1. Explanation of the Gibbs Paradox within the Framework of Quantum Thermodynamics
Theo M. Nieuwenhuizen
Physikalisches Kolloquium Johann Wolfgang Goete Universitaet
Frankfurt am Main
31-01, 2007

2. Outline
Who was Josiah Willard Gibbs? What is the Gibbs Paradox? On previous explanations: mixing entropy
Crash course in Quantum Thermodynamics
Maximal work = ergotropy
Application of mixing ergotropy to the paradox

3. Josiah Willard Gibbs
1839 – 1903
Carreer in Yale
: Travel to Paris, Berlin, Heidelberg
Gustav Kirchhoff, Hermann von Helmholtz
Gibbs free energy
Gibbs entropy Gibbs ensembles Gibbs Duhem relation
Gibbs distribution
Gibbs state
Gibbs paradox
Copley Medal 1901

4. The Gibbs Paradox (mixing of two gases) Josiah Willard Gibbs 1876
mixing entropy
But if A and B identical, no increase.
The paradox: There is a discontinuity, still k ln 2 for very similar but non-identical gases.

5. Gibbs --------------------------------------------------------------------
1876
= N log 2

6. Proper setup for the limit B to A
Isotopes: too few to yield a good limit
Let gases A and B both have translational modes at equilibrium at temperature T, but their internal states (e.g. spin) be described by a different density matrix or Then the limit B to A can be taken continuously.

7. Current opinions:
The paradox is solved within information theoretic approach to classical thermodynamics
Solution has been achieved within quantum statistical physics due to feature of partial distinguishability
Quantum physics is right starting point. But due to non-commutivity, the paradox is still unexplained.

8. Quantum mixing entropy argument
Von Neuman entropy
After mixing
Mixing entropy
ranges continuously from 2N ln 2 (orthogonal) to 0 (identical) . Many scholars believe this solves the paradox.
Dieks & van Dijk ’88: thermodynamic inconsistency, because there is no way to close the cycle by unmixing. If nonorthogonal to any attempt to unmix (measurement) will alter the states.

9. Another objection: lack of operationality
The employed notion of ``difference between gases’’ does not have a clear operational meaning.
If the above explanation would hold, certain measurements would not expose a difference between the gasses. So the ``solution’’ would depend on the quality of the apparatus.
There is something unsatisfactory with entropy itself. It is non-unique. Its definition depends on the formulation of the second law.
To be operational, the Gibbs paradox should be formulated in terms of work. Classically: Also in quantum situation??

10. Quantum Thermodynamics = Thermodynamics applying to:
System finite (small, non-extensive)
Bath extensive
Work source extensive (e.g. laser)
No thermodynamic limit
Bath has to be described explicitly Non-negligible interaction energy

11. Caldeira-Leggett model: particle + harmonic bath
Langevin equation (if initially no correlation between S and B)

12. First law: is there a thermodynamic description, though the system is finite?
where H is that part of the total Hamiltonian,
that governs the unitary part of (Langevin) dynamics in the small Hilbert space of the system.
Work: Energy-without-entropy added to the system by a macroscopic source.
1) Just energy increase of work source 2) Gibbs-Planck: energy of macroscopic degree of freedom.
Energy related to uncontrollable degrees of freedom
Picture developed by Allahverdyan, Balian, Nieuwenhuizen ’00 -’04

13. Roger Balian (1933-) CEA Saclay; Academie des Sciences
B phase = Balian –Werthamer phase
(p-wave pairing)
- Eigenfrequencies of Schroedinger operators in finite domain
- Casimir effect: Balian-Duplantier sum rule
- Book: From microphysics to macrophysics - Quantum measurement process

14. Second law for finite quantum systems
No thermodynamic limit Thermodynamics endangered
Different formulations are inequivalent
-Generalized Thomson formulation is valid:
Cyclic changes on system in Gibbs equilibrium cannot yield work
(Pusz+Woronowicz ’78, Lenard’78, A+N ’02.)
Clausius inequality may be violated
due to formation of cloud of bath modes
- Rate of energy dispersion may be negative Classically: = T * ( rate of entropy production ): non-negative
A+N: PRL 00 ; PRE 02, PRB 02, J. Phys A 02
Experiments proposed for mesoscopic circuits and quantum optics.

15. Armen Allahverdyan
Yerevan, Armenia
statistical mechanics quantum thermodynamics quantum measurement process
astrophysics, cosmology, arrow of time
adiabatic theorems quantum optics
quantum work fluctuations
Gibbs paradox
> 35 common papers

16. Work extraction from finite Q-systems
Couple to work source and do all possible work extractions
Thermodynamics: minimize final energy at fixed entropy Assume final state is gibbsian: fix final T from S = const. Extracted work W = U(0)-U(final)
But: Quantum mechanics is unitary,
So all n eigenvalues conserved: n-1 constraints, not 1.
(Gibbs state typically unattainable for n>2)
Optimal final situation: eigenvectors of become those of H

17. Maximal work = ergotropy
Lowest final energy: highest occupation in ground state, one-but-highest in first excited state, etc (ordering )
Maximal work:
(divine action, Aristotle)
Allahverdyan, Balian, Nieuwenhuizen, EPL 03.

18. Aspects of ergotropy -non-gibbsian states can be passive
-Comparison of activities:
Thermodynamic upper bounds: more work possible from
But actual work may be largest from
- Coupling to an auxiliary system : if is less active than
Then can be more active than
-Thermodynamic regime reduced to states that majorize one another
- Optimal unitary transformations U(t) do yield, in examples, explicit Hamiltonians for achieving optimal work extraction

19. Resolution of Gibbs paradox
Formulate problem in terms of work: mixing ergotropy = maximal extractable work before mixing – ( idem, after mixing)
Consequence: limit B to A well behaved: vanishing mixing ergotropy Paradox explained.
Operationality: difference between A and B depends on apparatus: extracted work need not be maximal
More mixing does not imply more work, and vice versa. Counterexamples given in A+N, PRE 06.

20. Luca Leuzzi, Rome PhD in Amsterdam 2002, cum laude
L+N book: Thermodynamics of the glassy state

21. Summary
Gibbs paradox not solved up to now Mixing entropy argument has its own drawbacks
Explanation by formulation in terms of work Mixing ergotropy = loss of maximal extractable work due to mixing
Operational definition: less work from less good apparatus
More mixing does not imply more work and vice versa Many details in Allahverdyan + N, Phys. Rev. E 73, (2006)

22. Are adiabatic processes always optimal?
One of the formulations of the second law:
Adiabatic thermally isolated processes done on an equilibrium system are optimal (cost least work or yield most work)
In finite Q-systems: Work larger or equal to free energy difference
But adiabatic work is not free energy difference.
A+N, PRE 2003:
-No level crossing : adiabatic theorem holds
-Level crossing: solve using adiabatic perturbation theory.
Diabatic processes are less costly than adiabatic Work = new tool to test level crossing.
Level crossing possible if two or more parameters are changed. Review expts on level crossing: Yarkony, Rev Mod Phys 1996