The Second Law ofQuantum ThermodynamicsTheo M. Nieuwenhuizen NSA-lezing 19 maart 2003
OutlineQuantum ThermodynamicsTowards the first lawSteam age Birth of the Second LawThe First LawThe Second LawFirst LawAtomic structure GibbsH, Holland, HQuantum mechanicsStatistical thermodynamicsJosephson junctionEntropy versus ergotropyMaxwells demon
Towards the first lawLeonardo da Vinci (1452-1519):The French academy must refuse all proposals for perpetual motion17th-century plan to both grind grain (M) and lift water. The downhill motion of water was supposed to drive the device.Prohibitio ante legem
Steam age 1769 James Watt: patent on steam engineimproving design of Thomas NewcomenIndustrial revolution: England becomes world power
Birth of the Second LawSidi 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: Emile Clapeyron (1799-1864) 1834: diagrams for Carnot processThe superiority of England over France is due to its skills to use the power of heat
The First LawHeat and work are forms of energy(there is no caloric, phlogiston)
Julius Robert von Mayer (1814-1878) 1842 James Prescott Joule (1818-1889) 1847 Herman von Helmholtz (1821-1894) 1847Germain Henri Hess (1802-1850) 1840Energy is conserved: dU=dQ+dW heat + work added to system
The Second LawRudolf Clausius (1822-1888)1865: Entropy related to Heat: Clausius inequality: dS dQ/T William Thomson (Lord Kelvin of Largs)(1824-1907) Absolute temperature scale Thomson formulation: Making a cyclic change costs workMost common formulation:Entropy of a closed system cannot decreaseClausius formulation: Heat goes from high to low temperature
First LawSecond Law = the way it movesThermodynamics according to Clausius:
Die Energie der Welt ist konstant; die Entropie der Welt strebt einen Maximum zu.Perpetuum mobile of the first kind is impossible: No work out of nothingPerpetuum mobile of the second kind is impossible: No work from heat without loss= the harness of natureThe 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 WBring vor, was wahr ist;Schreib so, da klar istUnd verfichts, bis es mit dir gar istOnthul, wat waar isSchrijf zo, dat het zonneklaar isEn vecht ervoor, tot je brein gaar isBoltzmann equation for molecular collisionsMaxwell-Boltzmann weight
GibbsJosiah Willard Gibbs (1839-1903)
Papers in 1875,1878 Ensembles: micro-canonical, canonical, macro-canonicalGibbs free energy F=U-TSGibbs-Duhem relation for chemical mixturesCanonical equilibrium state described bypartition sum Z = _n Exp ( - E_n / k T)
H, Holland, HJohannes Diederik van der Waals (1837-1923)1873: Equation of state for gases and mixturesAttraction between molecules (van der Waals force)Theory of interfaces
Nobel laureate 1910Jacobus Henricus van t Hoff (1852-1911)Osmotic pressure Nobel laureate chemistry 1901
Quantum mechanicsObservables are operators in Hilbert spaceNew parameter: Plancks constant hEinstein: Statistical interpretation: Quantum state describes ensemble of systemsArmen Allahverdyan, Roger Balian, Th.M. N. 2001; 2003:Exactly solvable models for quantum measurements.Ensemble of measurements on an ensemble of systems.explains: solid state, (bio-)chemistry high energy physics, early universeInterpretations: - 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 Max Born (1882-1970) 1926: Quantum mechanics is a statistical theoryJohn von Neumann (1903-1957) 1932: Wave function collapses in measurementClassical measurement specifies quantum measurement.Collapse is fast; occurs through interaction with apparatus.All possible outcomes with Born probabilities.
Quantum thermodynamicsQuantum partition sum: Z = Trace Exp( - H / k T ) as classically
Armen Allahverdyan + Th.M. N. 2000 Quantum particle coupled to bath of oscillators.Classically: standard thermodynamics example Hidden assumption: weak coupling with bath.
Clausius inequality violated.Several other formulations violated, but not allQuantum mechanically (low T)Coupling non-weak: friction = build up of a cloud.
Josephson junctionStep edge Josephson junction
Two Super-Conducting regions with Normal region in between: SNS-junctionElectric circuit with Josephson junction:Non-weak coupling to bath if resistance is non-small. SC1 N SC2One ingredient for violation of Clausius inequality measured in 1992
Photo-Carnot engine Scully groupVolume is set by photon pressure on pistonAtom beam phaseonium interacts with photonsEfficiency exceeds Carnot value, due to correlations of atoms
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 2portion 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>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
Entropy versus ergotropyMaximum 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 In-transformation work-transformation
Maxwells demonJames Clerk Maxwell (1831-1879)Quantum entanglement acts as a Maxwell demon in certain circumstancesNo work solely from heat:Maxwell demons should be exorcized! 1867: A tiny fingered being selects fast and slow atoms by moving a switch.Theory of electro-magnetism Maxwell-distribution
SummaryThermodynamics is old, strong theory of natureQuantum thermodynamics takes into account: quantum nature precise coupling to bathNew borders arise from: experiments model systems exact theoremsApplications in other fields of science