The Second Law of
Quantum Thermodynamics
Theo M. Nieuwenhuizen
NSA-lezing 19 maart 2003
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
Maxwell’s demon
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.
Steam age
1769 James Watt: patent on steam engine
improving design of Thomas Newcomen
Industrial revolution:
England becomes world power
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
The First Law
Julius Robert von Mayer (1814-1878)
James Prescott Joule (1818-1889)
Herman von Helmholtz (1821-1894)
Germain Henri Hess
(1802-1850)
1842
1847
1847
1840
Heat and work
are forms of energy
(there is no “caloric”, “phlogiston”)
Energy is conserved: dU=dQ+dW
heat + work
added to system
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
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.
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 verficht’s, bis es mit dir gar ist
Onthul, wat waar is
Schrijf zo, dat het zonneklaar is
En vecht ervoor, tot je brein gaar is
Gibbs
Josiah Willard Gibbs (1839-1903)
Papers in 1875,1878
Ensembles: micro-canonical, canonical, macro-canonical
Gibbs free energy F=U-TS
Gibbs-Duhem relation for chemical mixtures
Canonical equilibrium state described by
partition sum Z = _n Exp ( - E_n / k T)
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
Quantum mechanics
explains: solid state, (bio-)chemistry
high energy physics, early universe
Observables are operators in Hilbert space
New parameter: Planck’s 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.
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.
Classically: 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
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
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
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
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
Maxwell’s demon
James Clerk Maxwell (1831-1879)
Theory of electro-magnetism
1867: A “tiny fingered being”
Maxwell-distribution
selects fast and slow atoms
by moving a switch.
No work solely from heat:
Maxwell demons should
be exorcized!
Quantum entanglement
acts as a Maxwell demon
in certain circumstances
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