Transcript Thermodynamics at the Nanoscale (version Prague, July 2004)

Quantum thermodynamics:
Thermodynamics at the nanoscale
Armen E. Allahverdyan (Amsterdam/Yerevan)
Roger Balian (CEA-Saclay; Academie des Sciences)
Theo M. Nieuwenhuizen (University of Amsterdam)
Session in memory of Vlada Capek
Frontiers of Quantum and Mesoscopic Thermodynamics
Prague, 26 July 2004
Outline
Introduction to quantum thermodynamics
(Amsterdam-Paris-Yerevan view).
Position of works of Vlada Capek within quantum thermodynamics.
First law of thermodynamics: what is work, heat, system energy.
Second law: confirmation versus violations.
Maximal extractable work from a quantum system.
Are adiabatic changes always optimal?
Introduction to quantum thermodynamics
Standard thermodynamics: large system + large bath + large work source
Classical thermodynamics: of bath only temperature T needed
(and timescale for heat exchange)
But consider for example:
Mesoscopic ring: metal ring with size between micron and nanometer
1/10 000 cm 1/10 000 000 cm
0.1 hair
0.000 1 hair
Mesoscopic ring still has many atoms: many degrees of freedom
Study the electric current of such a ring at low temperature:
one interesting degree of freedom coupled to many uninteresting ones
Quantum thermodynamics: small system, large bath, large worksource
whole spectral density of coupling to bath needed
System-bath models: small quantum systems + large bath
see book Uli Weiss 1993; 1998
Caldeira-Leggett model 1983 (van Kampen’s thesis 1951;
Ullersma’s thesis 1965)
H tot
p2 b 2

 x  x ci xi
2m 2
i
particle

pi2 mi 2 2
i ( 2m  2  i xi )
i
 interactio n  bath of many harmonic oscillator s
Spin-boson model: spin ½ + harmonic oscillator bath
Leggett model + 10 coauthors: review 1983
Spin ½ : spin up or spin down = two level system
Capek models: coupled 2,3,4,5 two-level systems + their baths
rich class of models
rich amount of physical phenomena
Excursion to hill of Celts, April 2001
Is there a thermodynamic description?
First law: Change in energy = work added + heat added
dU  dW  dQ
U  H 
dW
dQ
where H is that part of the total Hamiltonian,
that governs the unitary part of (Langevin) dynamics
Work: Energy-without-entropy added to the system
1) Caratheodory: increase average energy of work source
2) Gibbs-Planck: energy of macroscopic degree of freedom
The rest: energy-without-work from the bath
Energy related to uncontrollable degrees of freedom
Internal energy in Caldeira-Leggett model
H tot
p2 b 2

 x  x ci xi 
2m 2
i
pi2 mi 2 2
i ( 2m  2  i xi )
i
particle  interactio n  harmonic oscillator bath
(phonons or photons)
Ohm’s law for resistor: V = I R.
quasi-Ohmic
spectral density
 ci2
2
J ( )  i
 ( i   )   2
2mi i
 2
Taking together effects of bath yields: Langevin equation for particle
m x  (_______
b   ) x   x   (t ),
  (t ) (t ' )  K (t  t ' )
p2 a 2
Newton force defines system Hamiltonian: H 
 x ,
___________
2m 2
Internal energy: U=<H>
a  b  
phonons: b renormalized to a
photons: a is the physical parameter
“All” about work
Work = change of averge energy of system + bath
= minus (change of energy of work source)
= time-integral of rate of change of energy of system alone
What is special about macroscopic work source?
It produces time-dependent parabeters e.g. m(t), b(t), V(t)
so it does not enlarge dimension of Hilbert space.
Why does the average energy enter this definition?
Thermodynamics does not apply to single systems
Quantum mechanics does not apply to single systems
The second law of thermodynamics
Heat goes from high temperatures to low temperatures
No cycles of work from bath (no perpetuum mobile): Thomson formulOptimal changes are adiabatically slow
ation
Entropy of closed system cannot decrease
Rate of entropy production is non-negative
Finite quantum systems: Thermodynamics endangered
No thermodynamic limit : Different formulations become inequivalent
Some may apply, others not
But: Generalized Thomson formulation is valid:
Cyclic changes on system in Gibbs equilibrium cannot yield work
(Pusz+Woronowicz ’78, Lenard’78, A+N ’02.)
The Linus effect:
The cloud goes where Linus goes
The appearence of clouds (“the Linus effect”)
In small quantum systems at not very high temperatures
a cloud of bath modes surrounds the central particle
Kondo cloud, polaron cloud
Such clouds must be attributed to bath
Not part of standard thermodynamics: new effects in quantum thermo
A+N: 2000, 2002 Clausius inequality
Caldeira-Leggett at T = 0:
dQ  TdS may be violated
 dm
dQ 
 0 if dm  0
2
2 m
Negative rate of energy dispersion, though starting from equilibrium
Out of equilibrium: work extraction cycles constructed (Finite yield)
A+N, PRB 02, experiments proposed for mesoscopic circuits
J. Phys A 02 expts for quantum optics.
Capek: electric currents, heat currents going in “wrong” direction
Work extraction from finite quantum systems
Couple to work source and do all possible work extractions
Thermodynamics: minimize final energy at fixed entropy
Assume final state is gibbsian: fix final T from S = const.
 (t )  U (t )  (0)U (t )
But: Quantum mechanics is unitary,
So all n eigenvalues conserved: n-1 constraints:
(Gibbs state typically unattainable for n>2)
Optimal: eigenvectors of
become those of H,
  / Z
if ordering 1   2  ...   d , 1   2  ...   d , as in   e

