Density matrix renormalisation applied to nonlinear dynamical systems [Elektronische Ressource] / vorgelegt von Thorsten Bogner
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English

Density matrix renormalisation applied to nonlinear dynamical systems [Elektronische Ressource] / vorgelegt von Thorsten Bogner

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Published 01 January 2007
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Bielefeld
he
Ph
ysik
K
ondensierte
Materie
Densit
Univ
or
Thorsten
an
F
für
ysik
y
v
Matrix
August
Renormalisation
der
Applied
akultät
to
Ph
Nonlinear
der
Dynamical
ersität
Systems
v
Dissertation
gelegt
zur
on
Erlangung
Bogner
des
2007
Doktorgrades
TheoretiscPrin
ted
on
pap
er
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9706
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ation
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erage
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of
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Results
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10.
8.1.
V
.
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n
.
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.
w
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83
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of
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Results
.
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101
.
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79
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83
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77
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v
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10.3.
.
cal
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tegration
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Correlation
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cal
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8.2.2.
Snapshots
79
the
D
w
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Mo
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ts
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er
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Sim
.
.
.
.
.
.
108
.
of
.
Sw
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
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.
.
.
.
.
Eects
.
Dieren
80
Num
.
ers
.
R
.
tained
-Mo
.
Sim
.
.
.
of
111
.
of
KPZ
Spatial
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
78
V
.
onal
.
v
.
the
dels
ectral
.
arian
.
.
.
.
Neig
.
9.
.
.
113
er
aluation
.
the
Decomp
e
.
.
DMRG
.
b
.
9.1.
.
.
ten
8.
73
.
.
the
.
.
.
Linear
.
Diusion
.
Equation
.
.
Visualisation
.
the
.
and
.
Mo
.
.
.
.
.
.
.
.
.
Visualisation
.
the
.
Ev
.
.
.
.
.
.
.
.
.
.
.
.
.
iii
.
Con
σ
ηlds
.
124
D.2.
.
.
.
.
V
Represen
Scal
.
ection
.
.
C.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Kernel
.
.
.
.
ects
D.
.
.
.
.
.
.
t
.
122
.
.
.
.
.
.
.
.
.
.
.
.
.
.
POD
.
Systems
.
POD
.
.
.
.
143
.
.
.
144
123
.
.
Schur
.
139
uture
.
.
.
.
.
.
.
G
.
Precision
.
.
.
.
.
.
.
.
.
ects
.
125
.
.
Axioms
.
.
.
.
.
.
.
Norm
y
.
.
.
.
Basis
.
.
.
123
.
.
D.2.3.
.
.
Reduction
.
.
.
Time
ar
.
.
.
.
.
.
.
.
.
.
.
.
.
.
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.
ur
.
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.
.
uture
.
121
of
.
osition
.
A
.
Basic
.
.
.
.
.
.
.
.
ects
.
.
.
.
.
.
D.1.1.
121
.
Finite
.
.
.
Fixed-P
.
.
.
tation
.
.
.
.
139
.
.
.
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uture
.
.
.
.
.
.
.
.
.
oin
.
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ector
.
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.
.
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.
140
.
pro
125
.
.
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.
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.
141
.
.
.
.
.
.
.
.
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.
.
.
V
.
.
.
.
.
B.
.
.
.
Linea
.
122
.
B.1.
.
.
D.2.4.
Pro
ransformations,
.
.
.
.
.
.
.
142
G
Rank
.
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.
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.
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.
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T
.
.
.
.
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.
Conclusions
.
.
.
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.
.
.
.
.
F
.
.
.
Prosp
130
.
Ordering
.
the
.
Decomp
.
135
F
Mathematical
.
ddenda
.
D.1.
.
notation
DMR
.
.
.
.
.
.
.
Prosp
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
139
.
Fie
ts
.
A.
.
.
.
Numerical
.
.
.
125
.
.
.
oin
.
.
.
Represen
.
.
.
.
.
.
.
.
.
.
.
.
D.1.2.
.
ectors
.
.
F
.
.
.
.
.
.
.
Prosp
.
.
.
.
.
.
.
.
.
.
.
.
.
Floating-P
.
.
.
t
140
11.1.
V
tation
Space
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
D.2.1.
.
ar
.
duct,
.
.
Stabilit
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
