Quantum thermodynamics
Introduction, papers, conferences, scientific press.
Introduction
Quantum thermodynamics.
A fundamental question is: What remains
of thermodynamics if one goes to the extreme limit of small quantum systems
with a few degrees of freedom? If it does survive, are the many formulations
of the second law (entropy of a closed system cannot decrease, heat goes from
high to low temperatures, the optimal changes are adiabatic ones,..) still equivalent,
or is there a universal formulation?
On this subject our group found many fundamental results, partly discovering
and defining the subject itself.
The first law can still be defined, provided the workscource and the bath are macroscopic.
This step is not trivial, it is a unique decomposition of energy change
in heat and work, [L38], [P51], [C29], [C27].
It has been realized that there is one formulation of the second law
which holds in general: Thomson's formulation (making cycles
costs work) when starting in equilibrium. It applies to systems without
bath, or systems coupled to a bath. The statement about Thomson's formulation
was known; we reproduced it to draw attention to it and eleborated on
it [P58], [P62].
Several formulations of the second law appear to be violated. The Clausius
inequality $dQ \le T dS$
was shown to be invalid at $T=0$. The physical reason is the formation of a cloud
of bath modes around the central particle. Such clouds are well known in the
Kondo problem and for polarons.
This effect has been termed ``the Linus effect: the cloud goes where Linus goes''
after a figues in the Charlie Brown cartoons.
The energy of that cloud must be attributed
to the bath, and some of it can be taken, because changing the parameters of
the particle changes the cloud [L38], [P51].
A test for the violation of the Clausius inequality was proposed for nanoscale
electric circuits [P59] and in quantum optics [L42], [C27].
The Thomson formulation can be violated
when the system is still coupled to a single heat bath, but starts out of equilibrium.
Setups for such cycles were derived analytically [P51].
The rate of energy dissipation can be negative, even when starting in equilibrium
[P51]. Classically this would be forbidden, because there it is, after dividing
by temperature, equal to the rate of entropy production. Positivity of the latter
is another formulation of the second law.
The Landauer bound for information
erasure, sometimes said to be another formulation of the second law,
has founded a whole field in computation science. We showed that it
can be violated in the quantum regime [P52].
Also in quantum optics the manipulation
of the surrounding cloud can lead to surprising effects [L42], such as bath
assisted work extraction and bath assisted cooling.
The maximal amount of work that can be extracted from a system is dictated by
thermodynamics. But for small quantum systems it is an unattainable upper bound.
The proper maximal amount, {\rm ergotropy}, was derived, [L46].
Another formulation of the second law is that optimal changes are adiabatically slow.
We proved that for small quantum systems this remains valid, provided that no level
crossing occurs. In the presence of level crossing, non-adiabatic changes can be more optimal, [L46].
Introduction
A fundamental question is: What remains
of thermodynamics if one goes to the extreme limit of small quantum systems
with a few degrees of freedom? If it does survaive, are the many formulations
of the second law (entropy of a closed system cannot decrease, heat goes from
high to low temperatures, the optimal changes are adiabatic ones,..) still equivalent,
or is there a "universal" formulation? It has been realized that there is
one formulation which holds in general: Thomson's formulation (making cycles
costs work) for a system starting in equilibrium. It applies to systems without
bath, or systems coupled to a bath. The statement about Thomson's formulation
was known before; we reproduced it to draw attention to it and eleborated on
it [P58], [P62]. The first law can still be defined,
provided the workscource and the bath are macroscopic [L38], [P51], [C29], [C27]. The Thomson formulation can be violated
when the system is still coupled to a single heat bath, but starts out of equilibrium.
Setups for such cycles were derived analytically [P51]. The Landauer bound for information
erasure, sometimes said to be another formulation of the second law, can be
violated in the quantum regime [P52]. Also in quantum optics the manipulation
of the surrounding cloud can lead to surprising effects [L42], such as bath
assisted work extraction and bath assisted cooling.
Papers
[P70] A.E. Allahverdyan and Th.M. N., Resolution of the Gibbs paradox via quantum thermodynamics,
Phys. Rev. E, submitted.
quant-ph/0507145
[P62] A.E. Allahverdyan, R. Balian
and Th. M. N., Thomson's formulation of the second law for macroscopic and
finite work sources Entropy 6, (2004) 30-37 Entropy 6, (2004) 30-37 [C29] A.E. Allahverdyan, R. Balian
and Th. M. N., Quantum thermodynamics: thermodynamics at the nanoscale
Proceedings of Physics of Quantum Electronics XXXIV (PQE 2004), Journal of Modern
Optics (2004); cond-mat/0402387
[C27] Th. M. N., Thermodynamics
and small quantum systems, Proceedings of Physics of Quantum Electronics
XXXIII (PQE 2003), Journal of Modern Optics 50, 2433-2442 (2003); cond-mat/0311582
[C30] Th.M. N., Armen E.
Allahverdyan, and Roger Balian, Mesoscopic perpetuum mobile of the second
kind, preprint ITFA-2002-30
[P60] A.E. Allahverdyan,
R. Balian, Th.M. N., Extracting work from a macroscopic thermal bath via
a mesoscopic work source, preprint ITFA-2002-20 [L42] Armen E. Allahverdyan
and Th.M. N., Bath-generated work extraction and inversion-free gain in
two-level systems, J. Phys. A: Math.
Gen. 36 , (2003) 875-882 [P59] A.E. Allahverdyan
and Th.M. N., On testing the violation of the Clausius inequality in nanoscale
electric circuits, Phys.
