Download Self-thermalization in isolated many-body quantum systems

QCD Hadronization and the Statistical Model
ECT, Trento, October 2014
Self-thermalization
in isolated many-body quantum systems
Mark Srednicki
UCSB
How do isolated quantum systems
come to thermal equilibrium?
How do isolated quantum systems
come to thermal equilibrium?
Simple example:
Dilute gas in a box
!
Assumptions:
Observables Ai e.g., Ai = a†p ap
!
"
"
†
bles Ai e.g.,NA!
=
a
degrees f freedom
i 1 degrees
ofofreedom
p ap
a†p ap
"
.,egrees
Ai = of freedom
†
Quantum
stateAof
complete
system:
Observables
e.g.,
A
=
a
Observables
i
i
p ap |ψ(t)"
!
"
eedom
Instantaneous
expectation value: #A(t)" ≡ #ψ(t)
N ! 1system:
degrees
of
freedom
m state of complete
|ψ(t)"
Quantum statequantum
omplete system:
|ψ(t)"
Total
energy:
E
≡
#ψ(t)|H|ψ(t)"
Quantum
state
of
complete
system:
neous quantum
expectationquantum value:
#A(t)"
≡ #ψ(t)|A|ψ(t)"
Instantaneous expectation v|ψ(t)"
alue
2
um expectation
value: energy
#A(t)" uncertainty:
≡ #ψ(t)|A|ψ(t)"
Quantum
∆E
≡
#ψ(t)|(H
−
E)
|ψ(t)
Instantaneous
quantum
expectation
value:
#A(t)"
≡
#ψ
ergy: E ≡ #ψ(t)|H|ψ(t)"
ψ(t)|H|ψ(t)"We assume ∆E is “small”; e.g., ∆E ∼ N −1/2 E
Total energy: E ≡ #ψ(t)|H|ψ(t)"2
1/2
m energy uncertainty: ∆E ≡ #ψ(t)|(H − E) |ψ(t)"
−βH
certainty: ∆E ≡ #ψ(t)|(H − E)2 |ψ(t)"1/2
Tr
Ae
2
Quantum
energy
uncertainty:
∆E
≡
#ψ(t)|(H
−
E)
|ψ
Thermal expectation value:
−1/2 [A]th ≡
Instantaneous
quantum
expectation
value:
#A(t)"
≡
#ψ(t)
Instantaneous
quantum
expectation
value:
#A(t)"
≡
#ψ(t)|A|ψ(t)
Instantaneous
quantum
expectation
value:
#A(t)"
≡
#ψ(t)|A|ψ
Instantaneous quantum expectation value: �A(t)� ≡ �ψ(t)|A|
Total
energy:
E
≡
#ψ(t)|H|ψ(t)"
Total
energy:
E
≡
#ψ(t)|H|ψ(t)"
TotalTotal
energy:
E ≡E#ψ(t)|H|ψ(t)"
Total eenergy:
nergy
≡ �ψ(t)|H|ψ(t)� ∼ N
2 1/2
2
�
2
1/2
Quantum
energy
uncertainty:
∆E
#ψ(t)|(H
− E)
|ψ(t)
Quantum
energy
uncertainty:
∆E
≡≡
#ψ(t)|(H
E)
|ψ(t)"
Quantum
energy
uncertainty:
∆E ≡
#ψ(t)|(H
− E)− |ψ(t)"
Quantum energy
energy uncertainty:
uncertainty ∆E ≡ �H 2 � − �H�2 ∼ N 1−ν ,
Quantum
!
1−ν−1/2
2
2
−1/2
∆E
≡
"H
#
−
"H#
∼
N
,
ν E> E
0
−1/2
We
assume
∆E
is
“small”;
e.g.,
∆E
∼
N
We
assume
∆E
“small”;
e.g.,
∆E
∼
N
We assume ∆E is “small”; e.g., ∆E ∼ N
E
We assume ∆E is “small”; e.g., ∆E ∼ N −1/2 E
−βH
−βH
Tr
Ae
−βH
Tr
Ae
"ψ|A
|ψ#
&
[A
]
Tr≡Ae
f [A]th
f th −βH
Thermal
expectation
value:
Thermal e
xpectation v
alue
−βH
Thermal
expectation
Thermal
expectation
value: value:
[A]th ≡[A]th ≡
Tr
Ae
Tr
e
e−βH
Thermal expectation value: [A]th Tr
≡ e−βHTr−βH
Tr e
Inverse temperature β determined
Af by [H]th = E
Inverse
temperature
β determined
E th = E
Inverse temperature
fixed bbyy [H]by
Inverse
temperature
β determined
[H]
th =
Inverse temperature β determined by [H]th = E
t
|ψ#
1 3
1 2
1 3
y 2) + x21y −
y
=
r
+
r sin(3θ)
V (x, 1y)
3
2
3
1
1
3
2
3
2y − y
2 = 1 r 3 + 1r 2sin(3θ)
1 3
(x
+
y
)
+
x
V (x, y)
=
V (x, 2y) = (x + y ) + x3 y − 2 y = 3 r + r sin(
2
1
(x
=
2 21 2 2+
2
2
�Ai (t)� = [Ai ]th
3
t
(X " | U |X) = δ 2f(X " − M (X))
|cα |
2
2
3
Instantaneous
quantum
expectation
value:
#A(t)"
≡
#ψ(t)
Instantaneous
quantum
expectation
value:
#A(t)"
≡
#ψ(t)|A|ψ(t)
Instantaneous
quantum
expectation
value:
#A(t)"
≡
#ψ(t)|A|ψ
Instantaneous quantum expectation value: �A(t)� ≡ �ψ(t)|A|
Total
energy:
E
≡
#ψ(t)|H|ψ(t)"
Total
energy:
E
≡
#ψ(t)|H|ψ(t)"
TotalTotal
energy:
E ≡E#ψ(t)|H|ψ(t)"
Total eenergy:
nergy
≡ �ψ(t)|H|ψ(t)� ∼ N
Restrictions
on ψ 2
2E)1/2
1/2
�
2 E)
Quantum
energy
uncertainty:
∆E
≡
#ψ(t)|(H
−
|ψ(t)
Quantum
energy
uncertainty:
∆E
≡
#ψ(t)|(H
−
|ψ(t)"
Quantum
energy
uncertainty:
∆E ≡ #ψ(t)|(H 2− E) |ψ(t)"
2 ∼ N 1−ν ,
Quantum e
nergy u
ncertainty
�H
�
−
�H�
Quantum energy uncertainty:
∆E
≡
!
