Quantum Approach to a Derivation of the Second Law of Thermodynamics

4 Pages·0.12 MB·English

by  Jochen Gemmer

Quantum Approach to a Derivation of the Second Law of Thermodynamics
Jochen Gemmer, Alexander Otte and Gu¨nter Mahler
Institut fu¨r Theoretische Physik,
Universit¨at Stuttgart, Pfaffenwaldring 57,
70550 Stuttgart, Germany
andpast–shouldberesponsibleforthe entropyincrease
We re-interprete the microcanonical conditions in the
according to the second law. This assertion has never
quantumdomainasconstraintsfortheinteractionofthe“gas-
been proved, though.
subsystem” under consideration and its environment (“con-
E. Schr¨odinger [5] argued that the conventional canoni-
tainer”). The time-average of a purity-measure is found to
caldistributionsofthermodynamicscanbeobtainedalso
equal the average over the respective path in Hilbert-space.
1 for a system described by a single wavefunction, if one
0 Wethenshowthatfortypical(degenerateornon-degenerate) invokes “the old crux of molecular disorder”.
0 thermodynamicalsystemsalmostallstateswithintheallowed
J. von Neumann [3] was able to show that a non-
2 region of Hilbert-space have a local von Neumann-entropyS
degenerate system should, indeed, obey a quantum-
closetothemaximumandapurityP closetoitsminimum,re-
n mechanical version of ergodicity under fairly weak con-
spectively. Typicallythermodynamicalsystemsshouldthere-
a
ditions. In addition he felt obliged to introduce “macro-
J fore obey thesecond law.
scopic observers”(coarsegraining)in order to come into
0
contact with standard thermodynamics.
3
This latter aspect has then been taken up also by W.

1 Pauli and M. Fierz [6]. These authors insisted that the
v The second law has been formulated in a number of 2nd law should be explained without reference to exter-
0 different ways [1]: According to Clausius’ postulate heat nal perturbations like those induced by quantum mea-
4 cannot flow spontaneously from a colder to a hotter sys- surements.
1 tem. Thomson’s formulation reads: “It is impossible to G. Lindblad [7] observed that the entropy of a multi-
1
constructaperpetuummobile ofthe secondkind.” Such partite system defined as the sum of the respective par-
0
rules define, essentially, some sort of irreversibility, i.e. tial entropies of the subsystems would tend to increase
1
0 the existence of a state function usually called entropy, due to the neglect of correlations (entanglement). This
/ S, which can only increase in closed systems. fact could be taken as a quantum mechanical justifica-
h
The secondlaw is arguablyone ofthe mostfundamental tion of Boltzmann’s Stosszahlansatz.
p
and far-reaching laws of physics; nevertheless its origin W.Zurek[8]andhiscoworkeshavediscussedingreatde-
-
t remains puzzling. tailhowtheinteractionofaquantumsystemwithitsen-
n
a On the macroscopic level the second law enables us to vironmentmayinducequasi-classicalbehaviorinthefor-
u calculate state equations from the requirement that en- mer(“environment-inducedsuperselectionrules”). They
q tropy has to be maximized under the condition of given arguedthatthe secondlawshouldeventuallyresultfrom
v: extensive variables in order to get a thermodynamical the impossibility of isolating macroscopic systems from
i potential as a function of those. their environment [9].
X
On the microscopic level irreversibility comes in conflict Many of these ideas are pertinent to our present investi-
r with the notorious reversibility of all fundamental phys- gation. ItisthepurposeofthisLettertoshowthatunder
a
ical laws. It has been the challenge for statistical me- (appropriately redefined) microcanonical conditions the
chanics to reconcile dS/dt ≥ 0 with the underlying mi- second law of thermodynamics follows from a quantum
croscopic dynamics. mechanical analysis of the total system partitioned into
Irreversibilityin classicalmechanics is conventionallyin- the object and the environment.
