≡ 0, that L = 1, and that E1, d1, E2, d2 are given by
E1 = −2µBB, E2 = 2µBB, d1 = d2 = N. (A.10)
Schroeder ([6], p.99) points out that 2µBB is the amount of energy needed
to ‘flip a single electron from the up state to the down state.’ It follows
from (A.10), and the assumption N ≫ 1, that E1 < µ = 0 < E2 and
N =∑2
i=1 Ni = d1 = d2 ≫ 1, and therefore (H1)-(H3) are satisfied.
Next, it follows from (A.1) and (A.4) that
U = −µBBN tanh
(
µBB
kT
)
and M = µBN tanh
(
µBB
kT
)
∀T > 0.
(A.11)
Finally, it follows from (A.11) that
limT→0+
U = −µBN and limT→0+
M = µBN. (A.12)
Thus, (A.12) predicts that the system of electrons becomes totally magne-
tized at absolute zero.
The Mathematical Error. The derivation of (A.4) uses Stirling approx-
imations ln(N !) = N ln(N) − N, ln(N1!) = N1 ln(N1) − N1 and ln((N −
N1)!) = (N − N1) ln(N − N1) − (N − N1), where
N ≫ 1, N1 ≫ 1 and N − N1 ≫ 1. (A.13)
However, it follows from (A.4) that N1 → N as T → 0+. Thus, (A.13) is
violated when T → 0+. Therefore, it is erroneous to let T → 0+ in (A.12).
To correct this error, we replace each term in (A.2) of the form ln(M !) with
ln(M !) = ln(Γ(M + 1)), and (A.2) becomes
S = k ln
(
Γ(N + 1)
Γ(N1 + 1)Γ(N + 1 − N1)
)
, 0 < N1 < N. (A.14)
Low Temperature Properties of Quantum Physics Models 14
It follows from (A.1), (A.14) and the relationship 1T = ∂S
∂U = dS/dN1
dU/dN1that
T = −2µBB
k(
Γ′(N+1−N1)Γ(N+1−N1) − Γ′(N1+1)
Γ(N1+1)
) > 0 ⇐⇒N
2< N1 < N. (A.15)
Because the digamma function Γ′(x)Γ(x) is increasing for all x ≥ 1, we conclude
from (A.15) that T is a decreasing function of N1 ∈(
N2 ,N
)
, and
T1 = limN1→N−
T =2µBB
k(
Γ′(N+1)Γ(N+1) − Γ′(1)
) > 0. (A.16)
It follows from (A.16) that T1, the lowest temperature of electrons in the
up state, is strictly positive, and this result suggests that electrons in the
up state cannot be cooled below T1. Property (A.16) contrasts with predic-
tion (A.6) that the lowest temperature of particles in the up state is zero.
Next, since equation (A.15) gives T as a decreasing function of N1 ∈(
N2 ,N
)
,
it can be inverted to give N1 as a decreasing function of T, and although
we do not have an explicit form of the inversion, we can conclude that the
converse of property (A.16) holds:
limT→T+
1
N1 = N. (A.17)
It follows from (A.1) and (A.17) that
limT→T+
1
U = −µBBN and limT→T+
1
M = µBBN. (A.18)
Thus, the system becomes totally magnetized at the positive temperature T1.
We refer to the transition to total magnetization at the positive temerature
T1 as the GPT effect. This result contrasts with prediction (A.12) that the
system becomes totally magnetized at T = 0.
Remark 5. To test the accuracy of (A.16) we use data from Grobet al [7],
who investigated magnetization of the organic free radical DPPH, a two
state paramagnet (also see [6], pp.105-106). Here
N = 2.3 × 1023, B = 2.06, k = 1.38 × 10−23, µB = 9.27 × 10−24 (A.19)
Low Temperature Properties of Quantum Physics Models 15
Substituting (A.19) into (A.16) gives T1 = .0545K. This theoretical predic-
tion is strictly positive, and satisfies the basic requirement that it lies below
the lowest experimental temperature value 2.2K It follows from (A.16) that
T1 can actually become arbitrarily large if B is large. This is easily tested.