We
1−ν−1/2
−1/2
∆E
≡
−
∼
N
,
ν E> E
0
−1/2
We
assume
∆E
is
“small”;
e.g.,
∆E
∼
N
We
assume
∆E
“small”;
e.g.,
∆E
∼
N
assume ∆E is “small”; e.g., ∆E ∼ N
E
−1/2
"H 2 #
"H#2
We assume ∆E is “small”; e.g., ∆E ∼ N
E
−βH
−βH
Tr
Ae
−βH
Tr
Ae
"ψ|A
|ψ#
&
[A
]
Tr≡Ae
f [A]th
f th −βH
Thermal
expectation
value:
Thermal e
xpectation v
alue
−βH
Thermal
expectation
Thermal
expectation
value: value:
[A]th ≡[A]th ≡
Tr
Ae
Tr
e
e−βH
Thermal expectation value: [A]th Tr
≡ e−βHTr−βH
Tr e
Inverse temperature β determined
Af by [H]th = E
Inverse
temperature
β determined
E th = E
Inverse temperature
fixed bbyy [H]by
Inverse
temperature
β determined
[H]
th =
Inverse temperature β determined by [H]th = E
t
|ψ#
1 3
1 2
1 3
y 2) + x21y −
y
=
r
+
r sin(3θ)
V (x, 1y)
3
2
3
1
1
3
2
3
2y − y
2 = 1 r 3 + 1r 2sin(3θ)
1 3
(x
+
y
)
+
x
V (x, y)
=
V (x, 2y) = (x + y ) + x3 y − 2 y = 3 r + r sin(
2
1
(x
=
2 21 2 2+
2
2
�Ai (t)� = [Ai ]th
3
t
(X " | U |X) = δ 2f(X " − M (X))
|cα |
2
2
3
Tr Ae
Thermal
expectation
value:
Thermal expectation value: [A]th ≡
Tr e
Inverse
deter
[H]
We will sInverse
ay that temperature
the system temperature
is βin determinedβ by
thermal equilibrium t at time t if
#Ai (t)" = [Ai ]th
2
1
(x
2
"
2
V (x, y) =
+y
to within experimental e
rror
!
†
2
2
2
1
1 3
Observables
A
e.g.,
A
=
a
a
for each
V (x, iy) = 2 (x i + y p) +
p x y − 3y
N ! 1 degrees of freedom
|cα |2
Quantum state of complete system: |ψ(t)"
Instantaneous quantum expectation value:
!
e.g., Ai = a†p ap
"
Quantum stat
#Ai (t)"
=xpect:
[Ai ]th
What w
e e
of freedom
Instantaneous
†
= ap ap|ψ(t)"
completeAsystem:
of
m: |ψ(t)"
Total energy:
uantum expectation value: #A(t)" ≡ #ψ(t)|A|ψ(t)"
ion value: #A(t)" ≡ #ψ(t)|A|ψ(t)"
2
2
2
1
≡ #ψ(t)|H|ψ(t)"
V (x, y) = 2 (x + y ) + x y −
2
1/2
y uncertainty: ∆E ≡
#ψ(t)|(H
−
E)
|ψ(t)"
E ≡ #ψ(t)|(H − E)2 |ψ(t)"1/2
is “small”; e.g., ∆E ∼ N −1/2 E
∆E ∼ N −1/2 E
Tr Ae−βH
ation
Trvalue:
Ae−βH[A]th ≡ Tr e−βH
≡
Tr e−βH
ture β determined by [H]th = E
d by [H]th = E
1 3
y
3
=
2
1 Quantum
1 3 ener
r + 3 r sin(
2
We assume ∆E
|cα |2
Thermal expe
Inverse tempe
Aαα
t
#Ai (t)" = [Ai ]t
!
e.g., Ai = a†p ap
"
Quantum stat
#Ai (t)"
=xpect:
[Ai ]th
What w
e e
of freedom
Instantaneous
†
= ap ap|ψ(t)"
completeAsystem:
of
m: |ψ(t)"
Total energy:
uantum expectation value: #A(t)" ≡ #ψ(t)|A|ψ(t)"
ion value: #A(t)" ≡ #ψ(t)|A|ψ(t)"
2
2
2
1
≡ #ψ(t)|H|ψ(t)"
V (x, y) =
2
1 3
1 Quantum
1 3 ener
(x + y ) + x y − 3 y =from r +QM r sin(
2 Can we get this picture 2
3
with no further restrictions on assume
ψ? ∆E
We
2
1/2
y uncertainty: ∆E ≡
#ψ(t)|(H
−
E)
|ψ(t)"
E ≡ #ψ(t)|(H − E)2 |ψ(t)"1/2
is “small”; e.g., ∆E ∼ N −1/2 E
∆E ∼ N −1/2 E
Tr Ae−βH
ation
Trvalue:
Ae−βH[A]th ≡ Tr e−βH
≡
Tr e−βH
ture β determined by [H]th = E
d by [H]th = E
|cα |2
Thermal expe
Inverse tempe
Aαα
t
#Ai (t)" = [Ai ]t
A = ap ap
Energy eigenstates: H|α" = Eα |α"
V (x, y) = 21 (x2 + y 2 ) + x2 y −
|cα |
2
Aαα
1
H|α" = Eα |α"
A = ap ap
Energy eigenstates: H|α" = Eα |α"
Time evolution:
|ψ(t)" =
#
cα e−iEα t |α"
α
V (x, y) = 21 (x2 + y 2 ) + x2 y −
V (x, y) = 21 (x2 + y 2 ) + x2 y − 31 y 3 = 12 r 2 + 13 r 3 sin(3θ)
|cα |
|cα |
2
2
Aαα
1
1
A = ap ap
H|α" = Eα |α"
Energy eigenstates:#H|α"−iE
= αEt α |α"
|ψ(t)" =
cα e
|α"
Time evolution:
α
|ψ(t)" =
#
cα e−iEα t |α"
α
2
2
2
1
(x
+
y
)
+
x
y
−
V
(x,
y)
=
#
#
2
∗
i(Eα −Eβ )t2
#A(t)" =
|cα | Aαα +
cα cβ e
Aαβ
α
V (x,
y) = 21 (x2 +α"y=2β) + x2 y − 31 y 3 = 12 r 2 + 13 r 3 sin(3θ)
V (x, y) =
2
1
(x
2
2
2
+y )+x y−
1
1 3
y
3
1 2
= 2r
2
|cα |
+
1 3
r
3
sin(3θ)
|cα |
2
Aαα
1
1
A = ap ap
H|α" = Eα |α"
Energy eigenstates:#H|α"
|ψ(t)" =
†
† Ai e.g.,
Time e
les
A
=
a
ap
i
ap ap
pvolution:
"!
"
= αEt α |α"
−iE
!
