Timesymmetry Breaking and Complex Spectral Representation of the Liouvillevon Neumann Equation: I

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Time-symmetry Breaking and Complex Spectral
Representation of the Liouville-von Neumann
Equation I
Tomio Petrosky Center for Complex Quantum
Systems, The University of Texas at Austin
Tay Buang Ann (Universiti Putra Malaysia) Valeri
Bersegov (Boston University) Naomichi Hatano
(University of Tokyo) Gonzalo Ordonez, Butler
University Kazuk Kanki (Osaka Prefecture
University) Gonzalo Ordonez (Butler
University) Satoshi Tanaka (Osaka Prefecture
University)
Aug. 17, 18 2009
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Comples Spectral Representation of the Liouville
Operator LH

• General formulation
• 2) Examples
• 2a) Quantum Lorentz Gas
• Exact solution of the eigenvalue problem of LH
• 2b) Protein Chain

Alien Baltan
Hofstadters Butterfly (Spectrum of the
Hamiltonian)
Protein Chain (Spectrum of the Liouvillian)
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Example 1 Lorentz gas
Pachinko Laboratory (New model)
Old model of the Lorents gas (Pachinko Japanese
pinball)
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Example 2 Protein Chain
?-helix
3D structure of Myoglobyn
Primary Structure One-dimensional Molecular Chain
S.V. E. Philips, J. Mol. Bio. 142, 531 (1980)
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General formulation
Poincarés Nonintegrability
Complex spectral analysis of Liouville-von
Neumann operators in a non-Hilbert space that
includes d-function singularities
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The Liouville - von Neumann Eq.
Fourier analysis
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Poincarés Nonintegrable Systems

(Liouvillian)
Elimination of intaractions by a Canonical
Transfomation (Small denominator problem)
Perturbation Analysis
Resonance singularity
Continuous spectrum
odd
New symmetry
even
odd
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Time symmetry
Inner product in the Liouville space
Hilbert norm in Thermodynamic limit(T-lim)
Ideal gases in the high-temperature limit The
Boltzmann gas
Not in the Hilbert space
The room temperature
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Thermodynamic limit
Extended function space
Broken time-symmetry
Complex eigenvalue problem for Liouvillian
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Thermodynamic limit and Non-local distribution in
space
Number density in space
n
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Class of distribution functions in the
thermodynamic limit (T-limt)
?-function singularities
Volume dependence of Fourier coefficients
Complicated ?-function singularities
Not in the Hilbert space
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Class of distribution functions in nonequilibrium
?-function singularities gt Intensitive variables
Not in the Hilbert space
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Solusion of the eigenstates (Brillouin-Wigner
type)
Creation- and destruction-of-correlation opetators
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Collective modes (Subdynamics)
Complete orthonoumal system
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Summary

• For unstable dynamical systems with Porincarés
  resonane and
• with the ? singularities in the wave-vectors,

irreversibility is described by the resonance
states of LH outside the Hilbelt space
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2) Technically the most difficult part is to
solve the eigenvalue problem of the collision
operator.
Creation and destruction parts of the correlation
component are relatively easy to calculate.
3) The evolution in each correlation-subspace is
described by Markov equation without any
approximation.
4) Non-Markov evolution is described by a
superposition of the Markov evolutions in
each correlation-subspaces.
5) Examples will be presented in the several
lectures in this work shop, as well as my
next lecture.
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Time-symmetry Breaking and Complex Spectral
Representation of the Liouville-von Neumann
Equation II
Tomio Petrosky Center for Complex Quantum
Systems, The University of Texas at Austin
Aug. 17, 18 2009
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Examples

• Quantum Lorentz Gas
• 2) Protein Chain

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Quantum Lorentz Gas
A test particle with light mass m Many
scatterers with heave mass M
a) m/M ltlt 1
b) The scatterers are uniformly distributed in
space
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Quantum Lorentz Gas (d-dimensional weakly coupled
case)
1D case
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Exact solution of the eigenvalue problem of
1D case
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Exact right- and left-eigenstates of
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Expansion by the phase mixing
Expansion by the diffusion process
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Time evolution of observables
In the Wigner representation
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Example Quantum Kinetic Equation for 1D Protein
Chain
??helix
3D structure of Myoglobyn
Primary Structure One-dimensional Molecular Chain
S.V. E. Philips, J. Mol. Bio. 142, 531 (1980)
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Exciton
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Dimensionless Hamiltonian
Reduced density matrix
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Eigenvalue problem of the collision operator
H-theorem
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resonance condition
2m 10
m 6
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t lt 0
t gt 0
Alien Baltan
Rich structure of the entropy production
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Eigenvalue Problem for Hamiltonian
Hofstadters butterfly
2D tight-binding model Magnetic field
(reversible process)
(dimensionless) rational or irrational
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Hofstadters butterfly
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Band spectrum for fixed value of R
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Davydovs interaction
(kBT)/(2J) 1
Band width of the exciton

(kBT)/(2J) 1
(kBT)/(2J) 0.1
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Quantum Hydrodynamic Sound Mode
Kinetic equation for spatially inhomogeneous
system
Movie for the sound propagation
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Summary
Rich structure of the spectrum of a simple
quantum kinetic operator Rational-irrationality
dependence of the spectrum, Band
structure Rich structure of the entropy
production Characteristic behavior of 1D
system Appearance of a kinetic sound mode in
addition to the diffusion mode, such as in
classical gas Energy transport using quantum
hydrodynamic sound mode in a protein chain
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???
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Change of the phase volume (in the sense of the
Wigner function)
Jacobian