D.2.2.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
11.3.
.
.
.
.
.
.
142
ariational
Subspaces
.
.
.
.
127
.
.
.
Optimal
.
.
.
of
.
.
.
r
.
.
.
129
.
.
.
Long
.
11.2.
142
Optimised
Line
.
T
j
Matrixes
h
.
.
.
.
.
.
.
DMR
.
.
.
.
.
.
Range,
.
and
.
.
.
.
.
.
.
.
.
.
.
ten
11.
Sc
.
.
.
.
.
.
.
.
.
.
.
.
.
.
129
D.2.5.
B.2.
ensors
Short
.
Time
.
Optimised
.
Pro
.
jec
.
t
.
i
.
on
.
.
.
.
.
.
.
.
.
.
.
.
.
.
iv
.
ConMicroscopic
t
from
142
30
2
P
um
olynomial
Discretised
erator
103
e
ar
t
resp
26
ct
b
e
matrix
Kronec
(ma-
step
um
21
with
and
uid
on
with
equal.
8
n
deriv
ansatz-functi
16
conjugated
e
Matrix
basis
transformed
precision
columns
Characteristic
er),
asymptotic
and
er
from
Landau
The
17
143
step
er
Nabla
ativ
in
ect
conjugated
y
ha
140
142
ativ
b
ect
op
scal
Scalar
ultiplicativ
eld
-th
0
e
Complex
to
Global
In
um
deriv
ld
resp
(n
state
formed
y
Notation
Mac
the
eld
b
b
of
p
ers
9
e
for
adjungated
k
matrix
delta
ourier
notation
.
Time
trix)
size
n
26
F
Spatial
b
size
Elemen
op
of
T
deriv
ws
e
Complex
resp
26
to
27
13
eld
viscosit
5
Dynamic
v
-th
teger
deriv
to
e
Laplace
resp
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to
ansatz-function
16
69
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erator
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142
(m
partial
ry
Description
In
to
26
ro
otal
Absolute
ativ
v
with
alue
ect
13
erse
complemen
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Orthogonal
36
18
artial
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ativ
viour
with
Time
e
dep
to
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13
t
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co
103
ecien
13
t
hine
26
126
a
Scalar
h
v
total
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i :j i j
†·
∗·
Δ
[A,B] A
B A B
|·|
α (t)i
χ(x)
δij
Δt
Δx
d t
dtnd n tndt
n∂ n xn∂x
∂ x
∂x
V
M
φ
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λ
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Unit
metho
a
Cellular
ector
38
t
tiplied
o
pro
extended
F
in
Arc
all
orticit
of
31
Op
Sum
erator
oating
summand
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-th
ransform
in
(also
t
1
v
141
the
matrix
POD
cir-
51
26
b
40
resp
t
cot
in
erator
G
t
Sto
inhomo-
whic
deriv
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and
oi
state
um
m
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[10,
after
y
69
matrix
Extended
1
v
of
d
matrix
ordinate
c
in
T
tional
wn
69
unit
est
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v
DFT
d
of
correlation
region
Set
v
n
Pseudo-scalar
125
constan
35
103
e
State
ativ
v
second
circumference
for
40
matrix
normalisation
condition
DMR
op
Eucl
Diric
in
t
c
the
h
e
Basis
rst
a
system
p
last
n
space
n
is
b
37
125
ul-
DE
diameter
basis
b
24
r
iteration
30
dimensions
93]
w
30
basis
Scalar
the
1
ariational
Ratio
53
y
69
du
in
142
Orthonormalised
kno
basis
1
the
t
aria-
ourier
POD
a
d
Discrete
co
y
T
y
basis
as
the
142
ariational
of
metho
9
69
Description
Spatial
of
matrix
state
ect
matrix
of
himedes'
complex
a
um
ector
ers
v
CA
cle
automaton
y
cos
Densit
with
of
Cosine
t)
32
Notation
group
and
idean
hastic
the
tial
Determinan
deriv
V
the
ortici
of
t
32
y
Diagonal
103
9
det
26
General
Left/righ
matrix
diusion
143
erator
13
hlet
Creation
geneous
op
h
erator
Op
38
for
equation
second
General
ativ
basis
wi
142
vi
metho
Glossary
π
Ψ
h·,·i
h·|
|·i
X0
1
2
k×kk
∂V V
ω
ω
A
a
B
B
iB i
i0B
i00B
Bnew
C
C
(x) x
(x) x
D
Dl/r
2Dd,inhom
2DDx/y
x/y
(A) A13
um
wind
26
g
w
op
to
constan
ativ
deriv
um
with
t
wn
in
e
space
142
30
8
29
ativ
Imaginary
ec
kron
p
wind
1
21
arisi-Zhang
to
p
hlet
KPZ
the
space
e
ourier
Matrix
F
the
F
22
T
Second
ev
tred
functi
ativ
step
with
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with
e
to
onen
a
triangular
t
to
er
ativ
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ous
103
ec
Exp
li
a
21
t
erator
er
Matrix
functio
deriv
and
deriv
143
26
of
24
de
resp
n
do
of
Natural
F
ativ
T
21
56
General
function
deriv
of
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16
deriv
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deriv
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ln
d
e
91
e
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resp
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e
ativ
t
22
re
Exp
x
t
of
matrix
oating
ect
oin
Up
n
inhomogene
b
deriv
125
x
Kinetic
e
r
Op
y
resp
(equation)
near
Kardar-P
t
onen
Diric
of
x
oating
Do
32
er
n
logarithm
b
Lo
125
wind
General
for
n
condition
phase
second
parameter
ativ
15
ativ
erator
for
an
22
F
with
mo
duct
31
Litre
ull
for
Kernel,
ect
unction
wn
Kern
Description
FFT
deriv
ast
x
ourier
e
ransform
transformation
t
Notation
of
35
with
sp
21
bias
pro
est
er
26
Matrix
Generator
for
time
cen
olution
tr
Boltzmann
e
matrix
d,
General
symmetric
on
deriv
F
ativ
140
e
Time
22
size
k
el
Kronec
unit,
Matrix
20
for
op
up
5
wind
vii
oin
Glossary
Dd,inhom
2Dx
Dx,d
Dx,s
Dx,u
2Dd
Dd
Ds
Du
E
E
e
F
F (t)k
f(y) y

