Quantum Szilard Engine

1.1 MB·

by  Sang Wook Kim

Quantum Szilard Engine
Sang Wook Kim1,2, Takahiro Sagawa2, Simone De Liberato2, and Masahito Ueda2
1Department of Physics Education, Pusan National University, Busan 609-735, Korea and
2Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
(Dated: January 11, 2011)
TheSzilardengine(SZE)isthequintessenceofMaxwell’sdemon,whichcanextracttheworkfrom
a heat bath by utilizing information. We present the first complete quantum analysis of the SZE,
and derive an analytic expression of the quantum-mechanical work performed by a quantum SZE
containing an arbitrary number of molecules, where it is crucial to regard the process of insertion
or removal of a wall as a legitimate thermodynamic process. We find that more (less) work can be
extracted from thebosonic (fermionic) SZE dueto theindistinguishability of identical particles.
1
PACSnumbers: 03.67.-a,05.30.-d,89.70.Cf,05.70.-a
1
0
2
Maxwell’sdemonisahypotheticalbeingofintelligence

n that was conceived to illuminate possible limitations of
a the second law of thermodynamics [1, 2]. Leo Szilard
J
conducted a classical analysis of the demon, considering
8
an idealized heat engine with a one-molecule gas, and

] directlyassociatedthe informationacquiredbymeasure-
h ment with a physical entropy to save the second law [3].
p
The basic working principle of the Szilard engine (SZE)
-
t is schematically illustrated in Fig. 1. If one acquires the
n
informationconcerningwhichsidethemoleculeisinafter
a
u dividing the box, the information can be utilized to ex-
q tractwork,e.g.,viaanisothermalexpansion. Thecrucial
[
question here is how this cyclic thermodynamic process

2 is compatible with the second law; the entropy of kBln2
v (k is the Boltzmann constant) that the engine acquires
B FIG. 1: (Color online) Schematic diagram of the thermo-
1 duringtheisothermalprocessisnotreturnedtothereser- dynamic processes of the classical SZE. Initially a single
7
voirbutseemstobeaccumulatedinsidetheengine. Now molecule is prepared in an isolated box. (A) A wall depicted
4
it is widely accepted that the measurement process in- as a vertical gray bar is inserted to split the box into to two
1
. cluding erasure or reset of demon’s memory requires the parts. The molecule is represented by the dotted circles to
6 indicate that at this stage we do not know in which box the
minimum energy cost of at least k ln2, associated with
0 B molecule is. (B) By the measurement, we find where the
0 the entropy decrease of the engine, and that it saves the molecule is. (C) A load is attached to the wall to extract a
1 second law [4–7]. work via an isothermal expansion at a constant temperature
:
v Although the SZE deals with a microscopic object, T.
i
X namely an engine with a single molecule, its fully quan-
tum analysis has not yet been conducted except for the
r
a measurement process [8, 9]. In this Letter we present gradually occurs as the temperature increases. We as-
the first complete quantum analysis of the SZE. In the sume that the measurement is performed perfectly. The
previous literature, it takes for granted that insertion or case of imperfect measurement is discussed in terms of
removal of the wall costs no energy. This assumption is mutual information in Ref. [7].
justified in classical mechanics but not in quantum me- To define the thermodynamic work in quantum me-
chanics [10] because the insertion or removal of the wall chanics, let us consider a closed system described as
alters the boundary condition that affects the eigenspec- Hψ =E ψ , whereH, ψ andE arethe Hamiltonian
n n n n n
trumofthesystem. Asshownbelow,acarefulanalysisof ofthesystem,itsntheigenstateandeigenenergy,respec-
this process leads to a concise analytic expression of the tively. The internal energy U of the system is given as
total net work performed by the quantum SZE. If more U = E P ,whereP isthemeanoccupationnumber
n n n n
thanoneparticleispresentintheSZE,weencounterthe ofthePntheigenstate. InequilibriumPn obeysthecanon-
issue of indistinguishability of quantum identical parti- ical distribution. From the derivative of U, one obtains
cles. Indeed,howmuchworkisextractedfromthequan- dU = (E dP + P dE ). Analogous to the classi-
n n n n n
tum SZE strongly depends crucially on whether it con- cal therPmodynamic first law, TdS = dU +dW, where S
sists of either bosons orfermions. We also showthat the and W are the entropy and work done by the system,
crossover from indistinguishability to distinguishability respectively, the quantum thermodynamic work (QTW)
2
can be identified as dW = − P dE [11, 12]. Note
n n n
that nEndPn shouldbe assoPciatedwithTdS sincethe
entroPpy S is defined as S =−kB nPnlnPn.
