Physical Limits on the Notion of Very Low Temperatures

12 Pages·0.14 MB·English

by  Juhao Wu

Physical Limits on the Notion of Very Low Temperatures
8 Juhao Wu and A. Widom
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Department of Physics, Northeastern University, Boston, Massachusetts 02115
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h Standard statistical thermodynamic views of temperature fluctuations
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predict a magnitude ( < (∆T)2 >/T) (k /C) for a system with heat
e B
≈
m capacity C. The extent to which low temperatures can be well defined is
p p
- discussed for those systems which obey the thermodynamic third law in the
t
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t form lim(T→0)C = 0. Physical limits on the notion of very low temperatures
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areexhibitedforsimplesystems. Application oftheseconceptstoboundBose
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a condensed systems are explored, and the notion of bound Boson superfluidity
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is discussed in terms of the thermodynamic moment of inertia.
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n PACS numbers: 05.30.Ch, 03.75.Fi, 05.30.Jp, 05.40.+j
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1
I. INTRODUCTION
A problem of considerable importance in low temperature physics concerns physical limi-
tations on how small a temperature can be well defined in the laboratory[1]. In what follows,
we shall consider temperature fluctuations which define a system temperature “uncertainty”
δT = < (T < T >) >2 , (1.1)
s s
−
q
whenever a finite system at temperature T is in thermal contact with a large reservoir
s
at bath temperature T. Only on the thermodynamic average do we expect the system
temperature to be equal to the bath temperature; i.e. < T > T. The fluctuation from
s
≈
thisaverageresult (quotedinthebetter textbooksdiscussing statisticalthermodynamics[2]),
has the magnitude
δT k
B
, (1.2)
T ≈ s C
(cid:16) (cid:17)
where C is the heat capacity of the finite system, and k is Boltzman’s constant. Since C
B
is an extensive thermodynamic quantity, one expects the usual small fluctuation in temper-
ature (δT/T) (1/√N) in the thermodynamic limit N , where N is the number of
∝ → ∞
microscopic particles. However, in low temperature physics (for systems with finite values
for N), temperature fluctuations are by no means required to be negligible.
For those finite sized systems which obey the thermodynamic third law
lim C = 0, (1.3)
T→0
one finds from Eqs.(1.2) and (1.3) that
δT
as T 0 with N < . (1.4)
T → ∞ → ∞
(cid:16) (cid:17)
Eq.(1.4) sets the limits on what can be regarded as the ultimate lowest temperatures for
finite thermodynamic systems; i.e. the temperature must at least obey δT << T.
In Sec.II, the theoretical foundations for Eq.(1.4) will be discussed. In brief, the micro-
canonical entropy of a thermodynamic system is given by
S(E) = k lnΓ(E), (1.5)
B
where Γ(E) is the number of system quantum states with energy E. The micro-canonical
entropy defines the system temperature T via
s
1 dS
= . (1.6)
T dE
s
(cid:16) (cid:17) (cid:16) (cid:17)
The thermal bath temperature T, which is not quite the system temperature T , enters into
s
the canonical free energy
F(T) = k T lnZ(T), (1.7)
B
−
where the partition function is defined as
Z(T) = Tr e−H/kBT = Γ(E)e−E/kBT. (1.8)
(cid:16) (cid:17) XE
2
The probability distribution for the energy of the system, when in contact with a thermal
bath at temperature T, is given by
Γ(E) E F(T) E +TS(E)
P(E;T) = exp − = exp − , (1.9)
Z(T) k T k T
B B
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
as dictated by Gibbs. Thus, the temperature T (of the thermal bath) does not fluctuate
while system energy E does fluctuate according to the probability rule of Eq.(1.9). On the
other hand, the system temperature T (E) in Eq.(1.6) depends on the system energy and
s
thereby fluctuates since E fluctuates. It is only for the energy E∗ of maximum probability
Max P(E;T) = P = P(E∗;T) that the system temperature is equal to the bath tem-
E max
∗
perature T (E ) = T. If the energy probability distribution is spread out at low thermal
s
bath temperatures, then temperature fluctuations are well defined in the canonical ensemble
of Gibbs.