troduced via two different schemes (cf. [3]): The Boltz- A purity measure that is formulated within standard
mann approach, based on the hypothesis of molecu- quantummechanicsbuthasalsoaninterpretationwithin
lar chaos (Stosszahlansatz) and the Gibbs ensemble ap- thermodynamics is the von Neumann-entropy
proach,basedonthehypothesisofquasi-ergodicity. Both
these attempts have to acknowledge some additional as- S(ρˆ):=−kTr{ρˆlnρˆ}. (0.1)
sumptions which do not follow from the underlying mi-
croscopic laws. If this entropy is zero, the system is in a pure state, if it
This fact has led to continuing controversies as to takesonits maximumvalue Smax =kln(N), where N is
whether this situation can be the last word. Several the systemsize (number ofaccessiblestates), the system
researchers suggested quantum mechanics as a possible is in the maximally mixed state. Another measureis the
remedy. “purity” P:
L. D. Landauand E.M. Lifshitz [4] expected that quan-
tum measurements – making a difference between future P(ρˆ):=Tr ρˆ2 . (0.2)
(cid:8) (cid:9)
1
IfP takesonitsmaximumvalue1thesystemisinapure by some “box” potential Vˆ, neglecting Wˆ′ altogether.
state, if P takes on its minimum value 1 the system is But however small, starting from first principles Wˆ′ will
N
in the maximally mixed state. always be present, and represents a coupling.
For those extreme cases the two measures uniquely map We consider the gas-system to be closed in the thermo-
ontoeachother. Forgeneralcases,however,thisdoesnot dynamicalsense, i. e., controlledby microcanonicalcon-
hold true anymore. Nevertheless, states with P(ρˆ) ≈ 1 ditions (E,V,N = const.); this is clearly an idealization
willhaveS(ρˆ)≈0andstateswithP(ρˆ)≈P willhave but can routinely be realized in an approximate way.
min
S(ρˆ)≈S . Inthe followingwemainly considerP and A system that is closed on the macroscopic level (ther-
max
get back to S in the end. modynamicallyclosed)doesnotneedtobe closedonthe
Using the von Neumann-equation for the density opera- microscopic (quantum-) level (i. e. not interacting with
tor of the total system it can easily be shown that the any other system).
entropy and the purity of a closed system that does not Thefactthatnoextensivequantitiesaretobeexchanged,
interact with any other system are conserved. however, puts constraints on its Hamiltonian, especially
This fact mightbe consideredacontradictionto the sec- on the interactions a system can have with its surround-
ond law which demands that entropy should be maxi- ing, in order for it to be thermodynamically closed.
mized during the evolution of a closed system. The strict conservation of particles N is taken into ac-
If the full system is being regarded as divided into two count simply by the way the system is partitioned. To
subsystems(IandII),thereduceddensityoperatorsare: which accuracy the volume V stays fixed is set by Lˆ .
c
Microcanonicalconditionsthencorrespondtoaboxwith
ρˆI =TrII{ρˆ}, ρˆII =TrI{ρˆ} (0.3) veryhigh,inthelimitofthevolumeV beingexactlycon-
served, infinitely high potential walls.
Entropy and purity may be defined for each subsystem
The condition that no energy is to be exchangedfurther
as before, using ρˆ (ρˆ ) instead of ρˆ.
I II constrainstheHamiltonian. Theenergycontainedinthe
Since those are not the density operators that appear
gas is given by:
in the von Neumann-equation, S (S ) and P (P ) de-
I II I II
finedonbasisoftheseoperatorsarenolongerconserved. E :=hLˆ′i. (0.7)
g g
Those are the quantities we are going to examine.
Althoughthe followingideasapply toallsortsofsubsys- If this is to be conserved, it follows that
tems,wewanttorefertothesystemofwhichtheentropy
istobecalculatedasthe“gas-system”,g,andallthesur- Lˆ′,Hˆ =0 Lˆ′,Wˆ′ =0. (0.8)
g g
rounding as the “container-system”,c.
h i h i
The full Hamiltonian is now divided according to the
Except for these constraints we need not specify Wˆ′ in
same scheme:
more detail.