Remark 6. As in Remark 4 we examine the effect of replacing Γ′(1) with
zero in prediction (A.16) of lowest temperature. This gives the higher value
T new1 = .055K, and therefore the magnitude of the relative change between
predictions T1 and T new1 satisfies
|Relative Change| =|T1 − T new
1 |
T1= .011 (A.20)
We conclude from (A.20) that replacing Γ′(1) with zero in (A.16) has a
negligible effect on lowest temperature prediction.
Part (II). Recall from (0.5), (0.7) and (0.8) that the Boltzman equation is
Ni = die−
Ei−µ(T )
kT , 0 < T < ∞, i ≥ 1, (A.21)
that µ(0) = 0, and that
limT→0+
Ni = 0 and limNi→0+
Ei − µ
kT= ∞, i ≥ 1. (A.22)
We claim that, for each i ≥ 1, predictions (A.22) are erroneous. To prove
this claim we examine the derivations of (A.21) and (A.22). We follow Grif-
fiths ([1], Ch. 5) and assume that the particles are distinguishable, and
di ≫ 1, i ≥ 1. Then Q = N !∏
∞
n=1dNn
n
Nn! and (0.11) becomes
G = ln
(
N !
∞∏
n=1
dNnn
Nn!
)
+ α
(
N −
∞∑
n=1
Nn
)
+ β
(
E −
∞∑
n=1
NnEn
)
. (A.23)
Applying Stirling’s approximation ln(Nn!) = Nn ln(Nn)−Nn to (A.23) gives
G =∑
∞
n=1 [Nn ln (dn) − Nn ln (Nn) + Nn]
+ ln (N !) + α [N −∑
∞
n=1 Nn] + β [E −∑
∞
n=1 NnEn] .(A.24)
Low Temperature Properties of Quantum Physics Models 16
Solve ∂G∂Ni
= 0 for Ni, set α = − µkT , β = 1
kT , and get (A.21). Note that
Ni ≫ 1 ∀i ≥ 1 is required in (A.24). Thus, since Ni → 0+ as T → 0+
in (A.22), it is erroneous to let T → 0+ in (A.22). To correct this error we
again replace N ! with Nn! = Γ(Nn + 1) in (A.23), set ∂G∂Ni
= 0, α = − µkT
and β = 1kT , and obtain the new formula
Ei − µ
kT= ln(di) −
Γ′(Ni + 1)
Γ(Ni + 1), i ≥ 1. (A.25)
An important consequence of (A.25) is that
limNi→0+
Ei − µ(T )
kT= ln(di) − Γ′(1) < ∞, i ≥ 1. (A.26)
Property (A.26) contrasts with the Boltman equation based prediction (A.22)
that limNi→0+Ei−µ(T )
kT = ∞, i ≥ 1.
Remark 7. As in Remarks 4 and 6 we examine the effect of replacing Γ′(1)
with zero in (A.26), in which case (A.26) becomes
limNi→0+
Ei − µ(T )
kT= ln(di) < ∞, i ≥ 1. (A.27)
The relative change between predictions (A.26) and (A.27) satisfies
|Relative Change| =|Γ′(1)|
ln(di) − Γ′(1))<
|Γ′(1)|
ln(di)≪ 1, i ≥ 1, (A.28)
since Γ′(1) < 0 and di ≫ 1. Thus, replacing Γ′(1) with zero in (A.26) has a
negligible effect on the predicted behavior of Ei−µ(T )kT as Ni → 0+.
Next, for i ≥ 1 let Ti denote the lowest value of temperature of particles with
energy Ei. We posit, as for the Fermi-Dirac model, that limT→Ti+Ni = 0.
