The validity of the concept of negative temperature has been recently challenged by arguing that the Boltzmann entropy (that allows negative temperatures) is inconsistent from a mathematical and statistical point of view, whereas the Gibbs entropy (that does not admit negative temperatures) provides the correct definition for the microcanonical entropy. We numerically study a lattice system to show that negative temperature equilibrium states are accessible and obey standard statistical mechanics prediction.

1. Bose-Einstein Condensation
Negative Temperatures
and the
Dispute between
Boltzmann e Gibbs
Entropy
R. Franzosi
QSTAR CNR-INO
Dept. of Physics Siena University December 21 2016

2. Bose Einstein condensation
in a gas of bosonic

3. Bose Einstein condensation
in a gas of bosonic quantum

4. Bose Einstein condensation
in a gas of bosonic quantum

5. Bose Einstein condensation
in a gas of bosonic quantum

6. Bose Einstein condensation
in a gas of bosonic quantum
identical: particles

7. Bose Einstein condensation
in a gas of bosonic quantum
identical: particles

8. Bose Einstein condensation
in a gas of bosonic quantum
identical: particles

9. Bose Einstein condensation
in a gas of bosonic quantum
identical: particles

10. Bose Einstein condensation
in a gas of bosonic quantum
identical: particles

11. Bose Einstein condensation
in a gas of bosonic quantum
identical: particles

12. Bose Einstein condensation
in a gas of bosonic quantum
identical: particles

13. Bose Einstein condensation
in a gas of bosonic quantum
identical: particles

14. Bose Einstein condensation
in a gas of bosonic quantum
identical: particles

15. Bose Einstein condensation
in a gas of bosonic quantum
identical: particles

16. Bose Einstein condensation
in a gas of bosonic quantum
identical: particles

17. Bose Einstein condensation
in a gas of bosonic quantum
identical: particles

18. Bose Einstein condensation
in a gas of bosonic quantum
identical: particles

19. Bose Einstein condensation
in a gas of bosonic quantum
identical: particles

20. Bose Einstein condensation
in a gas of bosonic quantum
identical: particles
Energy
Populationperenergystate
T=Tc
Bose-Einstein distribution

21. Bose Einstein condensation
in a gas of bosonic quantum
identical: particles

22. Bose Einstein condensation
in a gas of bosonic quantum
identical: particles
Energy
Populationperenergystate
Bose-Einstein distribution
T<TcCondensate!

23. Bose Einstein condensation
in a gas of bosonic quantum
identical: particles

24. Bose Einstein condensation
in a gas of bosonic quantum
identical: particles
Energy
Populationperenergystate
Bose-Einstein distribution
T<Tc Condensate!

25. Bose Einstein condensation
in a gas of bosonic quantum
identical: particles
Energy
Populationperenergystate
Bose-Einstein distribution
T<Tc Condensate!

26. Bose-Einstein condensates in optical
lattices
Laser Beam Laser Beam
^H=∫ d r ^ψ
†
(r)[−
ℏ2
2m ∇ 2
+Vext (r)] ^ψ(r)+
4π ℏ2
as
2m ∫ d r ^ψ
†
(r) ^ψ
†
(r) ^ψ(r) ^ψ(r)
H=∑j=1
M
[Λ
2
^nj( ^nj−1)+ξj ^nj
]
[^aj , ^ak
†
]=δj , k
−
1
2
∑j=1
M−1
(^aj
†
^aj+1+h.c.)
^nj=^aj
†
^aj
^N=∑j
^nj
[H, ^N]=0
H=∑j=1
M
[Λ
2
(N−1)∣zj ∣4
+ϵj∣zj∣2
]
N=∑ nj
−
1
2
∑j=1
M
[zj
*
zj+1+c.c.]
nj=|zj|
2
{zj
*
,zk}=i δjk
{H, N}=0
R. Franzosi, V. Penna and R. Zecchina, “Quantum Dynamics of coupled Bosonic Wells within the Bose-Hubbard Picture”,
Int. Jour. of Mod. Phys. B Vol. 14, No. 9 (2000) 943-961

27. Localized States via Boundary Dissipation
i
d
d τ
zj=Λ|zj|2
zj−
1
2
(zj+1+zj−1)−i γ zj(δj,1+δj, M)
Output
Coupler Output
Coupler
BEC
Output
Output
Effect of the Boundary Dissipation
Localized
solutions
Long-lived
solutions:
Breathers States
- static or moving
 
j j j
(a) (b) (c)
Figs. Time evolution of
the atomic density.
Panel (c) is the
continuation of (b).
R. Livi, R. Franzosi and G.-L. Oppo, “Selflocalization of Bose-Einstein condensates in optical lattices via boundary
dissipation”, Phys. Rev. Lett. 97, 060401 (2006)
R. Franzosi, R. Livi and G.-L. Oppo, “Probing the dynamics of Bose-Einstein condensates via boundary dissipation”,
Journal of Physics B 40, 1195 (2007)

28. Discrete Breathers: Long-Living Localized Excitations
The sink that does not drain
Water
u(x, τ)=v(x)e
−i E τ
v(x)⇒e
−|x|
for x⇒±∞
BEC

29. Thermodynamic Phase Diagram
Single Static Breather State
Ground state

30. Thermodynamic Phase Diagram
Single Static Breather State
Ground state
C

31. Thermodynamic Phase Diagram
Single Static Breather State
Ground state
C

32. Thermodynamic Phase Diagram
Single Static Breather State
Ground state
C

33. Thermodynamic Phase Diagram
Single Static Breather State
Ground state
C

34. Thermodynamic Phase Diagram
Single Static Breather State
Ground state
C

35. Thermodynamic Phase Diagram
Single Static Breather State
Ground state
C

36. Thermodynamic Phase Diagram
Single Static Breather State
Ground state
C

38. Recent work

39. Recent work
and counting ...

40. Boltzmann
# microstates at energy density h
Gibbs
# microstates up to energy density h
Boltzmann vs Gibbs

41. The winner is … Boltzmann

42. The winner is … Boltzmann
inequivalence of the Gibbs MC picture
for high energies (bG = 0, bB < 0)
P Buonsante, RF, A Smerzi, Annals of Physics 2016

43. The winner is … Boltzmann
inequivalence of the Gibbs MC picture
for high energies (bG = 0, bB < 0)
P Buonsante, RF, A Smerzi, Annals of Physics 2016

44. The winner is … Boltzmann
inequivalence of the Gibbs MC picture
for high energies (bG = 0, bB < 0)
P Buonsante, RF, A Smerzi, Annals of Physics 2016
phase transitions –
“standard BEC” (extended
state)

45. The winner is … Boltzmann
inequivalence of the Gibbs MC picture
for high energies (bG = 0, bB < 0)
P Buonsante, RF, A Smerzi, Annals of Physics 2016
phase transitions –
“standard BEC” (extended
state)
3D lattice, cubic nonlinearity, self-
focusing case:

46. The winner is … Boltzmann
inequivalence of the Gibbs MC picture
for high energies (bG = 0, bB < 0)
P Buonsante, RF, A Smerzi, Annals of Physics 2016
phase transitions –
“standard BEC” (extended
state)
3D lattice, cubic nonlinearity, self-
focusing case:
P Buonsante, RF, A Smerzi, arXiv:1506.01933

47. Conclusions

48. Conclusions

49. Conclusions
The Boltzmann's epitaph is correct!