In Sec.III, temperature fluctuations are illustrated using the example of black body
radiation in a cavity of volume V. For this case, it turns out that the thermal wave length
Λ of the radiation in the cavity,
T
h¯c
Λ = , (1.10)
T
k T
B
(cid:16) (cid:17)
must be small on the scale of the cavity length V1/3; i.e. Λ << V1/3. For example, in a
T
cavity of volume V 1 cm3, the lowest temperature for the radiation within the cavity is
∼
of order T 1 oK. It is of course possible to cool the conducting metal walls of a cavity
min
∼
with a length scale 1 cm to well below 1 oK. However, this by no means implies that the
∼
radiation within the cavity can have a temperature well below 1 oK. The point is that at
temperatures T < 1 oK, there are perhaps only a few photons (in total) in the cavity. The
s
total number of photons are far too few for the cavity radiation system temperature to be
well defined.
In Sec.IV, a confined system of atoms obeying ideal gas Bose statistics is discussed.
N
Such systems can be Bose condensed, and are presently (perhaps) the lowest temperature
systems available in laboratories. In the quasi-classical approximation, the free energy is
computed in Sec.V. Questions concerning bounds on ultra-low temperatures are explored.
Whether or not such Bose condensed atoms can exhibit superfluid behavior is discussed
in Sec.VI. The superfluid and normal fluid contributions to the moment of inertia are com-
puted. In the concluding Sec.VII, another simple system with fluctuation limits on ultra-low
temperatures will be briefly discussed.
II. THEORETICAL FOUNDATIONS
Let φ(E) denote some physical quantity which depends on the energy E of a physi-
cal system. If the system is in contact with a thermal bath at temperature T, then the
thermodynamic average may be calculated from
< φ >= P(E;T)φ(E), (2.1)
E
X
3
where the probability P(E;T) has been defined in Eq.(1.9). Using the “summation by
parts”[3,4] formula
∂
P(E;T)φ(E) = 0, (2.2)
∂E
XE (cid:16) (cid:17)
i.e. with a strongly peaked P(E;T)
dφ(E) ∂P(E;T)
P(E;T) = φ(E) , (2.3)
− dE ∂E
XE (cid:16) (cid:17) XE (cid:16) (cid:17)
one finds
dφ(E) ∂P(E;T)
k = k φ(E) . (2.4)
B B
− dE ∂E
D E XE (cid:16) (cid:17)
Employing Eqs.(1.6) and (1.9),
∂P(E;T) 1 1
k = P(E;T). (2.5)
B
∂E T (E) − T
s
(cid:16) (cid:17) (cid:16) (cid:17)
Eqs.(2.4) and (2.5) imply the central result of this section
dφ 1 1
k = φ . (2.6)
B
− dE T − T
s
D E D(cid:16) (cid:17) E
If we choose φ(E) = 1, then Eq.(2.6) reads
1 1
= ; (2.7)
T T
s
D E
i.e. on average, the reciprocal of the system temperature is equal to the reciprocal of the
thermal bath temperature. Thus, with fluctuations from the mean
∆φ = φ < φ >, (2.8)
−
1 1 1 1 1
∆ = = , (2.9)
T T − T T − T
s s s
(cid:16) (cid:17) D E
Eq.(2.6) reads
dφ 1
k = ∆ ∆φ . (2.10)
B
− dE T
D E D (cid:16) (cid:17) E
If we choose in Eq.(2.10) the function φ to be
1 dφ 1 dT 1
s
φ = and = = (2.11)
T − dE T2 dE T2C
(cid:16) s(cid:17) (cid:16) (cid:17) s (cid:16) (cid:17) (cid:16) s (cid:17)
(where C = (dE/dT ) is the system heat capacity), then
s
1 2 1 kB
∆ = . (2.12)
T T2 C
D (cid:16) (cid:17) E D s (cid:16) (cid:17)E
4
The standard Eqs.(1.1) and (1.2) follow from the more precise Eqs.(2.7) and (2.12) in the
limit of small temperature fluctuations; i.e.
k T2
< (∆T)2 > B if δT = < (∆T)2 > << T. (2.13)
≈ C
(cid:16) (cid:17) q
The condition δT << T is required in order that the canonical thermal bath tempera-
ture be equivalent to the micro-canonical system temperature. If the micro-canonical and
canonical temperatures are not equivalent, then the statistical thermodynamic definition of
temperature would no longer be unambiguous. This raises fundamental questions as to the
physical meaning of temperature. The view of this work is that in an ultra-low temperature
limit, whereby δT T for sufficiently small T, the whole notion of system temperature is
∼
undefined, although the notion of a thermal bath temperature retains validity.