AlthoughtheprocessofinsertinPgawallisaccompanied
byneitherheatnorworkintheclassicalSZE,itisnotthe
casewiththequantumSZE.Thisprocesscanbemodeled
asthatofincreasingtheheightofthepotentialbarrier. In
quantummechanics,energylevelsthenvary,contributing
totheQTW.Thisprocesscanbeperformedisothermally
sothatthetemperatureiskeptconstantduringthewhole
processinconformitywiththe originalspiritofthe SZE.
If the insertion is performed in an adiabatic process de-
fined as dQ = E dP = 0, one can easily show that
n n n
the temperaturPe is either changed or not well-defined at
the end. The former is obvious considering the classical
thermodynamic adiabatic process. If the wall is inserted
in a quantum adiabatic manner, dPn = 0 is always sat- FIG. 2: Schematic diagram of the quantum SZE containing
isfied. Given that the temperature is defined from the threemolecules. (I)Threemoleculesarepreparedinaclosed
ratio of probabilities as P /P = e−(En−Em)/kBT, it is box with size L. (II) A wall, depicted by a vertical gray bar,
n m is isothermally inserted at location l. The process (I) → (II)
well-defined only if all energy differences are changed by
is called ‘insertion’. (III) The information on the number of
the same ratio [13]. However, this cannot be achieved
molecules,m,ontheleftisacquiredbythemeasurement. The
in the SZE because eachenergy level shifts in a different process (II) → (III) is called ’measurement’. (IV) The wall
manner [14]. movesandundergoesanisothermalexpansionuntilitreaches
To describe the quantum SZE it is indeed sufficient its equilibrium location denoted by lm. The process (III) →
eq
to know only the isothermal process for a whole cycle. (IV) is called ‘expansion’. Finally the wall is isothermally
If the external parameter X is varied from X to X removed to complete the cycle. The process (IV) → (I) is
1 2
called ‘removal’.
isothermally, the QTW is obtained as
X2 ∂lnZ ∂E
W = k T ndX (1) work required for the insertion process is thus expressed
B
Z ∂E ∂X
Xn X1 n as
= k T[lnZ(X )−lnZ(X )], (2)
B 2 1
W =k T[lnZ(l)−lnZ(L)]. (3)
ins B
where Z = e−βEn is the partition function with β =
n
1/kBT. P Then, the measurement is performed without any ex-
It is worth mentioning that the isothermal process in- penditure of work. The amount of work extracted
ducesthermalizationofthemoleculewiththereservoirat via the subsequent isothermal expansion is given as
every moment, which destroys all the coherence among Wexp = kBT Nm=0fm lnZm(lemq)−lnZm(l) , where
energy levels. Therefore, it is not necessary to describe fm = Zm(l)/ZP(l) repres(cid:2)ents the probability o(cid:3)f having
the dynamics of our system in terms of the full density m particles on the left at the measurement. The wall
matrix; it is sufficient to know its diagonal part, i.e. the moves until it reaches an equilibrium position lm deter-
eq
probabilities P . However,this thermalizationhas noth- mined by the force balance, Fleft+Fright =0, where the
n
ing to do with the measurement of the location of the generalizedforce F is defined as P (∂E /∂X), as il-
n n n
molecule since it proceeds regardless of which box the lustrated in Fig. 2(IV). We notePthat lm is not simply
eq
molecule is in. (m/N)L unlike classical ideal gases.