Observables Ai e.g.,
cα e
|α"
N ! 1 degrees of free
#
α
|ψ(t)" =
cα e−iEα t |α"
Quantum state of com
α
2
2
2
1
(x
+
y
)
+
x
y
−
V
(x,
y)
=
egrees of freedom
#
#
2
∗
i(Eα −Eβ )t2
Instantaneous quantu
#A(t)" =
|cα | Aαα +
cα cβ e
Aαβ
of complete
α |ψ(t)" 1 2
e state
system:
|ψ(t)" system:
V (x, y) = (x
2
+α"y=2β) + x2 y − 31 y 3 = 12 r 2 + 13 r 3 sin(3θ)
Total energy: E ≡ #ψ
eous quantum expectation value: #A(t)" ≡ #ψ(t)|A|ψ(t)"
pectation value: #A(t)" ≡ #ψ(t)|A|ψ(t)"
2
1
y) = 2 (x
rgy: EV≡(x,
#ψ(t)|H|ψ(t)"
|ψ(t)"
2
2
+y )+x y−
1 3
y
3
2
1/2
energy uncertainty: ∆E ≡
#ψ(t)|(H
−
E)
|ψ(t)"
2
1/2
1 2
= 2r
2
|cα |
2
|c
|
α
Quantum
energy
unce
1 3
+ 3 r sin(3θ)
We assume ∆E is “sm
ty: ∆E ≡ #ψ(t)|(H − E) |ψ(t)"
Thermal expectation
me ∆E is “small”; e.g., ∆E ∼ N −1/2 E
e.g., ∆E ∼ N −1/2 E
−βH
expectation
Trvalue:
Ae−βH[A]th
: [A]th ≡
Tr e−βH
Tr Ae
≡
Tr e−βH
mperature β determined by [H]th = E
rmined by [H]th = E
1
Aαα
1
Inverse temperature β
t
#Ai (t)" = [Ai ]th
1
α
H|α" = Eα |α"
A = ap ap
!
Observables Ai e.g.,
#
#
#−iE
Energy eigenstates:
H|α"
=
|α"−E
t αi(E
2
∗ αE
α
β )t
|ψ(t)"
=
c
e
|α"
#A(t)" =
|cα | Aαα + α cα cβ#
e
AαβN ! 1 degrees of free
" α
α α"=β
"!
−iEα t
†
†
|ψ(t)"
=
c
e
|α"
les
Ai = aepvolution:
ap
α
ap aApi e.g.,Time Quantum state of com
α
2
2
2
1
(x
+
y
)
+
x
y
−
V
(x,
y)
=
egrees of freedom
#
#
#
#
2
22
∗
i(E
−E
α
β )t
∗
i(E
−E
)t
α
β
Instantaneous quantu
#A(t)"
|c
ccααccββee
AA
#A(t)" =
=
|cαα| Aαα
αβ
αβ
αα +
of complete
α |ψ(t)" 1 2
e state
system:
|ψ(t)" system:
V (x,
= 2 (x'
$ y)%&
+$α"y=2β) + x2 y%&− 31 y 3 = 12'r 2 + 13 r 3 sin(3θ)
Total energy: E ≡ #ψ
eous quantum expectation value:
= #A(t)"
[A]th ≡ #ψ(t)|A|ψ(t)"
pectation value: #A(t)" ≡ #ψ(t)|A|ψ(t)"
2
1
y) = 2 (x
rgy: EV≡(x,
#ψ(t)|H|ψ(t)"
|ψ(t)"
2
2
+y )+x y−
1
1 3
y
3
1 2
= 2r
2
|c1α | 2
2
1/2
energy uncertainty: ∆E ≡
#ψ(t)|(H
−
E)
|ψ(t)"
V
(x,
y)
= 2 (x
2
1/2
ty: ∆E ≡ #ψ(t)|(H − E) |ψ(t)"
−βH
expectation
Trvalue:
Ae−βH[A]th
: [A]th ≡
Tr e−βH
Tr Ae
≡
Tr e−βH
mperature β determined by [H]th = E
rmined by [H]th = E
|c
|
α
Quantum
energy
unce
1 3
+ 3 r sin(3θ)
2
We assume ∆E is “sm
+ y ) + x2 y − 31 y 3 = 12 r 2
Thermal expectation
me ∆E is “small”; e.g., ∆E ∼ N −1/2 E
e.g., ∆E ∼ N −1/2 E
2
1
Aαα
1
Inverse2 temperature β
|cα |
t
#Ai (t)" = [Ai ]th
1
α
H|α" = Eα |α"
A = ap ap
#
#
#−iE
2
∗ αE
i(E
−Eβ )t
Energy eigenstates:
H|α"
=
|α"α−E
2
∗ t αi(E
)t
#
#
!
Observables Ai e.g.,
#A(t)"
|c|cαα|| AA
+
c
c
e
A
α
β
β
αβ
=αα
c
e
|α"
α
α
#A(t)"== |ψ(t)"
+
c
c
e
A
αα
β
αβ
α #
N ! 1 degrees of free
α
" α
=
β
α α"
"!
−iEα t
α"=β =
†
†
|ψ(t)"
c
e
|α"
les
Ai = aepvolution:
ap
α
ap aApi e.g.,Time Quantum state of com
α
2
2
2
1
(x
+
y
)
+
x
y
−
V
(x,
y)
=
egrees of freedom
#
#
#
#
#
#
i(E
−E
)t2
∗∗∗
i(E
−E
222
αα
β)t
β
i(E
−E
)t
α
β
Instantaneous quantu
#A(t)"
=
|c
Aαα
+ ccααccββee = [A]th AA
A
#A(t)"
#A(t)"=
= |c
|cααα|| A
αα
αβ
αβ
αβ
αα +
of complete
2
1 3
α |ψ(t)" 1 2 α"=2β
e state
system:
|ψ(t)" system:
(x
+
y
)
+
x
y
−
V$(x,
y)
=
%&
%& 3 y
%& 2 '' $$
%&
$
= 12'r'2 + 13 r 3 sin(3θ)
Total energy: E ≡ #ψ
eous quantum expectation value:
#A(t)"
≡ #ψ(t)|A|ψ(t)" = fluctuations
=
[A]
th
2
pectation value: #A(t)" ≡ #ψ(t)|A|ψ(t)"
2
1
y) = 2 (x
rgy: EV≡(x,
#ψ(t)|H|ψ(t)"
|ψ(t)"
2
2
+y )+x y−
112
1 3
y
3
1 2
= 2r
2
|c1α | 2
1/2
energy uncertainty: ∆E ≡
#ψ(t)|(H
−
E)
|ψ(t)"
V
(x,
y)
= 2 (x
2
1/2
ty: ∆E ≡ #ψ(t)|(H − E) |ψ(t)"
−βH
expectation
Trvalue:
Ae−βH[A]th
: [A]th ≡
Tr e−βH
Tr Ae
≡
Tr e−βH
mperature β determined by [H]th = E
rmined by [H]th = E
+ 3 r sin(3θ)
2
We assume
∆E is “sm
2
11 3 2 1 2
(x,x y)
= 32y(x =+2yr )
+ y )V+
y−
Thermal expectation
me ∆E is “small”; e.g., ∆E ∼ N −1/2 E
e.g., ∆E ∼ N −1/2 E
|c
|
α
Quantum
energy
unce
1 3
1
Aαα
1
Inverse2 temperature β
|cα |
t
#Ai (t)" = [Ai ]th
1
α
H|α" = Eα |α"
A = ap ap
#
#
#−iE
2
∗ αE
i(E
−Eβ )t
Energy eigenstates:
H|α"
=
|α"α−E
2
∗ t αi(E
)t
#
#
!