III. BLACK BODY RADIATION EXAMPLE
The heat capacity of black body radiation in a cavity of volume V with the walls of the
cavity at temperature T is given by[5]
4π2 V
C(Black Body) = k , (3.1)
B 15 Λ3
(cid:16) (cid:17)(cid:16) T(cid:17)
where Λ is given by Eq.(1.10). From Eqs.(1.2) and (3.1) it follows that the radiation
T
temperature of a black body cavity of volume V is
δT(Black Body) 15 Λ3 Λ3
T 0.6 T . (3.2)
T ≈ s 4π2 V ≈ s V
(cid:16) (cid:17) (cid:16) (cid:17)(cid:16) (cid:17) (cid:16) (cid:17)
In order to achieve a well defined radiation temperature inside the cavity, δT(Black Body)
must be small on the scale of T or equivalently Λ << V1/3. As stated in Sec.1, this implies
T
a minimum temperature of T 1 oK for a cavity of V 1 cm3.
min
∼ ∼
IV. CONFINED IDEAL BOSE GAS
The grand canonical free energy of an ideal Bose gas is determined by the trace[6]
Ξ(T,µ) = k T tr ln 1 e(µ−h)/kBT , (4.1)
B
−
(cid:16) (cid:17)
where h is the one Boson Hamiltonian and
dΞ = dT dµ (4.2)
−S −N
determines the number of Bosons . If the one Boson partition function is defined as
N
q(T) = tr e−h/kBT , (4.3)
(cid:16) (cid:17)
5
then the free energy obeys
∞
1 T
Ξ(T,µ) = k T q enµ/kBT. (4.4)
B
− n n
nX=1(cid:16) (cid:17) (cid:16) (cid:17)
The mean number of Bosons is
∞
T
(T,µ) = q enµ/kBT, (4.5)
N n
nX=1 (cid:16) (cid:17)
and the statistical entropy is given by
∞
Ξ(T,µ)+µ (T,µ) 1 T
(T,µ) = N +k T q′ enµ/kBT, (4.6)
B
S − T n2 n
(cid:16) (cid:17) nX=1(cid:16) (cid:17) (cid:16) (cid:17)
where q′(T) = dq(T)/dT .
{ }
Of considerable theoretical[7,8] experimental[9,10,11] interest is the bound Boson in an
anisotropic oscillator potential,
h¯2 1 h¯
h = 2 + Mr ωˆ2 r tr(ωˆ) (4.7)
− 2M ∇ 2 · · − 2
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
where
ω2 0 0
1
ωˆ2 = 0 ω2 0 . (4.8)
 2 
0 0 ω2
 3
 
Eqs.(4.3) and (4.7) imply
3 1
q(T) = . (4.9)
1 e−¯hωj/kBT
jY=1(cid:16) − (cid:17)
The heat capacity may be defined by
∂S
C = T , (4.9)
∂T N
(cid:16) (cid:17)
which must be calculated numerically.
Shown in Fig.1 is a plot of the heat capacity (in units of k ) versus temperature (in
B
N
unitsofthecriticaltemperatureT )forthecaseof = 2 103 and = 2 106 particles. We
c
N × N ×
choose, forexperimentalinterest[12], thefrequencyeigenvalues(ω /2π) = (ω /2π) = 320Hz,
1 2
and (ω /2π) = 18Hz.
3
6
FIGURES
Heat Capacity vs. Temperature
N=2,000 (Dotted) N=2,000,000 (Solid)
11
10
9
8
7
k) 6
N
C/
( 5
4
3
2
1
0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
(T/Tc)
FIG. 1. The heat capacity (in units of k ) is plotted as a function of temperature (in units
B
N
of T ) for = 2 103 atoms (dotted curve) and = 2 106 atoms (solid curve).
c
N × N ×
Figure 1.
For finite , there is strictly speaking no Bose-Einstein condensation phase transition.
N
The critical temperature T is therefore defined as that temperature for which the heat
c
capacity reaches the maximum value C = C(T ). Although phase transitions are defined
max c
in mathematics only in the thermodynamic limit , for all practical purposes, a
N → ∞
quasi-classical approximation of Eq.(4.1) in the form
d3rd3p
Ξ(T,µ) = k T ln 1 e(µ−h(p,r))/kBT (quasi classical), (4.10)
B
(2πh¯)3 − −
Z Z (cid:16) (cid:17) (cid:16) (cid:17)
where
p2 1
h(p,r) = + Mr ωˆ2 r, (4.11)
2M 2 · ·
does yield a Bose-Einstein condensation phase transition whose heat capacity is sufficiently
accurate for values of 105 or higher. Thus, we regardrecent experiments on Bose atoms
N ∼
confined in a magnetic bottle to be probing a physical Bose-Einstein ordered phase. Let us
consider Eq.(4.10) in more detail.