This property and (A.25) imply that
limT→Ti+
Ei − µ(T )
kT= ln(di) − Γ′(1) < ∞, i ≥ 1. (A.29)
It follows from (A.29) that Ti satisfies the implicit equation
Ti =Ei − µ(T i)
k (ln(di) − Γ′(1)), i ≥ 1. (A.30)
Low Temperature Properties of Quantum Physics Models 17
We conjecture that (A.30) uniquely defines Ti, and in accordance with the
property µ(0) = 0 for the Boltzman equation, that µ(Ti) = 0, i ≥ 1. In this
case (A.30) becomes
Ti =Ei
k (ln(di) − Γ′(1)), i ≥ 1. (A.31)
This result suggests that, for i ≥ 1, the lowest temperature Ti of particles
with energy Ei is strictly positive, and cannot be lowered below Ti.
Remark 8. If Γ′(1) is replaced with zero in (A.31) then T newi = Ei
k(ln(di)).
Since di ≫ 1, the relative change between Ti and T newi satisfies
|Relative Change| =|Ti − T new
i |
Ti=
|Γ′(1)|
ln(di)≪ 1, i ≥ 1. (A.32)
Thus, replacing Γ′(1) with zero has a negligible effect on the lowest temper-
ature prediction.
Remark 9. Mean Value Formulas. Boltzman functions are core com-
ponents of the partition function method of deriving the mean value Fermi-
Dirac and Bose-Einstein equations (0.37) and (0.38). This method requires
an exact formula for each Ni. However, it is difficult to invert (A.25) and
obtain an exact formula for Ni. This may prove especially true if significant
modifications to our continuation of N ! are made. It is a challenging prob-
lem to (i) prove the conjectures made above, and (ii) develop a practical
expression for Ni which allows us to derive new mean value formulas which
replace (0.37) and (0.38) at low T.
Part (III). Recall from (0.6)-(0.8) that the Bose-Einstein equation is
Ni =di − 1
e(Ei−µ)
kT − 1, 0 < T < ∞, i ≥ 1, (A.33)
that µ(0) = 0, and that
limT→0+
Ni = 0 and limNi→0+
Ei − µ
kT= ∞, i ≥ 1. (A.34)
Low Temperature Properties of Quantum Physics Models 18
We claim that prediction (A.34) is erroneous. To prove this claim we ex-
amine the derivation of (A.33). Again, we follow Griffiths ([1], Ch. 6). As-
sume that the particles are identical bosons, and that di ≫ 1, i ≥ 1. Then
Q =∏
∞
n=1(Nn+dn−1)!Nn!(dn−1)! and (0.11) becomes
G = ln
(
∞∏
n=1
(Nn + dn − 1)!
Nn!(dn − 1)!
)
+ α
(
N −∞∑
n=1
Nn
)
+ β
(
E −∞∑
n=1
NnEn
)
.
(A.35)
Applying Stirling’s approximation to (A.35 ) gives
G =∑
∞
n=1 [(Nn + dn − 1) ln (Nn + dn − 1) − Nn ln (Nn) − (dn − 1) ln(dn − 1)]
+ α [N −∑
∞
i=1 Ni] + β [E −∑
∞
n=1 NnEn] .
(A.36)
Set ∂G∂Ni
= 0, α = − µkT and β = 1
kT , and get (A.33). Note that Ni ≫ 1, i ≥ 1,
is required in (A.36). Thus, since Ni → 0+ as T → 0+ in (A.34), it is
incorrect to let T → 0+ in (A.34). This proves that properties (A.34) are
erroneous. To correct these errors we set
(Nn + dn − 1)! = Γ(Nn + dn), Nn! = Γ(Nn + 1) and (dn − 1)! = Γ(dn)
in (A.35). Set ∂G∂Ni
= 0, α = − µkT and β = 1
kT , and obtain the new formula
Ei − µ
kT=
Γ′(Ni + di)
Γ(Ni + di)−
Γ′(Ni + 1)
Γ(Ni + 1), i ≥ 1. (A.37)
It follows from (A.37) that the fundamental ratio Ei−µ(T )kT satisfies
limNi→0+
Ei − µ(T )
kT=
Γ′(di)
Γ(di)− Γ′(1) < ∞, i ≥ 1. (A.38)
Property (A.38) contrasts with the Bose-Einstein equation based predic-
tion (A.34) that limNi→0+Ei−µ(T )
kT = ∞, i ≥ 1.
