# The Second Law of Quantum Thermodynamics Theo M. Nieuwenhuizen NSA-lezing 19 maart 2003.

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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 17th-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 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: Emile Clapeyron (1799-1864) 1834: diagrams for Carnot process The superiority of England over France is due to its skills to use the power of heat”

The First Law Heat 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) 1847 Germain Henri Hess (1802-1850) 1840 Energy is conserved: dU=dQ+dW heat + work added to system

The Second Law Rudolf 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 work Most common formulation: Entropy of a closed system cannot decrease Clausius formulation: Heat goes from high to low temperature

First Law Second Law = the way it moves Thermodynamics 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 nothing Perpetuum mobile of the second kind is impossible: No work from heat without loss = the harness of nature 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 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 Boltzmann equation for molecular collisions Maxwell-Boltzmann weight

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 Observables are operators in Hilbert space New parameter: Planck’s constant h 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. explains: solid state, (bio-)chemistry high energy physics, early universe 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 Max Born (1882-1970) 1926: Quantum mechanics is a statistical theory John von Neumann (1903-1957) 1932: Wave function collapses in measurement 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 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 all Quantum mechanically (low T) Coupling non-weak: friction = build up of a cloud.

Josephson junction Step edge Josephson junction Two Super-Conducting regions with Normal region in between: SNS-junction Electric circuit with Josephson junction: Non-weak coupling to bath if resistance is non-small. SC1 N SC2 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

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 T rad 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 T h 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 T c provides the entropy sink. (B) Two-level atoms in a regular thermal distribution, determined by temperature T h, heat the driving radiation to T rad = T h 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 T rad > T h, 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 ergotropy 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 In-transformation work-transformation

Maxwell’s demon James Clerk Maxwell (1831-1879) Quantum entanglement acts as a Maxwell demon in certain circumstances No 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

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

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