G
g(x) x
ht

i i = −1
J
K
K
kB
(·)
(·)
L
L
l
(x) x3
equation
Source-annihilation-diusion
Set
ONB
1
Probabilit
sin
Re
Geometric
of
35
b
op
reduction
erator
(pro
conguration
erio
densit
y
35
29
125
y
Line
n
ector
tial
10
32
grees
P
op
Probabilit
ector
103
scal
n
to
er
t
er
op
renormalisation
Mass
at
6
constan
um
n
ultiplici
op
In
des
Diagonal
24
v
a
b
b
Ordinary
n
n
sinc
basis
of
pro
Mo
7
freedom
dieren
p
Sign
Normal
for
of
op
le
b
35
for
erp
microsc
142
POD
plane
decomp
tissa
G
b
decomp
m
y
ultiplicit
65
n
atmosphere
6
lev
matrix
part
matrix
35
of
all
m
b
32
143
t
of
b
no
6
in
teger
-direction
Dimensionalit
125
A
of
ogadro
um
um
v
er
125
ODE
space,
dieren
er
equatio
um
13
oating
Orthonormal
er
2
MR
General
de-
jection
Master
erator
of
PDE
del
artial
141
tial
Man
1
cess)
38
2
y
v
state
erator
40
with
Source
General
er
ngth
um
oin
y
p
the
ar
-th
endicular
opic
dic
36
some
Prop
Sine
orthogonal
26
osition
Description
POD-DMR
Num
Prop
Algebraic
orthogonal
er
osition
P
matrix
Notation
function
a
er
n
y
bit
erator
group
38
Reynolds
SAD
Standard
15
pressure
Num
sea
b
el
er
37
of
Gas
no
t
des
Real
in
of
t
real
-direction
um
24
ers
men
Range
ele
erator
Num
Range
b
an
er
viii
Glossary
M
M
M
m
μ
na
ng
N
nˆ 1
n
N x1
N y2
Na
P
P ii
p(|ii) i
Ps
∗R
R
(·)

S
s
s
Σ
(x) x
SN