Observables Ai e.g.,
#A(t)"
|c|cαα|| AA
+
c
c
e
A
α
β
β
αβ
=αα
c
e
|α"
α
α
#A(t)"== |ψ(t)"
+
c
c
e
A
αα
β
αβ
α #
N ! 1 degrees of free
α
" α
=
β
α α"
"!
−iEα t
α"=β =
†
†
|ψ(t)"
c
e
|α"
les
Ai = aepvolution:
ap
α
ap aApi e.g.,Time Quantum state of com
α
2
2
2
1
(x
+
y
)
+
x
y
−
V
(x,
y)
=
egrees of freedom
#
#
#
#
#
#
i(E
−E
)t2
∗∗∗
i(E
−E
222
αα
β)t
β
i(E
−E
)t
α
β
Instantaneous quantu
#A(t)"
=
|c
Aαα
+ ccααccββee = [A]th AA
A
#A(t)"
#A(t)"=
= |c
|cααα|| A
αα
αβ
αβ
αβ
αα +
of complete
2
1 3
α |ψ(t)" 1 2 α"=2β
e state
system:
|ψ(t)" system:
(x
+
y
)
+
x
y
−
V$(x,
y)
=
%&
%& 3 y
%& 2 '' $$
%&
$
= 12'r'2 + 13 r 3 sin(3θ)
Total energy: E ≡ #ψ
eous quantum expectation value:
#A(t)"
≡ #ψ(t)|A|ψ(t)" = fluctuations
=
[A]
th
2
pectation value: #A(t)" ≡ #ψ(t)|A|ψ(t)"
2
1
y) = 2 (x
rgy: EV≡(x,
#ψ(t)|H|ψ(t)"
|ψ(t)"
2
2
+y )+x y−
112
?
1 3
y
3
1 2
= 2r
2
|c1α | 2
1/2
energy uncertainty: ∆E ≡
#ψ(t)|(H
−
E)
|ψ(t)"
V
(x,
y)
= 2 (x
2
1/2
ty: ∆E ≡ #ψ(t)|(H − E) |ψ(t)"
−βH
expectation
Trvalue:
Ae−βH[A]th
: [A]th ≡
Tr e−βH
Tr Ae
≡
Tr e−βH
mperature β determined by [H]th = E
rmined by [H]th = E
+ 3 r sin(3θ)
2
We assume
∆E is “sm
2
11 3 2 1 2
(x,x y)
= 32y(x =+2yr )
+ y )V+
y−
Thermal expectation
me ∆E is “small”; e.g., ∆E ∼ N −1/2 E
e.g., ∆E ∼ N −1/2 E
|c
|
α
Quantum
energy
unce
1 3
1
Aαα
1
Inverse2 temperature β
|cα |
t
#Ai (t)" = [Ai ]th
1
A = ap ap
α −Eβ )t
H|α"
=AEαβα:|α"
with
#A(t)" =Suppose
|cα |2 Aαα varies +
cs∗αmoothly cβ ei(E
#
#A(t)" =
#
α
α"=β
#
#
$
2
|cα | Aαα +
%&
'
$
∗
cα cβ
i(Eα −Eβ )t
e
%&
|ψ(t)" =
Aαβ
'
#A(t)" =
#
2
|cα | A
α
1
#A(t)" =
#
$
2
|cα | A
%&
VA(x,
y)"α|A|β#
=A 2=
(xap+apy ) + x y − 3 y = 2 r
αβ ≡
# Suppose Aαα varies smoothly with α:
∗
i(Eα −Eβ )t
=AEαβα:|α" |cα |2
with
#A(t)" =Suppose
|cα |2 Aαα varies +
csαmoothly cβ e H|α"
#
α
8
α"=β
V (x, y) = 21 (x2 + y 2 ) + x2 y − 31 y 3 =
#A(t)"
=
6
#
2
|cα | Aαα +
%&
$
'
#
∗
cα cβ
$
i(Eα −Eβ )t
e
%&
Aαβ
'
4
|cα |2
#
#A(t)"
E =
1
2
|ψ(t)" =
Aαα
α
5
10
15
20
25
30
2
|cα | A
α
Aαα
#
|cα | A
$
%&
#A(t)" =
0
1
2
Eα
←− ∆E −→
2
A2 = ap ap
2
1
(x, y)#
= 2 (x
2
2
3
2
2
1 3
1 2
1 3
)#
+Suppose
x y −A3ααy varies
= 2 rsmoothly
+ 3 r with
sin(3θ)
α:
∗
i(E −E )t
+y
α
=AEαβα:|α" |cα |2
wβith
#A(t)" =Suppose
|cα | Aαα varies +
csαmoothly cβ e H|α"
V
α
α"=β
8
2V (x, y) = 1 (x2 + y 2 ) + x2 y − 1 y 3 = 1 r
|cα |
2
3
2
V (x,#
y) = 12 (x2 + y 2 ) #
+ x2 y − 31 y 3 = 12 r 2 + 13 r 3 sin(3θ)
2
∗
cα cβ
|cα | Aαα +
#A(t)"
=
6
%&
$
'
$
|cα |2
i(Eα −Eβ )t
e
Aαα
%&
Aαβ
'
4
1
Aαα
Aαα
|cα |2
#A(t)"
Eα =
#A(t)" =
Eα
Eα
0
5
10
15
#
2
|cα | A
α
Aαα
Eα
2
|ψ(t)" =
20
25
30
#
2
|cα | A
$
←− ∆E −→
←− ∆E −→
%&
quantum
expectation
value:
#A(t)"
#ψ(t)|A|ψ
p Instantaneous
p
αβ #A(t)"
tantaneous
quantum
expectation
value:
≡ ≡#ψ(t)|A|ψ(
= fluctuations
N ! 1 degrees of freedom
|α" = Total
Eα |α" energy: E ≡ #ψ(t)|H|ψ(t)" A
αα
≡ "α|A|β#
αβ #ψ(t)|H|ψ(t)"
tal energy: Conclusion:
EA≡
Quantum state of complete
syste
2
1/2
#
Quantum
energy
uncertainty:
∆E
≡
#ψ(t)|(H
−
E)
|ψ(t)"
−iE
t α,
α
If
s
moothly w
ith
A
αα varies |ψ(t)"
=
c
e
α
antum energy uncertainty: ∆E
≡ |α"
#ψ(t)|(H − E)2 |ψ(t)"1/2
α
Instantaneous −1/2
quantum expectat
“small”, e.g., ∆E ∼ N
We assume
and
α ∆E isis “small”;
−1/2
E
assume ∆E is “small”;
e.g.,#∆E ∼ N
E
#
2
1
2
∗
i(Eα −Eβ )t−βH
V
(x,
y)
=
(x
+
only d
epends o
n #A(t)"
=
|c
|
A
+
c
c
e
A
Total
energy:
E
≡
#ψ(t)|H|ψ(t)"
2
then
α
αα
α β
Tr Ae αβ
α
Thermal expectation
value:
α"=β[A]th ≡
−βH
−βH
2 Tr Ae
2 Tr
1
e
V (x, y) = 2 (x + y ) + x2 y − 31 y 3 = 12 r 2 + 13 r 3
ermal expectation
value:
and must equal [A]
th ≡
Quantum
energy
uncertainty:
∆E
−βH
#
#
Tr
e
∗
i(E
2
Inverse
temperature
β
determined
by
[H]
E
α −E
β )t
thA=
c cβ e
#A(t)" =
|cα | Aαα +
αβ
α
%&
' We
$ thermalization”
%& ∆E
' “small”; e.g., ∆
assume
is
erse temperature$ β“eigenstate
determined
by [H]
=
E
2
th |c |
α
V (x, y) =
(t)" = [Ai ]th
2
1
(x
2
2
+y )+
2
1 3
1 2
xDeutsch
y − ’91,
y = r
3 M.S. ’942
1Thermal
+
1 3
r
3
sin(3θ)
expectation value: [A]th
|cα |2
Aαα
= fluc
= [A]th
#A(t)"
=
#A(t)"
#A(t)"=
=
"!
