low temperature properties of the fermi-dirac, boltzman ...troy/papers/pla2012ddd.pdf · low...

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)


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)


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)


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?


(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,


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


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)


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,


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)


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)


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.


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)


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)


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+


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)


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)


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+


Ei − µ

kT= ∞, i ≥ 1. (A.34)


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)


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


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)


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

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Ed. (1996)

[2] P. Dirac. Proceedings of the Royal Society Series A. 112 (1926) 661-677

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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.

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