Rev. B 66 , 115309 (2002) Also in: Virtual Journal of Nanoscale
Science & Technology, September 23, 2002, Volume 6, Issue 13 [P58] A.E. Allahverdyan
and Th.M. N., A mathematical theorem as the basis for the second law: Thomson's
formulation applied to equilibrium, Physica A 305, (2002) 542-552 [P52] A.E. Allahverdyan
and Th.M. N., Breakdown of the Landauer bound for information erasure in
the quantum regime, Phys. Rev. E 64, 056117 (2001) (9 pages) [P51] Th.M. N. and A.E.
Allahverdyan, Statistical thermodynamics of quantum Brownian motion: Construction
of perpetuum mobile of the second kind, Phys. Rev. E 66, 036102 (2002) (52 pages) [L38] A.E. Allahverdyan
and Th.M. N., Extracting work from a single thermal bath in the quantum
regime, Phys. Rev. Lett. 85 (2000) 1799-1802
Conferences covering the subject of quantum thermodynamics Conference Prague, July 2004 I am chairman of the
scientific committee of the conference: Frontiers of Quantum and Mesoscopic
Thermodynamics, July 26-29 2004 in Prague. See the
webpage. Conference in San Diego, 2002 I was coorganizer of:
the First International conference on Quantum limits to the second law,
July 29-31 2002 at the University
of San Diego [C26] Armen E. Allahverdyan,
Roger Balian and Theo .M. N., Thomson's formulation of the second law: an
exact theorem and limits of its validity, in: Quantum Limits to the
Second Law, AIP Conf. Proc. Vol. 643 (2002), pp. 35-40,
cond-mat/0208563
[C25] Theo M. N. and Armen
E. Allahverdyan, Quantum Brownian motion and its conflict with the second
law, in: Quantum Limits to the Second Law, AIP Conf. Proc. Vol.
643 (2002), pp. 29-34,
cond-mat/0208564 [C24] Claudia Pombo, Armen
E. Allahverdyan, and Theo M. N., Bath generated work extraction in two-level
systems, in: Quantum Limits to the Second Law, AIP Conf. Proc.
Vol. 643 (2002), pp. 254-258,
cond-mat/0208565 [C23] Theo M. N. and Armen
E. Allahverdyan, Unmasking Maxwell's Demon, in: Quantum Limits to
the Second Law, AIP Conf. Proc. Vol. 643 (2002), pp. 436-441
Scientific press A number of accounts
have been devoted to our research. About an improved setup for the
classical heat engine: About perpetuum mobile of the second
kind: Over perpetuum mobile van de tweede
soort: About perpetuum mobile: Sul moto perpetuo:
On this subject our group found many fundamental results, partly discovering
and defining the subject itself.
Several formulations of the second law appear to be violated. The Clauisius
inequality dQ less or equal to T dS
was shown to be invalid at T=0. The physical reason is the formation of a cloud
of bath modes around the central particle. Such clouds are well known in the
Kondo problem and for polarons. The energy of that cloud must be attributed
to the bath, and some of it can be taken, because changing the parameters of
the particle changes the cloud [L38], [P51].
A test for the violation of the Clausius inequality was proposed for nanoscale
electric circuits [P59] and in quantum optics [L42], [C27].
The rate of energy dissipation can be negative, even when starting in equilibrium
[P51]. Classically this would be forbidden, because there it is, after dividing
by temperature, equal to the rate of entropy production. Positivity of the latter
is another formulation of the second law.
ITFA-2005-31: A.E. Allahverdyan and Th.M. Nieuwenhuizen,
In 1875 the founding father of statistical physics Josiah Willard Gibbs pointed at the following paradox:
Take two equal volumina of different gases
and mix them. Then the entropy increases by and amount k log 2 per particle.
But if the gases are equal, there is not such an increase. The paradox lies in the discontinuity:
there is an increase no matter how small the difference between the gases, but not when they are equal. This raises questions such as: if the gases are composed of similar balls, red ones for the first gas, blue ones for the second, then what should a color-blind experimentator conclude? In other words: the mixing entropy is not operational.
There has been a long effort to resolve the paradox, which shows a limit of phenemenological thermodynamics.
It was believed to be solved by the quantum mixing entropy argument, but that was shown to create a
new problem at almost the same spot.
Assuming that the translational degrees of freedom of both gases are in thermal equilibrium at the same temperature,
we express the differences between the gases by their internal (spin) structure. The latter involve a
few degrees of freedom. Therefore we approach the problem via quantum thermodynamics, the theory of thermodynamics for
small quantum systems connected to a macroscopic bath and a macroscopic work source.
In this field we notioced before that the notion of entropy is messy, the physical quantity is work.
The maximal amount of work that can generally be extracted from a finite quantum system was already
derived in a paper with R. Balian: the so-called ergotropy.
This allows to consider the maximal amount of work that can be derived before and after mixing.
The difference is the mixing work or mixing ergotropy. Unlike the
mixing entropy, the mixing work is continuous when the gasses become more and more equal.
And the extractable work is an operational concept, it depends on the work extraction process employed.
This solves the Gibbs paradox using quantum mechanics alone.
ITFA-2005-31: A.E. Allahverdyan and Th.M. Nieuwenhuizen,
Resolution of the Gibbs paradox via quantum thermodynamics,
Phys. Rev. E, submitted,
quant-ph/0507145 .
A new work ethic, Nature July 12, 2000;
New frontiers of thermodynamics, The American Institute of Physics July 17, 2000
New frontiers of thermodynamics, The American Institute of Physics July 17,
2000
``Mazen in Gods grondwet: Een duivelse machine'',
De Volkskrant, 22 juli 2000
``Breaking the law: Can quantum mechanics + thermodynamics = perpetual motion
?'',
Science News, October 7, pp 158, 2000
``Moto perpetuo'', Focus N. 100, Febbraio 2001, (Milano), pagina 36