CONDENSATION OF FLUCTUATIONS IN A TEMPERATURE QUENCH Marco Zannetti - U Salerno in collaboration...
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CONDENSATION OF FLUCTUATIONS IN A TEMPERATURE QUENCH Marco Zannetti - U Salerno in collaboration with Federico Corberi - U Salerno Giuseppe Gonnella - U Bari Antonio Piscitelli - U Napoli
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Transcript of CONDENSATION OF FLUCTUATIONS IN A TEMPERATURE QUENCH Marco Zannetti - U Salerno in collaboration...
CONDENSATION OF FLUCTUATIONS IN A TEMPERATURE QUENCH
Marco Zannetti - U Salerno
in collaboration with
Federico Corberi - U SalernoGiuseppe Gonnella - U BariAntonio Piscitelli - U Napoli
temperature quench
study of fluctuationsduring relaxation• energy• heat
t
T
IT
FT
0
temperature quench
• Brownian Oscillator• Gaussian Model : collection of independent oscillators condensation of fluctuations
t
T
IT
FT
0
temperature quench
• Brownian Oscillator• Gaussian Model• Large – N Model: mean field of interacting system
no equilibration!
t
T
IT
FT
CT
0
Oscillator2
2
1)( H
2
2
11
)(
IT
II e
ZP
t
T
IT
FT
0
2
2
11
)(
IT
II e
ZP
2
)(2
11
),(
tT
t
eZ
tP
2
)()(
2
1 2 tTtH
t
1Bk
effective temperature
J.R.Gomez-Solano, A.Petrosyan, S.Ciliberto – PRL 106, 200602 (2011)
2
2
11
)(
IT
II e
ZP
2
)(2
11
),(
tT
t
eZ
tP
2
)()(
2
1 2 tTtH
t
FI
Ft
TTT
TTetT
2)(
FT
effective temperature
Energy fluctuations
ete
eteP
tee
)(2),(
)(2
2
)()(
tTte
0 t
Heat fluctuations
),(),()(),( www PHHq
• q < 0 heat released • q > 0 heat absorbed
wt t0
w
)~()~(~)( qqPqPqdqP AR
)(2
)(
)(2
)(
2
2
qqq
eqP
qqq
eqP
A
A
R
R
A
R
2)]()([ wAR tTtTqqq
fluctuation relation
qqqqP
qP
AR
11
2
1
)(
)(ln
fluctuation relation
qqqqP
qP
AR
11
2
1
)(
)(ln
1
2 111111
2
1
wt
I
F
FIAR
eT
T
TTqq
FI TT
11
wt
collection of brownian oscillatorsGAUSSIAN MODEL
0,
2
1][
])[(2
1)]([
2
22
rrk
VH
rxdxH
k
kkk kk
V
Temperature Quench FI TT
kkkk
Temperature Quench FI TT
kkkk
k
tT
k
kkkke
tZtP
)(2
1
)(
1),(
Ft
k TTetT k 2)(
spectrum ofeffective temperatures
Fluctuations of extensive quantities: Energy
])[(),(),( HVetPdteP
Fluctuations of extensive quantities: Energy
])[(),(),( HVetPdteP
zxi
i
ei
dzx
2)(
Fluctuations of extensive quantities: Energy
),(
][),(2
])[(),(),(
tzQ
zHVezi
i
etPdei
dz
HVetPdteP
][),(),(
1),,( zH
C etPtzQ
tzP
partition function in the biased ensemble
Fluctuations of extensive quantities: Energy
),(ln
2
])[(),(),(
tzQVezi
i
ei
dz
HVetPdteP
biased free energy
Fluctuations of extensive quantities: Energy
),(ln
2
])[(),(),(
tzQVezi
i
ei
dz
HVetPdteP
large V),ˆ,( tzeVIe
zH
Ve
ˆ
1
zetzQV
tzeI ˆ),ˆ(ln1
),ˆ,(
),ˆ,( tzPC
saddle point ẑ
zetzQV
