Resonance states of a Liouvillian and Hofstadter’s butterfly type of singular spectrum of a collision operator for a protein chain August 10, 2010 Tomio Yamakoshi Petrosky Center for Complex Quantum Systems, University of Texas at Austin Naomichi Hatano (University of Tokyo) Kazuk Kanki (Osaka Prefecture University) Satoshi Tanaka (Osaka Prefecture University) Alien BaltanAsteroid belt

Eigenvalue Problem of Liouville-von Neumann Operators (Classical and Quantum) Still in poorly understood situation especially for unstable systems: Resonance states for the Louvillian Eigenvalue problem of the collision operators in Kinetic equations (complex eigenvalues and irreversibility) Other interesting properties (reversible dynamics): continuous spectrum, discrete spectrum, band structure, level repulsion, …

Myoglobyn Primary Structure: One-dimensional Molecular Chain 1) Resonance States of Quantum Liouvillian for Protein Chain  -helix 2) Level Repulsion and threefold degeneracy of eigenstates of the Liouvillian in the Kirkwood gaps in the asteroid belt Instability due to the resonance

・ Band spectrum in the relaxation modes for momentum disturibution Characteristic behavior of 1D system: ・ No classical limit just like the 1D classical gas ・ Rational-irrationality dependence of the spectrum in a physical parameter (Fractal structure) ⇒ Similarity to Hofstadter’s butterfly of a Hamiltonian spectrum for a 2D tight-binding model in a magnetic field I. Resonance States of Quantum Liouvillian for Protein Chain reversible process R irreversible process

Intrinsic degeneracy Liouville equation

Dispersion equation and Resonance states of Liouvillian Dispersion equation Operator equation The collision operator Kinetic equation Complex eigenvalues: Transport coefficient in irreversible process

Exciton Dimensionless Hamiltonian ratio of band width

Reduced density matrix Kinetic equation for the momentum distribution (weak coupling case) Planck distribution for the phonon resonance The collision operator in P 0 subspace: Momentum distribution function

Eigenvalue problem of the collision operator H-theorem rational irrational

qq 2m =10 m = 6 : resonance condition

Alien Baltan t > 0 t < 0 Rich structure of the entropy production

type-I type-II

type-I type-II

t -1/2 power law decayBand structure of the spectrum Spectra of the collision operator ! type-I

type-II Spectra of the collision operator

II. Level Repulsion and threefold degeneracy of eigenstates of the Liouvillian in the Kirkwood gaps in Asteroid belt Band spectrum of the Liouvillian in the restrict problem of three bodys Keplar’s third law

The sidereal coordinate The synodic coordinate Hamiltonian of the restricted three-body problem

Delaunay’s variables:

Dispersion relation for m-n mode: Threefold degeneracy

! Level repulsion The degenerate perturbation theory for resonances Mode coupling (Selection rule) Saturn’s ring?

Evidence of the threefold degeneracy

Summary (Instability due to the resonance) ・ Rich fractal structure of the resonance spectrum of the Liouvillian in a protein chain, Rational-irrationality dependence of the spectrum, Band structure ・ Rich structure of the entropy production, "Complex Spectrum Representation of the Liouvillan and Kinetic Theory in Nonequilibrium Physics," T. Petrosky, Prog. Theor. Phys. 123, 395 (2010) "Hofstadter’s butterfly type of singular spectrum of a collision operator for a model of molecular chains," T. Petrosky, N. Hatano, K. Kanki, and S. Tanaka, Prog. Theor. Phys. Supplement, No. 184, 457 (2010). ・ Quantum analogy of the spectrum of the Liouvillian in classical system, ・ Analysis of classical systems in terms of “states” in stead of “trajectories,” ・ The degenerate perturbation theory for resonance effect, level repulsion, band structure, … (classical quantization?)

2D tight-binding model + Magnetic field （ reversible process ） Dispersion relation: (dimensionless) rational or irrational Hofstadter’s butterfly Eigenvalue Problem for Hamiltonian

Hofstadter’s butterfly

Band spectrum for fixed value of R

Davydov’s interaction (k B T)/(2J) = 0.1 (k B T)/(2J) = 1 Band width of the exciton Type I Type II

Resonance states of a Liouvillian and Hofstadter’s butterfly type of singular spectrum of a collision operator for a protein chain SPQS2010, August 2, 2010 Tomio Yamakoshi Petrosky Center for Complex Quantum Systems, University of Texas at Austin Naomichi Hatano (University of Tokyo) Kazuk Kanki (Osaka Prefecture University) Satoshi Tanaka (Osaka Prefecture University) Alien BaltanAsteroid belt