Next, for i ≥ 1 let Ti denote the lowest value of temperature of particles with
energy Ei. Again, we posit that limT→Ti+ Ni = 0. Combining this property
with (A.37) gives
limT→Ti
+
Ei − µ(T )
kT=
Γ′(di)
Γ(di)− Γ′(1) < ∞, i ≥ 1. (A.39)
Low Temperature Properties of Quantum Physics Models 19
It follows from (A.39) that Ti satisfies the implicit equation
Ti =Ei − µ(T i)
k(
Γ′(di)Γ(di)
− Γ′(1)) , i ≥ 1. (A.40)
We conjecture that (A.40) uniquely defines each Ti, and consistent with the
property µ(0) = 0 for the Boltzman equation, that µ(Ti) = 0, i ≥ 1. In this
case (A.40) becomes
Ti =Ei
k(
Γ′(di)Γ(di)
− Γ′(1)) , i ≥ 1. (A.41)
This result suggests that, for i ≥ 1, the lowest temperature, Ti, of particles
with energy Ei is strictly positive, and cannot be lowered below Ti.
The Case µ(T ) ≡ 0. Our recent study [9] of (A.33) when µ(T ) ≡ 0 was mo-
tivated by a series of experiments in the development of quantum computing
devices [10, 12, 13, 14], where the goal is to lower the temperature of a solid
to a value where all quanta of thermal energy are drained off, leaving the
object in a quantum state. Related modeling investigations [11, 14] assume
that the Bose-Einstein equation
q =d
exp(
hνkT
)
− 1, 0 < T < ∞, (A.42)
for a single atom represents the entire solid: q is the most probable number
of quanta with energy hν and degeneracy d, h is Planck’s constant, ν is
frequency, k is Boltzman’s constant. In 2010 O’Connel et al [11] achieved a
widely acclaimed breakthrough when they reduced the number of quanta per
state in a quantum drum to qd = .07 at T = 20 mK. Their solid contains 1013
atoms which vibrate in three dimensions (i.e. d = 3× 1013) with frequency
ν = 6 × 109 Hz. Substituting these values into (A.42) gives the theoretical
prediction T = 105 mK, which is five times greater than the 20 mK exper-
imental value. To resolve this wide discrepancy between theory (105mK)
and experiment (20mK), we proved that (A.42) gives erroneous predictions
Low Temperature Properties of Quantum Physics Models 20
as T → 0+, and derived the new formula
T =hν
k(
Γ′(q+d)Γ(q+d) − Γ′(q+1)
Γ(q+1)
) . (A.43)
To find the lowest temperature, T0, let q → 0+ in (A.43), and obtain
T0 =hν
k(
Γ′(d)Γ(d) − Γ′(1)
) . (A.44)
Formula (A.44) predicts that all quanta have been drained off at the positive
temperature T0 > 0. This contrasts sharply with the prediction of (A.42)
that q → 0 only when T decreases to zero. We tested (A.44) in two ways:
(i) We substituted the O’Conell et al [11] experimental values into (A.44)
and obtained the improved estimate T0 = 9.8 mK, which is significantly
closer to the 20 mK experimental value than the 105 mK theoretical predic-
tion of the Bose-Einstein equation.
(ii) In 1907 Einstein [16] derived his classical formula for specific heat:
CV = 3NAk( ǫ
kT
)2 exp( ǫkT )
(
exp( ǫkT ) − 1
)2 , 0 < T < ∞, (A.45)
where ǫ = hν, NA is Avogadro’s number. Formula (A.45), which can also
be derived from a microcanonical ensemble approach based on (A.42), has
the following property:
CV is well defined for all T > 0 and limT→0+
CV = 0. (A.46)
Property (A.46) is widely quoted in textbooks in the statistical mechanics
and physics literature (e.g. see Prathia [8], p. 175 or Schroeder [12], p. 309).