aes†p aApi
e.g., Ai =
Observables Ai e.g.,
# ∗
#
2
i(E
−E
)t
α
β)t
∗ c i(E
i(Eαα−E
−E
22 A
∗
)t
β
|c
|
+
c
e
A
β
αα +
β
αβ
αβ
c
e
A
|c
α
ccα=
c
e
A
|cααα| Aαα
β
αβ
αβ
α β fluctuations
=
[A]
th
α
α"
=
β
N ! 1 degrees of free
"
#
#
#
$$
a†p ap
!
#
%&
%&
'
A
$$
%&
%&
≡"
''
Aααof com
fluctuations
Quantum state
Aαβ ≡ =
"α|A|β#
grees of freedom 1 2
V (x, y) = 2 (x + y 2 ) + x2 y − 31 y 3 = 12 r 2 + 13 r 3 sin(3θ)
Instantaneous quantum
11 if Aαα varies smoothly with α
state of complete system: |ψ(t)"
system: |ψ(t)"
= [A]th
?
V (x, y) =
2
1
(x
2
eous quantum expectation value: #A(t)" ≡ #ψ(t)|A|ψ(t)"
pectation value: #A(t)" ≡ #ψ(t)|A|ψ(t)"
rgy: E ≡ #ψ(t)|H|ψ(t)"
α
1
|ψ(t)"
2
1/2
energy uncertainty: ∆E ≡
#ψ(t)|(H
−
E)
|ψ(t)"
2
1/2
ty: ∆E ≡ #ψ(t)|(H − E) |ψ(t)"
e ∆E is “small”; e.g., ∆E ∼ N −1/2 E
+y
2
11 3 2 1 22
y)
)+
)V+(x,
xTotal
y −=energy:
y(x=E+2≡ry #ψ(
32
2
Quantum energy unce
We
∆E is “sm
|cαassume
|2
2
1
(x
2
V (x, y) =
2
2
+y )+x
Thermal expectation v
e.g., ∆E ∼ N −1/2 E
expectation
Trvalue:
Ae−βH[A]th
[A]th ≡
Tr e−βH
Tr Ae−βH
≡
Tr e−βH
mperature β determined by [H]th = E
rmined by [H]th = E
Inverse temperature β
Aαα
t
#Ai (t)" = [Ai ]th
|c
= fluc
= [A]th
#A(t)"
=
#A(t)"
#A(t)"=
=
"!
aes†p aApi
e.g., Ai =
Observables Ai e.g.,
# ∗
#
2
i(E
−E
)t
α
β)t
∗ c i(E
i(Eαα−E
−E
22 A
∗
)t
β
|c
|
+
c
e
A
β
αα +
β
αβ
αβ
c
e
A
|c
α
ccα=
c
e
A
|cααα| Aαα
β
αβ
αβ
α β fluctuations
=
[A]
th
α
α"
=
β
N ! 1 degrees of free
"
#
#
#
$$
a†p ap
!
#
%&
%&
'
A
$$
%&
%&
≡"
?
''
Aααof com
fluctuations
Quantum state
Aαβ ≡ =
"α|A|β#
grees of freedom 1 2
V (x, y) = 2 (x + y 2 ) + x2 y − 31 y 3 = 12 r 2 + 13 r 3 sin(3θ)
Instantaneous quantum
11 if Aαα varies smoothly with α
state of complete system: |ψ(t)"
system: |ψ(t)"
= [A]th
?
V (x, y) =
2
1
(x
2
eous quantum expectation value: #A(t)" ≡ #ψ(t)|A|ψ(t)"
pectation value: #A(t)" ≡ #ψ(t)|A|ψ(t)"
rgy: E ≡ #ψ(t)|H|ψ(t)"
α
1
|ψ(t)"
2
1/2
energy uncertainty: ∆E ≡
#ψ(t)|(H
−
E)
|ψ(t)"
2
1/2
ty: ∆E ≡ #ψ(t)|(H − E) |ψ(t)"
e ∆E is “small”; e.g., ∆E ∼ N −1/2 E
+y
2
11 3 2 1 22
y)
)+
)V+(x,
xTotal
y −=energy:
y(x=E+2≡ry #ψ(
32
2
Quantum energy unce
We
∆E is “sm
|cαassume
|2
2
1
(x
2
V (x, y) =
2
2
+y )+x
Thermal expectation v
e.g., ∆E ∼ N −1/2 E
expectation
Trvalue:
Ae−βH[A]th
[A]th ≡
Tr e−βH
Tr Ae−βH
≡
Tr e−βH
mperature β determined by [H]th = E
rmined by [H]th = E
Inverse temperature β
Aαα
t
#Ai (t)" = [Ai ]th
|c
= fluc
= [A]th
#A(t)"
=
#A(t)"
#A(t)"=
=
"!
aes†p aApi
e.g., Ai =
Observables Ai e.g.,
# ∗
#
2
i(E
−E
)t
α
β)t
∗ c i(E
i(Eαα−E
−E
22 A
∗
)t
β
|c
|
+
c
e
A
β
αα +
β
αβ
αβ
c
e
A
|c
α
ccα=
c
e
A
|cααα| Aαα
β
αβ
αβ
α β fluctuations
=
[A]
th
α
α"
=
β
N ! 1 degrees of free
"
#
#
#
$$
a†p ap
!