Let us now consider a quantum SZE in a generalsitu- The wall is then finally removedisothermally. In real-
ation. As shownin Fig.2, the whole thermodynamic cy- ity the wall is not impenetrable, and has a finite poten-
cle consists offour processes,namely insertion,measure- tial height, namely X∞. During the expansion process,
ment, expansion and removal among four distinct states X∞ is assumed to be large enough to satisfy τt ≫ τ ,
(I)-(IV).N idealidenticalmoleculesarepreparedinapo- where τ and τ are a tunneling time between the two
t
tentialwellofsizeL asshowninFig.2(I).Awallis then sides and an operational time of thermodynamic pro-
isothermally inserted at a certain position l. The parti- cesses,respectively,ensuringthatmiswelldefined. Dur-
tion function, at the moment when the wall insertion is ingthewallremoval,however,τ graduallydecreasesand
t
completed but the measurement is not performed yet, is becomes comparable with τ for a certain strength, X ,
0
given as Z(l) = N Z (l), where Z (l) denotes the where any eigenstate is delocalized over both sides due
m=0 m m
partition functionPfor the case in which m particles are to tunneling. It implies that the partition function is
on the left side and N −m on the right. The amount of given by Z(lm) = N Z (lm) rather than Z (lm).
eq n=0 n eq m eq
P
3
limit, ∆ is simply given as E (L/2) − E (L). If the
1 1
TABLE I: Total work measured in unitsof kBT of thequan- insertion process were ignored in the classical SZE, the
tumSZEwithl=L/2containingtwobosonsortwofermions
second law would be violated because ∆ ≫ k T in the
at the low and the high temperature limits (see [18] for de- B
tailed derivation). low-temperature limit. In fact, ∆ of the expansion pro-
cess is compensated by the work required for inserting
bosons fermions
the wall. In the end, a tiny difference of work between
T →0 (2/3)ln3 0
these two processes results in the precise classical value,
T →∞ ln2 ln2
k Tln2. As the temperature increases, the classical re-
B
sultsofindividualprocessesarerecovered,i.e. W →0,
ins
W → k Tln2 and W = 0 since ∆ approaches
The integral (1) for each m can be split into two parts; exp B rem
k Tln2 in this limit.
X0[∂lnZ (lm)/∂X]dX and 0 [∂lnZ(lm)/∂X]dX. It B
X∞ m eq X0 eq ForthequantumSZEwithmorethanoneparticledra-
iRsthenshownthattheformervRanishesasfarasthequasi-
matic quantum effects come into play. Let us consider a
static process, τ →∞ (i.e. X0,X∞ →∞), is concerned. two-particle SZE confined in a symmetric potential well
This leads us to ∗ ∗
and a wall at l = L/2. One also finds f = f = 1
0 2
N because of the same reason mentioned above. Since for
W =k T f lnZ(L)−lnZ(lm) . (4) m =1 the wall does not move in the expansion process,
rem B m eq
mX=0 (cid:2) (cid:3) implying l = le1q, one obtains f1 = f1∗. We thus end up
with
NotethatthesummationovermmustbemadeinZ(l)of
Eq.(3) forthe insertionprocessirrespectiveoftunneling Wtot =−2kBTf0lnf0, (6)
since no measurement has been performed yet.
where f =f is used.
0 2
Combining all the contributions the total work per-
To get some physical insights let us consider two lim-
formed by the engine during a single cycle is given by
iting cases of Eq. (6), which is summarized in Table I
(see [18] for a detailed derivation). For simplicity here
N
f
W =W +W +W =−k T f ln m , the spin of a particle is ignored. In the low-temperature
tot ins exp rem B mX=0 m (cid:18)fm∗ (cid:19) limit only the ground state is predominantly occupied,
(5) so that there exists effectively only one available state
where f∗ = Z (lm)/Z(lm). Equation (5) has a clear for each side. It is clear, as shown in Fig. 3(a), that for
m m eq eq
information-theoretic interpretation in the context of bosons f should become 1/3, i.e. W =k T(2/3)ln2,
0 tot B
the so-called relative entropy [15] or Kullback-Leibler since we consider two mutually indistinguishable bosons
∗
divergence even though f is not normalized, namely overtwoplaces. Ontheotherhand,twofermionsarepro-
m
f∗ 6= 1. It has been shown that the average diss- hibitedtobeinthesamesideortooccupythesamestate
m m
pPative work upon bringing a system from one equilib- due to the Pauli exclusion principle, which explains why
rium state at a temperature T into another one at the the work vanishes in the low-temperature limit. How-
same temperature is given by the relative entropy of the ever, the higher the temperature, the larger the num-
phasespacedistributionsbetweenforwardandbackward ber of available states. Thermal fluctuations wash out
processes[16]. Withfilteringorfeedbackcontrollikemea- indistinguishability since two identical particles start to
surement processes of the SZE that determine m, the be distinguished by occupying different states. This is
work is represented as a form equivalent to Eq. (5) [17]. why in the high-temperature limit we havef =1/4,i.e.