In [9] we proved that it is invalid to let T → 0+ in (A.46), hence pre-
diction (A.46) is erroneous. We derived a new formula for Cv which re-
places (A.45), and is valid only when T0 ≤ T < ∞.
Diamond. When d = 3NA (one mole), (A.44) reduces to
T0 = 8.508 × 10−13ν, 0 < ν < ∞. (A.47)
Low Temperature Properties of Quantum Physics Models 21
Set ν = 2.726 × 1013, the value given by Einstein [16] for diamond, and get
T0 = 23.25K This prediction, which could easily be tested, satisfies the
basic requirement that T0 = 23.25K lies below the lowest experimental data
point temperature 225K, and suggests that diamond cannot be cooled below
23.25K, In addition, our prediction that the lowest temperature is positive
is consistent with the results in [11], which show that a high frequency
solid such as diamond more easily reaches its ground state at a positive
temperature. Recent breakthrough experiments of Le et al [15] demonstrate
that two diamonds can exhibit quantum entanglement at room temperature.
Their result contradicts the long held belief that quantum effects can occur
only at extremely low temperatures near absolute zero. Further experiments
and theoretical studies are needed to determine if the diamond preparation
described in (A.47) exhibits quantum entanglement properties.
Remark 10 Replacing Γ′(1) with zero in (A.44) gives the new prediction
T new0 = 23.985K, and the magnitude of relative change is .03 Thus, re-
placing Γ′(1) with zero produces a negligible change in lowest temperature
predictions.
References
[1] D. Griffiths. Introduction To Quantum Mechanics, Prent. Hall, Second
Ed. (1996)
[2] P. Dirac. Proceedings of the Royal Society Series A. 112 (1926) 661-677
[3] E. Fermi. Rendiconti Lincei 3. (1926) 145-149
[4] S. N. Bose. Zeitschrift fur Physik. (1924) 178-181
[5] A. Einstein. Sitzungsberichte der Preussischen Akademie der Wis-
senschaften. (1925) 3-14
Low Temperature Properties of Quantum Physics Models 22
[6] D. V. Schroeder. An Introduction To Thermal Physics, Addison-Wesley
(1999)
[7] P. Grobet, L. Van Gerven, A. Van den Bosch. J. Ch. Phys. 68 (1978)
5225-5230
[8] D. K. Prathia. Statistical Mechanics, Addison-Wesley (1999)
[9] W. C. Troy. AMS Quarterly of Applied Mathematics. to appear.
[10] S. Groblacher, B. Hertzberg, M. Vanner, D. Cole, G. Gigan, S. Schwab,
M. Aspelmeyer. Nature Physics. 5 (2009) 485-488.
[11] A. D. O’Connell, M. Hofheinz, M. Ansmann, C. Bialczak, M. Lenander,
E. Lucero, E. M. Neeley, D. Sank, D., H. Wang, M. Weides, J. Wenner,
J. M. Martinis, A. N. Cleland, Nature. 464 (2010) 697-703
[12] A. Schliesser, O. Arcizet, R. Rivire1, G. Anetsberger, T. Kippenberg.
Nature Physics. 5 (2009) 509-514.
[13] T. Rocheleau, T. Ndukum1, C. Macklin, J. B. Hertzberg, A. A. Clerk,
K. C. Schwab, Nature. 463 (2010) 72-75.
[14] J. D. Teufel, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K.
W. Lehnert, R. W. Simmonds. Nature. 475 (2011) 359-363.
[15] K. C. Le, M. R. Sprague, N.K. Langford; X.-M. Jin; T. Champion; P.
Michelberger; K.F. Reim; D. England; D. Jaksch; I.A. Walmsley, B.J.
Sussman, X.-M. Jin; D. Jaksch. Science. 33 (2011) 6060.
[16] A. Einstein, Die plancksche theorie der strahlung und die theorie der
spezifischen warme, Annalen der Physik. 22 (1907) 180-190.