#
%&
%&
'
A
$$
%&
%&
''
≡"
?
Aααof com
fluctuations
Quantum state
Aαβ ≡ =
"α|A|β#
grees of freedom 1 2
V (x, y) = 2 (x + y 2 ) + x2 y − 31 y 3 = 12 r 2 + 13 r 3 sin(3θ)
Instantaneous quantum
11 if Aαα varies smoothly with α
state of complete system: |ψ(t)"
system: |ψ(t)"
= [A]th
?
V (x, y) =
2
1
(x
2
eous quantum expectation value: #A(t)" ≡ #ψ(t)|A|ψ(t)"
pectation value: #A(t)" ≡ #ψ(t)|A|ψ(t)"
rgy: E ≡ #ψ(t)|H|ψ(t)"
α
2
1/2
energy uncertainty: ∆E ≡
#ψ(t)|(H
−
E)
|ψ(t)"
2
1/2
ty: ∆E ≡ #ψ(t)|(H − E) |ψ(t)"
e ∆E is “small”; e.g., ∆E ∼ N −1/2 E
e.g., ∆E ∼ N −1/2 E
[A]th ≡
Tr e−βH
Tr Ae−βH
≡
Tr e−βH
mperature β determined by [H]th = E
rmined by [H]th = E
11 3 2 1 22
y)
)+
)V+(x,
xTotal
y −=energy:
y(x=E+2≡ry #ψ(
32
2
= [A]Quantum
energy unce
th
1
|ψ(t)"
expectation
Trvalue:
Ae−βH[A]th
+y
2
We
∆E is “sm
|cαassume
|2
= fluctuations
2
2
1
2
V (x, y) = 2 (x + y ) + x
Thermal expectation v
≡“"α|A|β#
if Aαβ is small”
Inverse temperature β
Aαα
α
Aαα
t
#Ai (t)" = [Ai ]th
|c
Aαβ ≡ "α|A|β#
= fluctuations
= [A]th
A
αα
A
≡
"α|A|β#
αβ
Is it ever reasonable to expect
= fluctuations
1) Aαα varies smoothly with α
is “"α|A|β#
small”
2) αAαβ ≡
Aαα
α
V (x, y) =
?
V (x, y) = 12 (x2 + y 2 ) + x2 y − 31 y 3 = 12 r 2
2
V (x, y) =
2
1
(x
2
2
|cα |2
+y )+x y−
Aαα
|cα |
2
1 3
y
3
=
= fluctuations
Aαβ ≡ "α|A|β#
= [A]th
A
αα
A
≡
"α|A|β#
αβ
Is it ever reasonable to expect
= fluctuations
1) Aαα varies smoothly with α
is “"α|A|β#
small”
2) αAαβ ≡
Aαα
α
?
V (x, y) =
V (x, y) = 12 (x2 + y 2 ) + x2 y − 31 y 3 = 12 r 2
Yes!
Known to be true
2
|c
α|
2
2
2
1
1 3
for
V (x, y) = 2 (x + y ) + x y − 3 y =
quantized chaotic systems
Aαα
|cα |
2
is independent of the cα ’s
A = h̄-independent operator
A = h̄-independent−1/2
operator
=⇒ ∆E ∼ N
E
Shnirelman’s
theorem:
"α|A|α! → classical, microcanonical average of A
"α|A|α! → classical,
average of A
=⇒microcanonical
A = �A�T
+ O(h̄1/2 )
if system is chaotic and A = h̄-independent operator
is independent
∆Ck,Rof
(s)the cα ’s
A = h̄-independent operator
A = h̄-independent−1/2
operator
=⇒H|α!
∆E =
∼E
Nα |α! E
Shnirelman’s
theorem:
"α|A|α! → classical, microcanonical average of A
"α|A|α! → classical,
average of A
=⇒microcanonical
A = �A�T
A = h̄-independent
operator
1/2
+ O(h̄
)
if system is chaotic and A = h̄-independent operator
"α|A|α! → classical, microcanonical average of A
1/2
+ O(h̄ )
But all we need is
"α|A|α! = O(h̄0 ), varies smoothly with Eα
Aαβ = "α|A|β! varies erratically with α and β
0
O(h̄ ) = "α|(A−"A!)2 |α! =
!
"α|A|β!"β|A|α!
SoE –SoE
Random
wave wave
modelmodel
– chaotic
systems
IV
–
Random
–
chaotic
syst
Physical intuition: Berry’s random-wave conjecture
for energy eigenfuctions
th
Berry ‘77
Baecker ‘03
th
6000 6000
eigenfunction,
eigenfunction,
0
"α|A|α!
=
O(h̄
) with Eα
�α|A|α�
varies
smoothly
Randomvaries
wavesmoothly
conjecture
�α|A|α�
with Eα
"α|A|β!
varies
erratically
with
α and
β β
αβ =
If A
A
=
�α|A|β�
varies
erratically
with
α
and
If αβ
Aαβ = �α|A|β� varies erratically with α and β
�
�
22
�α|(A−�A�)
�α|A|β��β|A|α�
�α|(A−�A�) |α�
|α� =
=
�α|A|β��β|A|α�
β�=
=α
α
=
=
�
�
β�=α
22
|A
|
|Aαβ |
αβ
β�=α
Ē)
2
|A
|
∼ eS(
αβ
S(Ē)
2
|Aαβ |
∼e
Ē)/2
=⇒ Aαβ ∼ e−S(
−S(Ē)/2
=⇒ Aαβ ∼ e
†
Let AVery
= asmall!
p ap
Let A =
t
†
ap ap
!
