The Second Law of Quantum Thermodynamics

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The Second Law ofQuantum Thermodynamics
Theo M. Nieuwenhuizen
NSA-lezing 19 maart 2003
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Outline
Quantum Thermodynamics Towards the first
law Steam age Birth of the Second Law The First
Law The Second Law First Law Atomic structure
Gibbs H, Holland, H Quantum mechanics Statistical
thermodynamics Josephson junction Entropy versus
ergotropy Maxwells demon
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Towards the first law
Leonardo da Vinci (1452-1519) The French academy
must refuse all proposals for perpetual motion
Prohibitio ante legem
17th-century plan to both grind grain (M) and
lift water. The downhill motion of water was
supposed to drive the device.
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Steam age
1769 James Watt patent on steam engine improving
design of Thomas Newcomen
Industrial revolution England becomes world
power
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Birth of the Second Law
Sidi Carnot (1796-1832) Military engineer in
army Napoleon 1824 Reflexions sur la Puissance
Motrice du Feu, et sur les Machines Propres a
Developer cette Puissance
The superiority of England over France is due to
its skills to use the power of heat
Emile Clapeyron (1799-1864)1834 diagrams for
Carnot process
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The First Law
Heat and work are forms of energy (there is no
caloric, phlogiston)

Julius Robert von Mayer (1814-1878) 1842James
Prescott Joule (1818-1889) 1847Herman von
Helmholtz (1821-1894) 1847 Germain Henri Hess
(1802-1850) 1840

      
Energy is conserved dUdQdW
heat work
added to system
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The Second Law
William Thomson (Lord Kelvin of
Largs) (1824-1907) Absolute temperature scale
Thomson formulation Making a cyclic change
costs work
Rudolf Clausius (1822-1888) 1865 Entropy related
to Heat Clausius inequality dS ? dQ/T
Clausius formulation Heat goes from high to low
temperature
Most common formulation Entropy of a closed
system cannot decrease
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First Law
the harness of nature
Second Law the way it moves
Perpetuum mobile of the first kind is
impossible No work out of nothing Perpetuum
mobile of the second kind is impossible No work
from heat without loss
Thermodynamics according to Clausius Die
Energie der Welt ist konstant die Entropie der
Welt strebt einen Maximum zu.
The energy of the universe is constant The
entropy of the universe approaches a maximum.
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Atomic structure
Ludwig Boltzmann (1844 -1906) Statistical
thermodynamics S k Log W
Boltzmann equation for molecular
collisions Maxwell-Boltzmann weight
Bring vor, was wahr ist Schreib so, daß klar
ist Und verfichts, bis es mit dir gar ist
Onthul, wat waar is Schrijf zo, dat het
zonneklaar is En vecht ervoor, tot je brein gaar
is
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Gibbs
Josiah Willard Gibbs (1839-1903) Papers in
1875,1878
Ensembles micro-canonical, canonical,
macro-canonical Gibbs free energy
FU-TS Gibbs-Duhem relation for chemical mixtures
Canonical equilibrium state described
by partition sum Z ?_n Exp ( - E_n / k T)
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H, Holland, H
Johannes Diederik van der Waals (1837-1923) 1873
Equation of state for gases and
mixtures Attraction between molecules (van der
Waals force) Theory of interfaces Nobel laureate
1910
Jacobus Henricus van t Hoff (1852-1911) Osmotic
pressure Nobel laureate chemistry 1901
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Quantum mechanics
explains solid state, (bio-)chemistry high
energy physics, early universe
Observables are operators in Hilbert space New
parameter Plancks constant h
Max Born (1882-1970) 1926
Quantum mechanics is a statistical theory John
von Neumann (1903-1957) 1932 Wave function
collapses in measurement
Interpretations - Copenhagen wave function
most complete description of the system
- mind-body problem mind needed for
measurement - multi-universe
picture no collapse, system goes into new
universe
Einstein Statistical interpretation
Quantum state describes ensemble of systems
Armen Allahverdyan, Roger Balian, Th.M. N. 2001
2003 Exactly solvable models for quantum
measurements. Ensemble of measurements on an
ensemble of systems.
Classical measurement specifies quantum
measurement. Collapse is fast occurs through
interaction with apparatus. All possible outcomes
with Born probabilities.
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Quantum thermodynamics
Quantum partition sum Z Trace Exp( - H / k T
) as classically
Hidden assumption weak coupling with bath.
Armen Allahverdyan Th.M. N. 2000 Quantum
particle coupled to bath of oscillators. Classical
ly standard thermodynamics example
Quantum mechanically (low T) Coupling non-weak
friction build up of a cloud.
Clausius inequality violated. Several other
formulations violated, but not all
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Josephson junction
Two Super-Conducting regions with Normal region
in between SNS-junction
Step edge Josephson junction
SC1 N SC2
Electric circuit with Josephson
junction Non-weak coupling to bath if resistance
is non-small.
One ingredient for violation of Clausius
inequality measured in 1992
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Photo-Carnot engine
                                                
           
Scully group
Volume is set by photon pressure on piston Atom
beam phaseonium interacts with
photons Efficiency exceeds Carnot value, due to
correlations of atoms
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Figure 1. (A) Photo-Carnot engine in which
radiation pressure from a thermally excited
single-mode field drives a piston. Atoms flow
through the engine and keep the field at a
constant temperature Trad for the isothermal 1 
   2 portion of the Carnot cycle (Fig. 2). Upon
exiting the engine, the bath atoms are cooler
than when they entered and are reheated by
interactions with the hohlraum at Th and "stored"
in preparation for the next cycle. The
combination of reheating and storing is depicted
in (A) as the heat reservoir. A cold reservoir at
Tc provides the entropy sink. (B) Two-level atoms
in a regular thermal distribution, determined by
temperature Th, heat the driving radiation to
Trad  Th such that the regular operating
efficiency is given by  . (C) When the field is
heated, however, by a phaseonium in which the
ground state doublet has a small amount of
coherence and the populations of levels a, b, and
c, are thermally distributed, the field
temperature is Trad gt Th, and the indicated in
Eq. 4. operating efficiency is given by    ,
where   can be read off from Eq. 7. (D) A free
electron propagates coherently from holes b and c
with amplitudes B and C to point a on screen. The
probability of the electron landing at point a
shows the characteristic pattern of interference
between (partially) coherent waves. (E) A bound
atomic electron is excited by the radiation field
from a coherent superposition of levels b and c
with amplitudes B and C to level a. The
probability of exciting the electron to level a
displays the same kind of interference behavior
as in the case of free electrons i.e., as we
change the relative phase between levels b and c,
by, for example, changing the phase of the
microwave field which prepares the coherence, the
probability of exciting the atom varies
sinusoidally, as
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Entropy versus ergotropy
In-transformation
work-transformation
Maximum thermodynamic work optimize among all
states with same entropy But best state need not
be reachable dynamically.
Ergotropy Maximum work for states reachable
quantum mechanically. Relevant
for mesoscopic systems
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Maxwells demon
James Clerk Maxwell (1831-1879)
Theory of electro-magnetismMaxwell-distribution
1867 A tiny fingered being selects fast and
slow atoms by moving a switch.
No work solely from heat Maxwell demons should
be exorcized!
Quantum entanglementacts as a Maxwell demonin
certain circumstances
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Summary
Thermodynamics is old, strong theory of nature
Quantum thermodynamics takes into account
quantum nature
precise
coupling to bath
New borders arise from experiments
model systems
exact theorems
Applications in other fields of science