tzeI ˆ),ˆ(ln1
),ˆ,(
fluctuations in the gaussian model
average properties in the biased ensemble
condensation• spherical model• BEC
condensation
k
zkHVe
1
condensation
k kkzk ztTV
HV
e)(
1
2
111
condensation
k kkzk ztTV
HV
e)(
1
2
111
1Mk
T z
e~
11 min kk TTM
hottestmode
condensation
k kkzk ztTV
HV
e)(
1
2
111
1Mk
T z
e~0Mk
condensation
k kkzk ztTV
HV
e)(
1
2
111
10T z
e~0Mk
Ce
condensation
k kkzk ztTV
HV
e)(
1
2
111
10Tz
Ce
z
0e
e~0Mk e
zC HV
eeeˆ00
1
finite contribution fromthe zero mode
phase diagram
Ce
t
condensed
normal
phase diagram
Ce
t
condensed
normal
e
Fluctuations of extensive quantities: Heat
])[][(),(),,( wwww HHVqPddttqP
Fluctuations of extensive quantities: Heat
])[][(),(),,( wwww HHVqPddttqP ),,ˆ,(),,( wttzqVI
w ettqP
)]ˆ1)(ˆ1(ln[2
1ˆ),,ˆ,( ,, zqzq
VzqttzqI kAkR
kw
0FT 0, kAq
k kRkzkR zqV
qV
q1,
ˆ,
1
2
1
2
1
mostreleasing
mode
1,
MkRq0
z
saddle point ẑ
0Mk
1,
MkRq
z
0Mk
10,
Rq
z
Cq
q 0q
phase diagram 0FT
q
tt
condensed
normal
N-VECTOR MODEL
0,0),,..,(
2)(
2
1)]([
1
2222
gr
N
grxdxH
N
V
02 CTd
22
N
gr
22
N
gr
Large N limitself-average
)(),(2 tStx
kkkk t ,,, )(
22
N
gr
)(),(2 tStx
Large N limitself-average
)()(
)()(
)(
0
02
,,,
tgSrt
tkt
t
k
kkkk
22
N
gr
)(),(2 tStx
Large N limitself-average
Temperature Quench 0 FI TT
kk tgSrk ,2
, )]([
Temperature Quench 0 FI TT
kk tgSrk ,2
, )]([
1t
0
Temperature Quench 0 FI TT
kk tgSrk ,2
, )]([
0,02 kk
0
1t
unstable modes
t1t
1t
kMk
kT
effective temperatures
)()()()( ttttT kkkk
0
)()()()( ttttT kkkk
effective temperatures
t2t
0
1t
k
2t
Mk
kT
t3t
1t
kMk
2t3t
kT
)()()()( ttttT kkkk
effective temperatures
0
k
kT
t
0)(2
10
k
k tTV
H
)(
)(
0)(
0
0
0
tT
tT
tT
k
k
ksystem permanently
out of equilibrium
0
«apparent» equilibrium
k
kT
k k ztTVe
)(
111
energy fluctuations
k k ztTVe
)(
111
1Mk
T z
k
kT
1Mk
T
normal fluctuations
e~
0Mk
k
kT
k k ztTVe
)(
111
k
1kT
1Mk
T
10T
k k ztTVe
)(
111
hottestmode
1Mk
T10T 0
z
coldestmode
k
1kT
1Mk
T
10T
condensation of negative energy
fluctuationsdue to unstable modes
k k ztTVe
)(
111
1Mk
Tz
10T
e~
Ce~
e0e
phase diagram 0FT
Ce
t
normal
condensednormal
LARGE-N
Ce
t
condensednormal
GAUSSIAN
J. Phys. A: Math. Theor. 46, 042001 (2013)
),,( wttqI
wttx
condensation
conclusions and outlook
• Fluctuations of energy and heat after a temperature quench non trivial even in trivial non interacting systems• Condensation of fluctuations is related to the duality fluctuations in unbiased ensemble ↔ average properties in biased ensemble• Effective temperature key quantity – relation with effective temperature from FDR• Mean field qualitatively different from real system (energy unbounded below) - go beyond mean field