"
^
tion. The long-time average of Aðt Þ is then
! " X
^ ~
A
jCa j2 Aaa
n(kx = 0)
Numerical investigations:
Coherence
If the quantum-mechanical mean of an observable behave
mally it should settle to the prediction of an appropriate stat
g
mechanical ensemble. For our numerical
experiments we ch
ð2Þ
Ea Thermal
Initial
monitor the marginal momentum distribution along the hor
a
axis n(kx) and its central component
n(kx 5 0) (see
Supplem
Initial
state
Eb Thermal
We note that if the system relaxes at all, it must be to this value. We Initial
state
E Information). In Fig. 1b, c we demonstrate that both relax t
find it convenient to think of equation (2) as stating the prediction of
Thermal
Ec diagonal-ensemble
microcanonical predictions. The
predictio
the same as these, but the canonical ones, although quite clo
w
b 2.0
a
not. This is an indication of the relevance of finite-size effectso
may be the origin of some of the apparent deviations from
S
modynamics seen
in the recent
numerical
studies of refs 4 an
Eigenstate
thermalization
Relaxation
n
1.5
The statement that the diagonal and
microcanonical ens
dynamics
^ reads
Figure 2 | Thermalization
in classical
versus
quantum
mechanics.
a, In[
give the same
predictions
for the
relaxed
value
of A
Diagonal
Initial
e
classical mechanics, time
constructs
the
thermal
state
from
an
Xevolution
X
Microcanonical
1
2
1.0
j
C
j
A
~
h
A
i
ð
E
Þ
:
Ain
a no
aaresemblance
0 the
a
b
,
In
quantu
initial state that generally bears
former.
microcan to
Canonical
N
E
,
DE
a
0
s
a
mechanics, according to the ETH, every eigenstate of the hamiltonian
alwa
jE0 { Ea j v DE
d
implicitly containswhere
a thermal
state.
The
coherence
between
the
eigenstat
E0 is the mean energy of the initial state, DE is the half
t
0.5
initially hides it, but
time
dynamics
reveals
it
through
dephasing.
chosen energy
window centred at tE
0
50of an appropriately
100
150
200
Supplementary
Discussion), and the normalization N E0 , DE
tJ
t
Relaxation
b 2.0
1.0
0.5
c 2.0
n(kx)
1.5
1.0
number of energy eigenstates with energies in the w
[E0 2 DE, E0 1 DE]. Thermodynamical universality is evident
Non-integrable
a
equality:
although the left-hand side depends on the detailst
2.0
Microcanonical
initial conditions through the set of coefficients Ca, the righ(
Canonical
state
1.5
H
side depends only on the total energy,Initial
which
is the same for
Diagonal/relaxation
different initial conditions. The following
three scenarios ts
dynamics
1.5
themselves as possible explanations of
this universality (ass
Diagonal/
p
Microcanonical
Microcanonical
that the initial state is sufficiently narrow
in energy, as is no
r
1.0
Canonical
Eigenstate a
the case—see Supplementary Discussion).
First, even for eigenstates close in energy,
there areblarge eigejC
Eigenstate
1.0
to-eigenstate fluctuations of both the eigenstate expectation
in
0.5
(EEVs)
A
and
the
eigenstate
occupation
numbers
(EONs)
aa
–2
–1
0
1
2
Initial state
However,
forxd)]physically interesting initial conditions, the f
kx[2π/(L
0.5
Diagonal/relaxation
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0.5
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Figure
1 | Relaxation
dynamics.
alattice
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on which
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Figure 1 | Relaxation
dynamics.
Figure
a1 ,| Two-dimensional
Relaxation
dynamics.
a,on
Two-dimensional
which fivelattice lattice
on which
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dynamics
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which,which,
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,
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the
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0
Deutsch
and Srednicki independently
proposed
Deutsch and Srednicki
r(E)
Canonical
r(E)
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The
ETH
suggests
that
classical
and quantum
thermal
states states
have
The
suggests
that
classical
The
ETHand
suggests
quantum
that
thermal
classical
states
and
quantum
have
thermal
have
distribution
function
in the
initial
state,
after
relaxation,
and inand
the
different
distribution
function
in the
distribution
initial
state,function
after
relaxation,
in the initial
and
state,
in the
after
different
relaxation,
inETH
the different
0)
of the
marginal
momentum
dynamics
of the
central
component
n(k
5
0)
of
the
marginal
momentum
5
0)
of
the
marginal
momentum
dynamics
of the
component
dynamics
of
n(k
the
central
component
n(k
x 5central
x
x
which, following Srednicki,
we call
the ‘eigenstate
th
which,
following
Srednick
wh
1.0
5 5 Lis
the5 lattice
width.
Here
dHere
is the
constant
and width.
Lxand
5 5lattice
is the constant
lattice
is very
thewith
lattice
ensembles. Here d isensembles.
the lattice
ensembles.
constant
and
d lattice
is
Lxthe
#
#
veryofas
different
natures,
as
depicted
inat
Fig.
Although
at
present
there
different
natures,
very
depicted
different
in Fig.
natures,
2.plotted
Although
depicted
present
in2.Fig.
there
2.plotted
Although
at
present
there
!
distribution,
compared
the width.
predictions
the
three
ensembles,
distribution,
compared
with
distribution,
the
predictions
compared
ofas
the
with
three
the
ensembles,
predictions
of
the
three
ensembles,
plotted
x5
12,13
12,13
#
#
^
thesis
(ETH)’
thesis
(ETH)’
th
:
the
expectation
value
Y
A
:
the
exp
against ‘dimensionless time’
(in our
conventions time’
J, the
hopping
parameter,time’
against
‘dimensionless
against
(in
our
‘dimensionless
conventions
J, the(in
hopping
our conventions
parameter,
J, the hopping
parameter, a
855 855
855
^ in Information).
^ in an
has
units
of inverse
time;
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Information).
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has
units
of inverse
time; see
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inverseInformation).
time; seeobservable
Supplementary
In the A
the A
0.5
eigenstate
t
an energy-E
observable
jY energy
a i ofob
aIn
Nature
Nature
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©2008see
©2008
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Publishing
Publishing
Publishing
Group
Group
Group
microcanonical case, we averaged
over all eigenstates
whose over
energies
lie we averaged
microcanonical
case, we averaged
microcanonical
all case,
eigenstates
whose
over
energies
all
eigenstates
lie
whose
energies
lie
large, interacting many-body
equals
thelar
th
large, system
interacting
many-bo
Diagonal
Many systems do not obey ETH:
Near-integrable systems
(require generalized Gibbs ensembles)
Rigol, Dunjko, Yurovsky & Olshanii (2007)
Many-body localized systems
For these, thermalization requires
initial state restriction
(or: external interactions)
SHM:
Becattini 2009
Urprinzip:
“Every multihadronic state
localized within the cluster
compatible
conservation
Figure 1: Highand
energy
collisions arewith
assumed
to give rise tolaws
multiple clusters at
is equally
likely.”is a colorless extended massive
hadronization stage [top]. Each
cluster [bottom]
the
object endowed with abelian charges (electric, strange, baryonic etc.), intrinsic angular
momentum and other quantum numbers such as parity, C-parity and isospin.
cluster
might
created
whose
state
is then
necessarily
a pure
cluster
might
be be
created
whose
state
is then
necessarily
a pure
one.onA
the
initial
conserved
quantities,
i.e.superposition of all l
postulate
of
the
SHM,
this
must
be
an
even
postulate of the SHM, this must be an even superposition of all loca
!
the
initial
conserved
quantities,
i.e.
State
inside
a
cluster
of
volume
V:
the initial conserved quantities, i.e.
|ψ!
=
c
with
|c
hV Pi |hV !
hV
!