0
It is straightforward to apply Eq. (5) to the original W = k T ln2, for both bosons and fermions, which
tot B
SZE consisting of a single molecule of mass M. For results from allocating two distinguishable particles over
simplicity let us consider an infinite potential well of two places as shown in Fig. 3(b). It is also shown in
∗ ∗
size L, and l = L/2. One finds f = f = 1 since Fig. 3 that f continuously varies from 1/3 (0) to 1/4,
0 1 0
in these cases the wall reaches the end of the box so exhibiting the crossover from indistinguishability to dis-
that Z(lm) = Z (lm) (m = 0,1) is satisfied. Note that tinguishability for bosons (fermions) as the temperature
eq m eq
∗
f = 1 is always true for m = 0 and N. Together with increases.
m
f = f = 1/2, we obtain W = k T ln2, implying The inset of Fig. 3 clearly shows that one can extract
0 1 tot B
the work performed by the quantum SZE is equivalent more work from the bosonic SZE but less work from the
to that of the classical SZE. However, consideration of fermionicSZEovertheentirerangeoftemperature. (See
individual processesrevealsan importantdistinction be- [18] for detailed discussions of the W (T).) While de-
tot
tweentheclassicalandquantumSZE’s. Forthequantum tailsofW (T)dependontheconfinementpotential,its
tot
SZE one obtains W = −∆+k T ln2, W = ∆ and low-temperaturelimitsgiveninTableIareuniversaland
ins B exp
W = 0 for each process, where ∆ = ln[z(L)/z(L/2)], have a deep physical meaning associated with the infor-
rem
z(l) = ∞ e−βEn(l), and E (l) = h2n2/(8Ml2) with mationcontentofquantumindistinguishableparticlesas
n=1 n
h beingPthe Planck constant. In the low-temperature mentioned above.
4
information in quantum physics.
We would like to thank Takuya Kanazawa and Juan
Parrondo for useful discussions. SWK acknowledges
JSPS fellowship for supporting his stay in University of
Tokyo. SWK was supported by the NRF grant funded
bytheKoreagovernment(MEST)(No.2009-0084606and
No.2009-0087261). TS acknowledgesJSPSResearchFel-
lowships for Young Scientists (Grant No.208038). SDL
acknowledges FY2009 JSPS Postdoctoral Fellowship for
Foreign Researchers. MU was supported by a Grant-in-
Aidfor Scientific Research(GrantNo.22340114)andthe
PhotonFrontierNetworkProgramoftheMinistryofEd-
ucation,Culture,Sports,ScienceandTechnology,Japan.
FIG.3: (Color online) f0 as afunction of T for bosons(solid
curve),fermions (dashed curve)and classical particles (dash-
dotted line) in the case of the infinite potential well. The
temperature is given in units of E1(L)/kB. (a) Three pos- [1] H. S. Leff and A. F. Rex, Maxwell’s Demons 2 (IOP
sible ways in which two identical bosons are assigned over
Publishing, Bristol, 2003).
two states. (b) Four possible ways in which two distinguish-
[2] K. Maruyama, F. Nori, and V. Vedral, Rev.Mod. Phys.
able particles are allocated over two places. The inset shows 81, 1 (2009).
Wtot/Wc as a function of T. [3] L. Szilard, Z. Phys.53, 840 (1929).
[4] L. Brillouin, J. Appl.Phys. 22, 334 (1951).
[5] R. Landauer, IBM J. Res. Dev.5, 183 (1961).