Hˆ =:Lˆ +Lˆ +Wˆ (0.4) Basedonthesecommutatorrelationswefindthatforany
g c
energy eigenspace A,B
Lˆ describes the energies arising from the gas parti-
clges alone, including their mutual interactions. Lˆ de- |ψiAjB(t)|2 = |ψiAjB(0)|2 (0.9)
c
i,j i,j
scribes the corresponding energies of the container par- X X
ticles alone. Wˆ describes the interaction terms that de-
is a conserved quantity, set by the initial state, where
pend on both, the coordinates of gas particles and the ψAB denotes the amplitudes of the degenerate product
ij
container particles. These are here basically the “wall”
energy eigenstates of Lˆ′ + Lˆ (“i” denoting the gas,
interactions that keep the gas particles inside the con- g c
“j” the container part of the product) that are associ-
tainer.
ated with the energy eigenvalues Eg (Ec) in the gas-
Theenergyeigenstatesofafreegasareunboundandcon- A B
(container-) system.
tinuous. It is far more convenient to have bound states
Since wewantto considercasesherethat havezerolocal
for the “separate” systems. Thus we modify the Hamil-
entropy in the beginning (product states), we get
tonian formally in the following way:
Hˆ =:Lˆ′g+Lˆc+Wˆ′ (0.5) |ψiAjB(0)|2 = |ψiA(0)|2|ψjB(0)|2 =PAgPBc (0.10)
i,j i,j
X X
with
where Pg(Pc) are the probabilities of finding the gas-
A B
Lˆ′g :=Lˆg+Vˆ and Wˆ′ :=Wˆ −Vˆ (0.6) (container-) sytem somewhere in the possibly highly de-
generate subspace characterized by the energy eigenval-
where Vˆ models the mean effect of the container on the ues Eg (Ec). If no energy is to be exchanged, clearly
A B
gas particles. Vˆ is an effective potential that only de- these probabilities haveto remainconserved. This is the
pendsonthecoordinatesofthegasparticlesandischosen constraintthat microcanonicalconditions impose on the
tominimizeWˆ′. Usuallythecontainerissimplymodelled accessible region of Hilbert-space.
2
Although we are interested in P(t → ∞), we start by where Ng is the degree of degeneracy of Eg.
A A
considering the time average of the purity P for reasons To calculate the Hilbert-space average of P denoted as
that will become clear later. < P > we need a parametrization for ψ ,ψ′ confined
ij ij
to the allowed region (0.9) that essentially consists of
1 T
P := P(|ψ(t)i)dt (0.11) hyperspheres in different parts of the Hilbert-space of
T
Z0 the total system. The Hilbert-space averagecan then be
Choosing a special parametrization for |ψi, we can con- written as
vertthetime integralintoanintegraloverthe trajectory P ψ ({φ }),ψ′ ({φ }) detF dφ
generated by the total system’s dynamics for given ini- <P >= ij n ij n n n
detF dφ
tial conditions. Parametrizing |ψi in terms of the real R (cid:0)(cid:8) n (cid:9)n(cid:1) Q
(0.17)
and imaginary parts of its amplitudes R Q
|ψ(t)i:= ψ (t),ψ′ (t) (0.12) whereφn istherespectivesetofparametersandF isthe
ij ij
corresponding functional matrix.
we can write instead of (0(cid:8).11) (cid:9) This integral can actually be solved analytically. The
techniques are essentially the same as those used to cal-
P = ||ψψ((0T)i)iP ψij,ψi′j ve1ff|d|ψi| (0.13) culate surface areas of hyperspheres in the classical sta-
R |ψ(cid:0)(cid:8)(T)i 1 |(cid:9)d(cid:1)|ψi| tistical analysis of the ideal gas. Since this calculationis
|ψ(0)i veff rather elaborate we do not want to present it in detail
where|d|ψi|denoteRsthe“length”ofaninfinitesimalstep here, but give and discuss the result:
along the trajectory in Hilbert-space.