7
V. QUASI-CLASSICAL BOSE CONDENSATION
In order to evaluate Eq.(4.10) we employ the quasi-classical form[13,14] of Eqs.(4.3) and
(4.11); i.e.
q(T) = d3rd3p e−h(p,r)/kBT = 3 kBT = kBT 3, (5.1)
(2πh¯)3 h¯ω h¯ω¯
Z Z (cid:16) (cid:17) jY=1(cid:16) j (cid:17) (cid:16) (cid:17)
where ω¯ = (ω ω ω )1/3. From Eqs.(4.4) and (5.1), it follows that Eq.(4.10) evaluates to
1 2 3
∞
Ξ(T,µ) = h¯ω¯ kBT 4 1 enµ/kBT. (5.2)
− h¯ω¯ n4
(cid:16) (cid:17) nX=1(cid:16) (cid:17)
From Eqs.(4.2) and (5.2), the number of particles obeys
∞
(T,µ) = kBT 3 1 enµ/kBT, (µ < 0). (5.3)
N h¯ω¯ n3
(cid:16) (cid:17) nX=1(cid:16) (cid:17)
With the usual definition of the ζ-function
∞
1
ζ(s) = , e(s) > 1, (5.4)
ns R
nX=1(cid:16) (cid:17)
the Bose-Einstein condensation critical temperature is
h¯ω¯ N 1/3
T = . (5.5)
c
k ζ(3)
B
(cid:16) (cid:17)(cid:16) (cid:17)
The non-zero number of Bosons (below the critical temperature) in the condensate state is
given by
T 3
(T) = 1 , (T < T ). (5.6)
0 c
N N − T
c
n (cid:16) (cid:17) o
Finally, the entropy below the critical temperature
kBT 3 4ζ(4) T 3
(T) = 4k ζ(4) = Nk , (T < T ), (5.7)
B B c
S h¯ω¯ ζ(3) T
c
(cid:16) (cid:17) (cid:16) (cid:17)(cid:16) (cid:17)
obeys the thermodynamic third law lim (T) = 0. The heat capacity in the Bose-
(T→0)
S
Einstein Condensed phase is then given by
C 12ζ(4) T 3 T 3
= 10.81 , (T < T ). (5.8)
c
k ζ(3) T ≈ T
B c c
(cid:16)N (cid:17) (cid:16) (cid:17)(cid:16) (cid:17) (cid:16) (cid:17)
Employing Eqs.(1.2)and(5.8)wefindthatthetemperatureuncertaintybelowthecritical
temperature obeys
δT 0.3 Tc 3/2
, (T < T ). (5.9)
c
T ≈ √ T
(cid:16) (cid:17) (cid:16) N(cid:17)(cid:16) (cid:17)
Thus, for 105 one may safely consider the temperature of the ordered phase to be well
N ∼
defined in the range T > T > T , where T 0.05 T . The open question as to whether
c min min c
∼
the ordered phase is a superfluid may now be considered.
8
VI. SUPERFLUID FRACTION OF THE BOUND BOSON SYSTEM
The notion of a superfluid fraction in an experimental Bose fluid (such as liquid He4)
may be viewed in the following manner: Suppose that we pour the liquid into a very slowly
rotating vessel and close it off from the environment. The walls of the vessel are at a bath
temperature T, and the vessel itself rigidly rotates at a very small angular velocity Ω. In
the “two-fluid” model[15,16], the normal part of the fluid rotates with a rigid body angular
velocity Ω, which is the same as the angular velocity of the vessel. On the other hand, the
superfluid part of the fluid does not rotate. The superfluid exhibits virtually zero angular
momentum for sufficiently small Ω. The total fluid moment of inertia tensor Iˆis defined by
the fluid angular momentum L = IˆΩ (as Ω 0). We here take the limit Ω 0, to avoid
· → →
questions concerning the effects of vortex singularities on the superfluid. The normal fluid,
which rotates along with the rotating vessel, contributes to the fluid moment of inertia. The
superfluid, which does not rotate with the vessel, does not contribute to the moment of
inertia. Thus, the geometric moment of inertia,
Iˆgeometric = d3rρ¯(r)(r2δ r r ), (6.1)
ij ij − i j
Z
where ρ¯(r) is the mean mass density of the fluid (at rest), overestimates the physical moment
of inertia eigenvalues when the fluid is actually a superfluid. The normal fluid contributes
to the moment of inertia and the superfluid does not do so in the limit Ω 0. Below, we
→
consider in detail the moment of inertia of the bound Bose gas.
