Rytis Paškauskas, alumnus of the School of Physics, Georgia Tech.

As a graduate student in physics, I studied chaos and the periodic orbit theory in Hamiltonian dynamical systems. A specific property of a chaotic dynamical system is exponential divergence in time of the distance between trajectories with very similar initial conditions. The only way to make sense of observable quantities in these circumstances is to average over an ensemble of trajectories whose initial conditions are sampled from a suitable distribution. The distribution evolves in time by the Liouville equation, and the spectrum of the associated transfer operator provides information about the decay of correlations. Provided that a dynamical system is hyperbolic, the spectrum of the transfer operator is dual to the spectrum of periodic orbits. The precise mathematical formulation of this statement reveals that statistical averages are stability-weighted contributions from the zero-measure set of unstable periodic orbits. In low-dimensional Hamiltonian systems, the assumption of hyperbolicity is effective whenever their regular dynamics is confined to ``pockets'', which would require either very small spatial or very large time scales to be resolved. Writing of a manuscript on periodic orbit theory, applied to chaotic scattering by the Coulomb potential in external electromagnetic fields is in progress.

Dynamical bottlenecks to energy flow in chemical reactions

Vibrational energy in some molecules flows unevenly among the degrees of freedom, and the relaxation towards the equilibrium proceeds slower than predicted by statistical theories. Since relaxation to equilibrium is related to rates of chemical reactions, it is necessary to understand the specific pathways of energy redistribution among the degrees of freedom in such molecules. The bottlenecks of transport have been related to transitions across the so-called dynamical barriers in phase space. The dynamical barriers are invariant geometric structures such as broken tori, saddle orbits, and the configuration of their heteroclinic manifolds. The familiar picture in phase space of two degree-of-freedom Hamiltonian systems of chaotic zones and regular islands, separated by robust tori and ``sticky'' structures breaks down as the number of degrees of freedom increases. Individual tori no longer partition the phase space. We have shown in a molecule with three vibrational degrees of freedom that trajectories, contributing to the slowest relaxation, ``live'' in a resonance channel and relax by travelling along this chanel, helped by a tiny normal instability, until it reaches the chaotic layer. The dynamical invariants that best describe this motion in the resonance channel are a family of normally hyperbolic invariant tori. They have two toroidal and one hyperbolic degrees of freedom. Averaging out the vibrational motion of an individual torus the Lyapunov exponent provides the local rate of relaxation.

Out-of-equilibrium long-range interacting systems of particles

Currently I work on particle-wave interactions in the free-electron laser (FEL) at Sincrotrone Trieste. In an FEL, electrons interact by coherent radiation, emitted as they are accelerated by a magnetic field in an undulator. The lasing regime of an FEL is a plasma-like collective instability. The saturation is achieved via a combination of kinetic mixing and collisionless relaxation. In the saturation regime, the electron distribution displays non-equilibrium quasi-stationarity and non-Maxwellian tails. It therefore provides an experimental laboratory to study many of the effects that are common to long-range interacting systems. In particular, since the single-particle distribution function obeys an equation $\partial_t f= \{ f, {\mathcal H}[f] \} + C[f]$, where $\mathcal H[f]$ is the Hamiltonian functional, $C[f]$ is the collision operator and $\{,\}$ is the symplectic bracket, it has well-known stationary solutions $f=\Phi(E)$, parametrized by the mean-field energy $E$. In many interesting cases $C\approx0$ in the sense that collective behaviour rather than collisions shape the evolution. At low temperature, strong instability enhances kinetic mixing, and equilibrium $\Phi\simeq\exp{(-\beta E)}$, is reached almost instantly. The non-equilibrium $\Phi$ are observed when the instability is weaker, i.e. at a higher temperature. In this case the intermediate, quasi-stationary state forms, which further relaxes towards an equilibrium, with diverging rate $\tau\sim N^\gamma$, $\gamma>0$, and $N$ is the number of particles. Most interestingly, weak instability can result in collective oscillations in $f$ which violate $f=\Phi(E)$. The latter effect appears to be dynamical. It has been conjectured that averaging over the mixing ``degrees of freedom'', and taking into account Casimir invariants as constraints of the theory (which are imposed by the structure of the symplectic bracket), could explain why and how do quasi-stationary distributions depend on the initial conditions. In the dynamical case, the oscillations are an effect of the trajectory (i.e. the evolution of $f$) settling in an out-of-equilibrium state before mixing becomes effective. Information about the directionality of the unstable manifold of the initial state becomes relevant in description of these self-sustained oscillations.