Finally, we briefly mention possible experimental real- [6] C. H.Bennett, Int.J. Theor. Phys. 21, 905 (1982).
izations of the quantum SZE. Although several experi- [7] T. Sagawa and M. Ueda, Phys. Rev. Lett. 102, 250602
ments [19–22] or proposals [23, 24] associated with the (2009).
classical SZE have been presented so far, its fully quan- [8] W. H. Zurek, Frontiers of Nonequlibrium Statistical
tum version has been elusive. There exist three impor- Physicsed.G.T.MooreandM.O.Scully,p.151(Plenum
Press, New York,1984).
tantingredientsfortheexperimentalrealization: (i)con-
[9] S. Lloyd, Phy.Rev.A 56, 3374 (1997).
trollability of the confinement potential, (ii) availabil-
[10] C. M. Bender, D. C. Brody, and B. K. Meister, Proc.
ity of a thermal heat bath to perform isothermal pro- Roy.Soc. London A 461, 733 (2005).
cesses,and(iii)measurabilityoftheworkperformed. For [11] T. D.Kieu, Phys. Rev.Lett. 93, 140403 (2004).
bosons,the systemoftrappedcoldatomsmaybeagood [12] M. Esposito and S. Mukamel, Phys. Rev. E 73, 046129
candidate since the confinement potential can be easily (2006).
controlled. Although such a system usually lacks a ther- [13] H. T. Quan, Y.-X. Liu, C. P. Sun, and F. Nori, Phys.
Rev. E76, 031105 (2007).
mal heat bath, it can be immersed in a different species
[14] Forexample,ifthewallislocatedatthenodeofthewave-
ofatomtrappedinawiderconfinementpotentialsothat
function,thecorrespondingenergylevelneverchangesas
they can play a role of a heat reservoir. For fermions,
the height of the wall increases even though other levels
two-dimensionalelectrongasesconfinedinquantumdots move substanially.
made of semiconductor heterostructures might be a can- [15] V. Vedral, Rev.Mod. Phys. 74, 197 (2002).
didateduetoitscontrollabilityoftheconfinementpoten- [16] R. Kawai, J. M. R. Parrondo, and C. Van den Broeck,
tialandthe existenceofaheatreservoirofthe Fermisea Phys. Rev.Lett. 98, 080602 (2007).
[17] J. M. R. Parrondo, C. Van den Broeck, and R. Kawai,
of electrons. In principle, the work is determined once
New J. Phys.11, 1 (2009).
both E and P are known during the entire isothermal
n n [18] See supplementary material.
processes for both bosons and fermions. [19] V.Serreli,C.-F.Lee,E.R.Kay,andD.A.Leigh,Nature
In summary, we have studied the quantum nature of 445, 523 (2007).
theSZE.ThetotalworkperformedbythequantumSZE [20] G.N.Price,S.T.Bannerman,K.Viering,E.Narevicius,
isexpressedasasimpleanalyticformulawhichisdirectly and M. G.Raizen, Phys.Rev.Lett.100, 093004 (2008).
associatedwiththerelativeentropyintheclassicallimit. [21] J.J.Thorn,E.A.Schoene,T.Li,andD.A.Steck,Phys.
Rev. Lett.100, 240407 (2008).
To correctly describe the quantum SZE the processes of
[22] S. Toyabe, T. Sagawa, M. Ueda, Eiro Muneyuki, and
inserting or removing the wall should be regarded as a
Masaki Sano, Nature Phys.doi:10.1038/nphys1821.
relevant thermodynamic procedure. The quantum SZE
[23] M. O. Scully, M. S. Zhubairy, G. S. Agarwal, and H.
consisting of more than one particle clearly shows the Walther, Science 299, 862 (2003).
quantum nature of indistinguishable identical particles. [24] S. W. Kim and M.-S.Choi, Phys. Rev.Lett.95, 226802
We believe our finings shed light on the subtle role of (2005).
EPAP: Quantum Szilard Engine
Sang Wook Kim1,2, Takahiro Sagawa2, Simone De Liberato2, and Masahito Ueda2
1Department of Physics Education, Pusan National University, Busan 609-735, Korea and
2Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
(Dated: January 11, 2011)
WepresentthederivationofTableIandthedetaileddiscussionofWtot(T)inthehightemperature
limit.
PACSnumbers:
I. DERIVATION OF TABLE I
1
1
Even though W obtained from Eq. (6) depends on what the confinement potential is, in the limits T → 0 and
0 tot
2 T →∞, it has a universalvalue under quite generalconditions discussedbelow. We firstconsider the two-bosoncase
for an arbitrary confinement potential V(x) with reflection symmetry, V(x)=V(−x). The wall is inserted at x=0,
n
i.e. l=L/2 separating V into two single-wells. The probability that two bosons are in the right well is given as
a
J
d+1
8 f = , (S1)
0
4d+2

]
h where
p
z(β)2
- d≡ (S2)
t
n z(2β)
a
u with z(β) is the partition function of a single particle in the single-well at temperature T ≡(k β)−1.