<P >= (0.18)
The advantage of this special parametrization derives
from the fact that the effective velocity (PAg)2 1− (Pc)2 + (PBc)2 1− (Pg)2
Ng B Nc A
v2 = ψ˙ 2+ψ˙′ 2 = 1 hψ(0)|Hˆ2|ψ(0)i (0.14) XA A XB ! XB B XA !
eff ij ij ~2 (Pg)2(Pc)2(Ng +Nc)
Xi,j (cid:16) (cid:17) + A NBgNc +A1 B
is constant on each trajectory and thus independent of AX,B A B
the time t or the special point on the trajectory. Hence, HereNg is the degreeofdegeneracyofthe energyeigen-
the integral (0.13) simplifies to value EBg.
B
1 |ψ(T)i If the degeneracy of the occupied energy levels is large
P = L P ψij,ψi′j |d|ψi| (0.15) enough so that
Z|ψ(0)i
(cid:0)(cid:8) (cid:9)(cid:1) 1 1
where L is the length of the path. So, the time average NgNc +1 ≈ NcNc (0.19)
of P equals the path average along a special trajectory A B A B
in this parametrizationof Hilbert-space. whichshould holdtrue for typicalthermodynamicalsys-
We are not able to compute this integral for we do not tems, (0.18) reduces to
know Wˆ′ in detail, and even if we did, we could never
hopetosolvetheSchr¨odinger-equationforasystemwith <P >≈ (PAg)2 + (PBc)2 (0.20)
about 1023 degrees of freedom. Ng Nc
A A B B
All we wantto provehere,is that for typicaltrajectories X X
staying within the region allowed by the microcanonical ThefirstsuminthisexpressionisobviouslyexactlyPmin
conditions,P isextremelyclosetoitsminimumvaluefor (0.16),sothatforsystemsandinitialconditionsinwhich
almost all points within this region. the second sum is small the allowed region almost only
We proceed as follows: consists of states for which P ≈ Pmin. The second sum
First we calculate P which is the smallest possible will be small if the container system occupies highly de-
min
value of P within the allowed region. Then we compute generate states, typical for thermodynamical systems.
the average of P over the total allowed region. If this To illustrate this result, we have plotted the relative fre-
averageis closeto P , wecanconclude thatP ≈P quency of P (see Fig 1.), which we calculated using a
min min
for almost all points within this region, which means for formulaby Lloyd,Pagels[10]andPage[11],that applies
almostallP(t), sinceanydistributionwithameanvalue to completely degenerate subsystems only. In this case
close to a boundary has to be sharply peaked. we find from (0.18)
From(0.8)itfollowsthatthePg remainconserved. Now
A Ng+Nc
thelowestpurityP ofanystateconsistentwiththiscon- <P >= (0.21)
NgNc+1
diton is:
(Pg)2 (Forthisspecialcasetheaveragehasalsobeencalculated
Pmin = NAg (0.16) by Lubkin [12].)
A A
X
3
P
P where K is a positive function that scales linearly with
the system size of the gas system. (0.24) is valid for
situations with
2x8 (Pg)2 (Pc)2
A ≫ B , (0.25)
2x4 Ng Nc
2x2 A A B B
X X
which is the thermodynamical regime in which the sec-
ond term in (0.24) will be small. (Again, for the special
0 1 1P caseofbothsubsystemsbeingcompletelydegenerate,our
n1 results are in perfect agreement with a result by S. Sen
[13])
FIG. 1. Probability density as a function of P = Tr{ρˆ2I} InconclusionwehaveshownthatthelocalvonNeumann-
forvariousn1xn2 systems. Withincreasingsizeofsubsystem entropy of a considered system will be maximized dur-
II thedensity becomes peaked near thevalue 1 .
n1 ing its evolution, even if the system is thermodynami-
cally closed, provided the energy eigenspaces occupied
Obviously our average < P > is in perfect agreement
by the surrounding are much bigger than the energy
with the more detailed distribution of P.