Rytis Paškauskas

Address

Research Interests

Periodic orbit theory

Dynamical bottlenecks to energy flow in chemical reactions

Out-of-equilibrium long-range interacting systems of particles

		Center for Nonlinear Science School of Physics, 837 State Street Atlanta, Georgia 30332-0430 Phone 404.894.5200 - Fax 404.385-2506

publications address
	Rytis Paškauskas Ph.D in Physics, Georgia Institute of Technology, 2007 Physicist, Sincrotrone Trieste Affiliate, Center for Nonlinear Science Address Sincrotrone Trieste, AREA Science Park, Basovizza, 34012 Trieste, Italy Research Interests Dynamical systems, Hamiltonian and dissipative; Periodic orbit theory; Transition state theory; Out-of-equilibrium in many particle systems with long-range interactions: statistical, collisionless relaxation and dynamical properties; Large deviation technique; Numerical multidimensional extremization and root searching; Periodic orbit theory As a graduate student in physics, I studied chaos and the periodic orbit theory in Hamiltonian dynamical systems. A specific property of a chaotic dynamical system is exponential divergence in time of the distance between trajectories with very similar initial conditions. The only way to make sense of observable quantities in these circumstances is to average over an ensemble of trajectories whose initial conditions are sampled from a suitable distribution. The distribution evolves in time by the Liouville equation, and the spectrum of the associated transfer operator provides information about the decay of correlations. Provided that a dynamical system is hyperbolic, the spectrum of the transfer operator is dual to the spectrum of periodic orbits. The precise mathematical formulation of this statement reveals that statistical averages are stability-weighted contributions from the zero-measure set of unstable periodic orbits. In low-dimensional Hamiltonian systems, the assumption of hyperbolicity is effective whenever their regular dynamics is confined to ``pockets'', which would require either very small spatial or very large time scales to be resolved. Writing of a manuscript on periodic orbit theory, applied to chaotic scattering by the Coulomb potential in external electromagnetic fields is in progress. Dynamical bottlenecks to energy flow in chemical reactions Vibrational energy in some molecules flows unevenly among the degrees of freedom, and the relaxation towards the equilibrium proceeds slower than predicted by statistical theories. Since relaxation to equilibrium is related to rates of chemical reactions, it is necessary to understand the specific pathways of energy redistribution among the degrees of freedom in such molecules. The bottlenecks of transport have been related to transitions across the so-called dynamical barriers in phase space. The dynamical barriers are invariant geometric structures such as broken tori, saddle orbits, and the configuration of their heteroclinic manifolds. The familiar picture in phase space of two degree-of-freedom Hamiltonian systems of chaotic zones and regular islands, separated by robust tori and ``sticky'' structures breaks down as the number of degrees of freedom increases. Individual tori no longer partition the phase space. We have shown in a molecule with three vibrational degrees of freedom that trajectories, contributing to the slowest relaxation, ``live'' in a resonance channel and relax by travelling along this chanel, helped by a tiny normal instability, until it reaches the chaotic layer. The dynamical invariants that best describe this motion in the resonance channel are a family of normally hyperbolic invariant tori. They have two toroidal and one hyperbolic degrees of freedom. Averaging out the vibrational motion of an individual torus the Lyapunov exponent provides the local rate of relaxation. Out-of-equilibrium long-range interacting systems of particles Currently I work on particle-wave interactions in the free-electron laser (FEL) at Sincrotrone Trieste. In an FEL, electrons interact by coherent radiation, emitted as they are accelerated by a magnetic field in an undulator. The lasing regime of an FEL is a plasma-like collective instability. The saturation is achieved via a combination of kinetic mixing and collisionless relaxation. In the saturation regime, the electron distribution displays non-equilibrium quasi-stationarity and non-Maxwellian tails. It therefore provides an experimental laboratory to study many of the effects that are common to long-range interacting systems. In particular, since the single-particle distribution function obeys an equation $\partial_t f= \{ f, {\mathcal H}[f] \} + C[f]$, where $\mathcal H[f]$ is the Hamiltonian functional, $C[f]$ is the collision operator and $\{,\}$ is the symplectic bracket, it has well-known stationary solutions $f=\Phi(E)$, parametrized by the mean-field energy $E$. In many interesting cases $C\approx0$ in the sense that collective behaviour rather than collisions shape the evolution. At low temperature, strong instability enhances kinetic mixing, and equilibrium $\Phi\simeq\exp{(-\beta E)}$, is reached almost instantly. The non-equilibrium $\Phi$ are observed when the instability is weaker, i.e. at a higher temperature. In this case the intermediate, quasi-stationary state forms, which further relaxes towards an equilibrium, with diverging rate $\tau\sim N^\gamma$, $\gamma>0$, and $N$ is the number of particles. Most interestingly, weak instability can result in collective oscillations in $f$ which violate $f=\Phi(E)$. The latter effect appears to be dynamical. It has been conjectured that averaging over the mixing ``degrees of freedom'', and taking into account Casimir invariants as constraints of the theory (which are imposed by the structure of the symplectic bracket), could explain why and how do quasi-stationary distributions depend on the initial conditions. In the dynamical case, the oscillations are an effect of the trajectory (i.e. the evolution of $f$) settling in an out-of-equilibrium state before mixing becomes effective. Information about the directionality of the unstable manifold of the initial state becomes relevant in description of these self-sustained oscillations.