For the bound Bose system, we consider a mesoscopic rotational state[17] with a thermal
angular velocity Ω. The rotational version of Eq.(4.2) reads
dΞ = dT dµ L dΩ, (6.2)
Ω
−S −N − ·
where L is the bound Boson angular momentum. Eq.(4.11) gets replaced by
p2 1
h (p,r) = + Mr ωˆ2 r Ω (r p), (6.3)
Ω
2M 2 · · − · ×
so that Eq.(5.1) now reads
d3rd3p
q (T) = e−hΩ(p,r)/kBT = q(T) Det(1 ωˆ−2Ωˆ2) , (6.4)
Ω
(2πh¯)3 −
Z Z (cid:16) (cid:17) (cid:16) .q (cid:17)
where the matrix ωˆ2 is written in Eq.(4.8) and
(Ω2 +Ω2) Ω Ω Ω Ω
Ωˆ2 = 2Ω Ω3 (Ω−2 +1 Ω22) −Ω1Ω3 . (6.5)
 − 1 2 1 3 − 2 3 
Ω Ω Ω Ω (Ω2 +Ω2)
 − 1 3 − 2 3 1 2 
 
From Eqs.(4.4) and (6.4), it follows that
Ξ (T,µ) = Ξ(T,µ) Det(1 ωˆ−2Ωˆ2) , (6.6)
Ω
−
(cid:16) .q (cid:17)
9
The fluid moment of inertia tensor has the matrix elements
∂L ∂2Ξ
Iˆ = lim i = lim Ω . (6.7)
ij
Ω→0 ∂Ω T,µ −Ω→0 ∂Ω ∂Ω T,µ
j j j
(cid:16) (cid:17) (cid:16) (cid:17)
Eqs.(4.8), (6.5), (6.6) and (6.7) imply (in the unordered phase)
1 + 1 0 0
ω22 ω32
Iˆ= Ξ(T,µ)(cid:16) 0 (cid:17) 1 + 1 0 , (T > T ). (6.8)
− ω12 ω32 c
 0 (cid:16) 0 (cid:17) 1 + 1 
 ω12 ω22 
 (cid:16) (cid:17)
In the unordered phase, obeying Eq.(5.2), one finds that Eq.(6.8) is precisely what would
be expected from a normal fluid with geometric moment of inertia
Iˆ = d3rρ¯(r)(r2δ r r ), (T > T ), (6.9)
ij ij i j c
−
Z
where ρ¯(r) is the mean mass density of the atoms.
In the ordered phase (T < T ), the moment of inertia of the particles over and above the
c
condensate is given by Eq.(6.8) with µ = 0, i.e.
1 + 1 0 0
Iˆexcitation = ζ(4)h¯ω¯ kBT 4(cid:16)ω22 0 ω32(cid:17) 1 + 1 0 , (T < T ). (6.10)
h¯ω¯ ω12 ω32 c
(cid:16) (cid:17)  0 (cid:16) 0 (cid:17) 1 + 1 
 ω12 ω22 
 (cid:16) (cid:17)
The question of superfluidity concerns the magnitude of the moment of inertia of those
particles within the condensate. For T < T , we use the notation that Iˆ denotes the mo-
c
ment of inertia of the excited Bosons, and Jˆ represents the moment of inertia of the Bose
condensate. If the moment of inertia of the particles in the condensate were zero, then the
condensate particles would all be “superfluid”.
Let ψ (r) be the normalized ( d3r ψ (r) 2 = 1) Bose condensation state. From the
0 0
| |
geometric viewpoint, the moment of inertia of the condensate would be given by
R
Jgeometric = d3r ψ (r) 2(r2δ r r ); (6.11)
ij N0 | 0 | ij − i j
Z
i.e.
1 + 1 0 0
h¯ ω2 ω3
Jˆgeometric = N0 (cid:16) 0 (cid:17) 1 + 1 0 . (6.12)
2 ω1 ω3
 0 (cid:16) 0 (cid:17) 1 + 1 
 
 ω1 ω2 
 (cid:16) (cid:17)
The physical Bose condensate moment of inertia tensor is in reality
< ψ l ψ >< ψ l ψ > + < ψ l ψ >< ψ l ψ >
Jphysical = 0| i| κ κ| j| 0 0| j| κ κ| i| 0 , (6.13)
ij N0 ǫ ǫ
Xκ (cid:16) κ − 0 (cid:17)
where l = ih¯(r ). One may derive Eq.(6.13) by treating the rotational coupling ∆h =
− ×∇
Ω l to second order perturbation theory in the energy ∆ǫ (Ω) as Ω 0. Eq.(6.13)
0
− · →
10