Transports in Protein Chain

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Transports in Protein Chain and Hofstadters
Butterfly type of Spectrum in Quantum Kinetic
Equation
Tomio Petrosky Center for Complex Quantum
Systems, University of Texas at Austin Tay Buang
Ann (Universiti Putra Malaysia) Valeri Bersegov
(Boston University) Naomichi Hatano (University
of Tokyo) Kazuk Kanki (Osaka Prefecture
University) Satoshi Tanaka (Osaka Prefecture
University)
Alien Baltan
July 31, 2009
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Eigenvalue Problem for Hamiltonian in Quantum
Mechanics
Well established
Ex. Discrete spectrum, Continuous spectrum, Band
spectrum, Resonance states for unstable systems
(e.g., spontaneous decays of excited states),
and so on.
Todays talk To show a rich structure of the
eigenvalue problem for the collision
operator for a simple quantum system.
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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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Eigenvalue problem for spatially homogeneous case
Band spectrum in the relaxation modes Finite
gap between collision invariance and the
relaxation modes
Local equilibrium, Quantum hydrodynamic modes
for spatially inhomogeneous case
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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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(kBT)/(2J) 10
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Shift of energy e
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Type II
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