B
q To obtain universal values of f in the high- and low-temperature limits, the key observation is that
0
[

d→∞ (T →∞) (S3)
2
v
and
1
7
d→1 (T →0) (S4)
4
1
. hold only with the following two assumptions:
6
0 1. The number of energy levels is infinite.
0
1 2. The single-particle ground state is not degenerate.
:
v
We first prove Eq. (S3). Since z(β) > z(2β) holds, we have d > z(β). On the other hand, we can show that
Xi z(β)→∞ holds with T →∞ from assumption 1. Therefore we obtain Eq. (S3). Next, from assumption 2, we have
d=(1+2e−β(E1−E0)+···)/(1+e−2β(E1−E0)+···),whereE isthenthenergylevelofthesingle-well. Itimmediately
r n
a leads us to Eq. (S4). From Eqs. (S3) and (S4), we obtain
f →1/4 (T →∞) (S5)
0
and
f →1/3 (T →0), (S6)
0
which are potential-independent universalresults anddescribe the distinguishability-indistinguishability crossoverby
varying temperature. The total work is then given as
W →k T ln2 (T →∞) (S7)
tot B
and
W →(2/3)k T ln3 (T →0), (S8)
tot B
which are also model-independent.
2
FIG. S1: Wtot/Wc as afunction of T for bosons (solid curve)and fermions (dashed curve)obtained from Eq. (5). Thedashed
dot line represents the classical work. This is the same as the inset of Fig. 3. The inset shows Wtot−Wc for bosons (solid
curve)andfermions(dashedcurve)asafunctionofT foran infinitepotentialwell(thickcurves)andforaharmonicpotential
(thin curves),which exhibits thedeviation proportional to±T1/2 for theinfinitepotential well. (Seethetextfor details.) For
theharmonicpotential~ω =10E1 isused,whereω andE1 representtheresonantangularfrequencyoftheharmonicoscillator
and the energy of theground state of theinfinite potential well, respectively, and ~=h/(2π).
A similar approachcan be done for the spinless two-fermion case. f is given as
0
d−1
f = , (S9)
0 4d−2
where d is also defined as (S2). After exactly the same assumptions and similar procedure done above one reaches
f →1/4 (T →∞) (S10)
0
and
f →0 (T →0). (S11)
0
Therefore, we have
W →k T ln2 (T →∞) (S12)
tot B
and
W →0 (T →0). (S13)
tot
II. Wtot AS A FUNCTION OF T
Figure S1 shows W /W is greater than one for bosons and smaller than one for fermions. As the temperature
tot c
increases, W /W approaches one for both bosons and fermions as expected. It is noted, however, that W itself
tot c tot
3
does not converge to W but exhibits rather clear deviations from W depending on the shape of the potential as
c c
shown in the inset of Fig. S1. More precisely W −W diverges as ±T1/2 (+ for bosons, and − for fermions) for an
tot c
infinite potential well, and keeps constant for a harmonic potential.
These deviations can be understood by expanding W perturbatively. From Eqs. (S1) and (S9) one obtains
tot
1±b
f = , (S14)
0 4±2b
where b≡d−1 of Eq. (S2), and + for boson and − for fermions. Note that b goes to zero, i.e. f →1/4, in the high
0
temperature limit. The total work is then expressed as
b 4
W ≈k T ln2± ln . (S15)
tot B
(cid:18) 4 e(cid:19)
It is shown that b→T−1/2 for an infinite potential well implying W −W ∼±T1/2, and b →T−1 for a harmonic
tot c
potential implying W −W ∼ (k/4)ln(4/e). In fact, one finds that b approaches T−1/α if the energy eigenvalues
tot c
are given as E ∼nα. For α→0, W approaches the classical result irrespective of temperature. Since the infinite
n tot
potentialwellisthesteepestpotential,α=2isthemaximumpossiblevalueinpractice. Therefore,atworstW −W
tot c
exhibits ±T1/2 deviation. Such a deviation is regarded as quantum effect of the SZE, and may be directly observed
in experiment. Note that b was ignoredin the previous section, so that we simply had W →k T ln2 as T →∞ in
tot B
Table I.