eigenspaces occupied by the considered system. This is
So far we have only shown that, except for negligible
typical for thermodynamical systems. We did not need
parts, the Hilbert-space-section in which the trajecto-
the additional assumptions underlying classical deriva-
ries can be found under microcanonical conditions, has
tions. Since we considered microcanonical conditions we
P ≈ P , provided the surrounding system is much
min get the maximum entropy as an explicit function of the
larger than the considered system.
initial energy distribution. This allows for a connection
To examine under what conditions trajectories will even
withstandardthermodynamics,entropybeingathermo-
“fill” the whole space they can possibly live in, we con-
dynamical potential.
sider the special case of no degeneracy in either subsys-
Generalizations to canonical conditions and the treat-
tem. Here the evolution of P can be calculated exactly:
mentof quantumcomputer systems asspecific opensys-
P(t)= e(i1~(EAB−ECB+ECD−EAD)t)PAgPBcPCgPDc tWemesthaarenkunDdre.r wIa.y.Kim, M. Stollsteimer, Dipl. Phys.
A,B,C,D
X F.TonnerandT.Wahlforfruitfuldiscussions. Oneofus
(0.22)
(A.O.) acknowledeges financial support by the Deutsche
where E are the energy eigenvalues of the respective Forschungsgemeinschaft.
IJ
energy eigenstates. Note that without any interaction
E =E +E and P(t)=const., as expected.
IJ I J
Assuming that all terms in (0.22) are oscillating except
the ones with A=C or B =D, we get
P = (Pg)2+ (Pc)2− (Pg)2 (Pc)2
A B A B
A B A ! B ! [1] Ya. P. Terlitskii, Statistical Physics (North Holland
X X X X
(0.23) Publ., Amsterdam, 1991)
[2] F. Schl¨ogl, Probability and Heat (Vieweg, Braunschweig,
which is exactly the same result as (0.18) with Ng =
A 1998)
NBc = 1. So, under this assumption the Hilbert-space [3] J. v.Neumann,Zeitschr. f. Physik 57, 30 (1930).
average is exactly equal to the time average.
[4] L.Landau,M.Lifshitz,Statistical Mechanics(Pergamon
Ingeneral,however,thisspecificergodicityisnotneeded:
Press, Oxford, 1978); Quantum Mechanics (Pergamon
Itsufficesthattypicalquantumtrajectories,eventhough
Press, Oxford, 1977)
startingwithP(0)=1,ventureoutintothevastHilbert- [5] E. Schr¨odinger, Statistical Thermodynamics (Dover
spaceregionscharacterizedby P =Pmin. Ofcourse,one Publ., NewYork, 1989).
cannot exclude that in special situations there might be [6] W. Pauli, M. Fierz, Zeitschr. f. Physik 106, 572 (1937).
trajectories that never leave the very tiny region with [7] G.Lindblad,Non-equilibriumEntropy andIrreversibility
P ≈ 1, but these situations become extremely rare as (D. Reidel Publ. Comp., Dordrecht 1983)
the surrounding gets big. [8] W. Zurek, Phys. Today 44, No.10, 36 (1991); Phys. To-
Finally,wereturntothelocalentropyS. Tryingtocom- day 46, No.12, 81 (1993).
pute<S >ratherthan<P >weget,aftersomelengthy [9] W. Zurek, J. Paz, Phys. RevLett. 72, 2508 (1994).
but straightforwardperturbative calculations [10] S. Lloyd, H.Pagels, Ann.Phys.(N.Y.) 188, 186 (1988).
[11] Don N.Page, Phys. Rev.Lett. 71, 1291 (1993).
(Pc)2 [12] E. Lubkin,Math. Phys. 19, 1028 (1978).
<S >≈Smax({PAg,NAg})−K NBc (0.24) [13] S. Sen,Phys. Rev.Lett. 77, 1 (1996).
B B !
X
4