The Second Law ofQuantum 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) 1842James Prescott Joule (1818-1889) 1847Herman von Helmholtz (1821-1894) 1847Germain 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 nothingPerpetuum 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 istUnd verficht’s, bis es mit dir gar ist

Onthul, wat waar isSchrijf zo, dat het zonneklaar isEn vecht ervoor, tot je brein gaar is

Boltzmann equation for molecular collisionsMaxwell-Boltzmann weight

Gibbs

Josiah Willard Gibbs (1839-1903)

Papers in 1875,1878

Ensembles: micro-canonical, canonical, macro-canonicalGibbs free energy F=U-TSGibbs-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 mixturesAttraction 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 spaceNew 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 theoryJohn 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 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 > 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 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 demonJames Clerk Maxwell (1831-1879)

Quantum entanglementacts as a Maxwell demonin 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-magnetismMaxwell-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