!
quantum mechanics
problems
of decoherence2 and measu
h
2h | = const .
|ψ!
= ch cPhVi |h
PVi |h
with
|c
V !V with
|ψ!
=
!
|c
|
=
const
.
hV inVmeasurement
stance here.
It suffices
to realize that
some
low-ener
V of decoherence
quantum mechanics
problems
and
hV
h
V
cluster
might
bemeasurement,
created that
whose
is then
s problems
decoherence
andto
we
take
a|f pragmatic
stanceofhere.
It suffices
realize
instate
some
low-energy
The
probability
of
observing
a final
state
!necessarily
is thencol
postulate
of the
SHM,state
thiscollision
must
be necessarily
an
even superpositio
ffices
to
realize
that
in
some
low-energy
events,
only
one
cluster
might
be
created
whose
is
then
a
pure
The
probability
of
observing
a
final
state
|f
!
is
then
!
The probability of
observing
a final
|f
! isi.e.
then
Probability
to
see astate
final
state
f
:
the
initial
conserved
quantities,
2 to the
2
eated postulate
whose state
is then
necessarily
a
pure
one.
According
of the
SHM,
this|"f
must
be
an
even
superposition
of a
"f
|P
|h
!c
|
|ψ!|
=
|
i
V
h
!
V
!
!
2
2
2
2
HM, this
an|"f
even
superposition
of
all
localized
with
themust
initialbeconserved
quantities,
i.e.
2
"f
|P
|h
!c
|
|"f
|ψ!|
=
|
"f |P
|hhVV Pi |hV ! states
|ψ!| = |
i |h=
Vi !cVhVhV
|ψ!
c
with
|c
|
=
hV
!
!
hV
d quantities, i.e.
2
hV!
h
V
=! const
|"f
|P!
i |hV !| +
2 "f |Pi |h
!
!
c|h
with
|chV!"h
const
hV2 P
i2|h
V!
" | ! "=|P
! = = const
const|ψ!|"f=|"f
|P
!|
+
"f
|P
|h
|f
!c
|P
|h
"f
|P
|h
|P
|f
!c
c
i2 !|VhV+
i !"h
VV V
i
h
=
!
h
V
i
V
i
V
i
h
V
V
The
probability
of
observing
a
final
state
|f
!
is
then
hV | = const .
|ψ! =
chV Pi |hIfVwe
! assume
with
|c
(11)
!
h
!
h
V
h
=
!
h
V
h
V
h !=h
V
V
V
!
hV
2random phases, the2 last term
If
the
coefficients
c
have
and
h
|"f
|ψ!|
= | |f"f! |P
V
V !chV |
The probability of observing a final state
isi |h
then
V
If the
coefficients
have
random
phases,
the
last
term
in
Eq.
(12)
If the
coefficients
chVchhave
random
phases,
the
last
term
in
Eq.
(12)
van
V
left with
the
same
expression
appearing
in
Eq.
(7);
in
hV
!!
observing
a the
final
state
|f
! is then
!
with
the
same
expression
appearing
in
Eq.
(7);
in
other
words
leftleft
with
same
expression
appearing
in
Eq.
(7);
in
other
words
an
2
2
2
description
is
recovered.
Hence
a
new
hypothesis
is
int
= = |const"f |P|"f
|P
|h
!|
+
"f
|P
|h
|h
!c
|
|"f
|ψ!|
i
V
i
V!
then
i
V
h
V
description
is recovered.
Hence
a
new
hypothesis
is
introduced
in
t
!iscluster
description
recovered.
Hence
a
new
hypothesis
is
introduced
in
the
S
!
is
a
pure
state,
the
superposition
of
multi-had
hV
hV !=hV
2
2
hV
"f
|Pistate,
|hVstate,
!chthe
| superposition
(12)
"f |ψ!|
= |is is
cluster
a pure
superposition
of multi-hadronic
localize
cluster
a pure
of multi-hadronic
localized
s
V the
!
!
Becattini
2009
random phases.
2
"
h
=
const
|"f
!| + phases,
"f |P
|
random
Vphases.
random
phases.
i |hV random
i |hlast
V !"h
If the coefficients
chV|Phave
the
term
V |Piin
!
!main goal of the model is! to "
Now the
determine the
2goal
h
Now
main
goal
of"f
the
is" determine
to
the
probabilitie
hV !=hthe
same
appearing
Eq.
(7); in oth
main
of the
the
model
is
to
(7
V model
const Now
|"fthe
|Pthe
|h
!|left
+with
|P |hexpression
!"h
|P determine
|f
!c
cV∗ in
.probabilities
Recast in ETH language:
Af
Observables Af
Assume |ψ! has “small” energy uncertainty
!
!
2f
!
t
(X | U |X) = δ (X − M (X))
2f
!
t
(X | U |X) = δ (X − M (X))
t
Xt = M (X0 )
t
Xt = M (X0 )
t
det ∇i Mj = 1
!
2f
t
det ∇i Mj
"
=1
#
t
d" X log of e’value of ∇i M
=
λ
t
i
j
#
2f
t
d X log of e’value of ∇i Mj = λi t
Recast in ETH language:
Af
Observables Af
Assume |ψ! has “small” energy uncertainty
!
2f
!
t
(X | U |X) = δ (X − M (X))
ETH
!
2f
! !ψ|At f |ψ" # [Af ]th
(X | U |X) = δ (X − M (X))
t
Xt = M (X0 )
t
Xt = M (X0 )
t
det ∇i Mj = 1
t
det ∇i Mj
"
Af
|ψ"
!
#
2f
t
d" X log of e’value of ∇i M
=
λ
t
i
j
#
!
2f
!
t
2f
t
(X |of
U |X)
=j δ =(Xλi t− M (X))
d X log of e’value
∇i M
=1
Recast in ETH language:
Af
Observables Af
Assume |ψ! has “small” energy uncertainty
!
2f
!
t
(X | U |X) = δ (X − M (X))
ETH
!
2f
! !ψ|At f |ψ" # [Af ]th
(X | U |X) = δ (X − M (X))
t
Xt = M (X0 )
t that ETH applies
Af to the cluster
But:
Showing
Xt = M (X0 )
t problem!
is
a
very
hard
det ∇ M = 1
i
t
det ∇i Mj
"
j
|ψ"
!
#
2f
t
d" X log of e’value of ∇i M
=
λ
t
i
j
#
!
2f
!
t
2f
t
(X |of
U |X)
=j δ =(Xλi t− M (X))
d X log of e’value
∇i M
=1
Conclusions:
Conclusions:
SHM consistent with
clusters obeying ETH
Conclusions:
SHM consistent with
clusters obeying ETH
Clusters obeying ETH is plausible
Conclusions:
SHM consistent with
clusters obeying ETH
Clusters obeying ETH is plausible
But will not be easy to prove!