Maximally extractable work:
ergotropy
n
W  U (0)    i i
i 1
  work
  turn, transformation
entropy   -   in - transformation (Clausius)
ergotropy   -   work - transformation
ABN, EPL 2004: Properties of ergotropy
• Majorization: defines set of states within which
thermodynamic relations are satisfied qualitatively.
• Other states: all kinds of thermodynamic surprises
Are adiabatic processes always optimal?
Minimal work principle (one of the formulations of the second law):
Slow thermally isolated processes (“adiabatic processes”) done on an
equilibrium system are optimal (cost least work or yield most work)
In finite Q-systems: Work larger or equal to free energy difference
But adiabatic work is not free energy difference.
A+N, 2003:
-No level crossing : minimal work principle holds
-Level crossing: solve using adiabatic perturbation theory.
Diabatic processes are less costly than adiabatic.
Work = new tool to test level crossing.
Level crossing possible if two or more parameters are changed.
Review expts on level crossing: Yarkony, Rev Mod Phys 1996
Summary
Q-thermodynamics: small system, macroscopicwork source+bath
Different formulations of the second law
have different ranges of validity
Experimental tests feasible e.g. in quantum optics
New results for thermodynamics of small Quantum-systems:
-violation of Clausius inequality
-optimal extractable work: ergotropy
-adiabatic changes non-optimal if level crossing
Vada Capek was strong forefighter of Quantum Thermodynamics
Summary
Vlada Capek was strong forefighter of Quantum Thermodynamics
Closing session
• Thanks to all those who contributed
and why
1) Why of those
2) All of those
In loving memory
Vlada Capek
was a strong forefighter of Quantum Thermodynamics
Capek models
Capek’s and our common issue in science
The Linus effect:
The cloud goes where Linus goes
damping
relaxation
entanglement
purity
quantum thermodynamics = classical thermodynamics + Linus
book with Daniel Sheehan
Thanks to all participants
• You all came here in good mood
Contributed to the extremely high level of the
meeting
• Even though we could provide no funding
• Even though we will ask you to contribute to the
proceedings = equally fine as the meeting
• Thanks, thanks and (thanks)^2
• Special thanks to Toni Leggett
Thanks to our sponsors
• Czech Senate, Wallenstein palace
Czech Academy of Sciences
Charles University
Masarykova kolej
• Local hotels, printing office, restaurant
• Czech press
Thanks to the scientific organizers etc
• Roger Balian
Marlan Scully
Daniel Sheehan
Milena Grifoni
Vladimir Zakharov + Alexei Nikulov
Vaclav Spicka
Theo Nieuwenhuizen + Armen Allahverydan
• Our international organizer: Peter Keefe
• All chairwomen and chairmen (chairhumans)
Thanks to our many local organizers
• Jiri Bok
Petr Chovsta
Michal Fanta
Sona Fialova
Pavel Hubik
Zdenek Kozisek
•Karla Kuldova
Jan Krajnik
Jiri Mares
Evzen Subrt
David Vyskocil
Karolina Vyskocilova
Thanks, thanks, thanks, thanks, thanks, thanks, thanks, thanks, than
There is
one special person to thank
• Our friend and main organizer
• Vaclav Vaclav Vaclav Vaclav
Vaclav Vaclav Vaclav Vaclav
Vaclav Vaclav Vaclav Vaclav
Vaclav Vaclav Vaclav Vaclav
Vaclav Vaclav Vaclav Vaclav
Vaclav Vaclav Vaclav Vaclav