bose-einstein condensation, negative temperatures and the dispute between boltzmann e gibbs entropy

Bose-Einstein CondensationNegative Temperatures

and theDispute between

Boltzmann e GibbsEntropy R. Franzosi

QSTAR CNR-INO

Dept. of Physics Siena University December 21 2016

Bose Einstein condensationin a gas of bosonic

Bose Einstein condensationin a gas of bosonic quantum


identical: particles



Energy

Pop

ulat

ion

per

ener

gy s

tate

T=Tc

Bose-Einstein distribution



Energy

Pop

ulat

ion

per

ener

gy s

tate


T<TcCondensate!



Energy

Pop

ulat

ion

per

ener

gy s

tate


T<Tc Condensate!

Bose-Einstein condensates in optical lattices

Laser Beam Laser Beam

H=∫ d r ψ†(r)[−

ℏ2

2 m ∇ 2+Vext (r)] ψ (r)+4π ℏ2 as

2 m ∫ d r ψ†(r) ψ

†(r) ψ (r) ψ (r)

H=∑ j=1

M

[ Λ2 n j( n j−1)+ξ j n j]

[ a j , ak† ]=δ j , k

−12∑ j=1

M−1

( a j† a j+1+h . c .)

n j= a j† a j

N=∑ jn j

[ H , N ]=0

H=∑ j=1

M

[ Λ2 (N−1)∣ z j ∣4+ϵ j∣ z j∣

2]

N=∑ n j

−12∑ j=1

M

[z j* z j+1+c. c .]

n j=|z j|2

{z j* , z k}=iδ jk

{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

Localized States via Boundary Dissipation

id

d τz j=Λ|z j|

2 z j−12 (z j+1+z j−1) − i γ z j (δ j ,1+δ j , M)

Output Coupler Output

Coupler

BEC

OutputOutput

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)

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

Thermodynamic Phase DiagramSingle Static Breather State

Ground state

Thermodynamic Phase DiagramSingle Static Breather State

Ground state

C

Recent work

Recent work

and counting ...

Boltzmann # microstates at energy density h

Gibbs # microstates up to energy density h

Boltzmann vs Gibbs

The winner is … Boltzmann

The winner is … Boltzmanninequivalence 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:




3D lattice, cubic nonlinearity, self-focusing case:

P Buonsante, RF, A Smerzi, arXiv:1506.01933

Conclusions

Conclusions

The Boltzmann's epitaph is correct!

bose-einstein condensation, negative temperatures and the dispute between boltzmann e gibbs entropy

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