Quantum Localization and Self-trapping

The average deBroglie wavelength of an itinerant electron or positron equilibrated in a fluid is usually much greater than the mean inter-atomic spacing, so either may simultaneously interact with many atoms. Consequently self-trapping may occur, where the light particle "digs" a potential well for itself in the fluid and localizes in the self-induced ground state. The stability of the trapped state depends sensitively on the thermodynamic properties of the fluid and is appreciable in the neighborhood of the critical point. Experimentally, localization alters the decay rates of the positron and positronium, and the mobility of electrons.

Starting in 1990 we employed the Feynman-Kacs path integral to explore the relation between the quantum states of the light particle and local fluctuations in the fluid. We used path integral Monte Carlo (PIMC) to study positron annihilation in fluids, and to develop simulated data that provides a benchmark for testing the predictions of various theories. We also derived a quantum virial expansion for the average properties of the particle at low density, and used PIMC to evaluate the coefficients. The method was used to study the temperature dependence of the positron lifetime in a dilute gas. Recently we have used PIMC to study self-trapping at the liquid-vapor critical point of a Lennard-Jones fluid.We have also adapted the path integral method to represent a quantum particle on a lattice where, in principle, we can study both critical phenomena and Anderson Localization. This results in rapid convergence and permits us to test theories of localization over a wide parameter range. We plan to use the lattice model to study the influence of critical point fluctuations on quantum localization.

Quantum Particle on a Lattice, Henry Guo and Bruce N. Miller, Journal of Statistical Physics 98, 347 (2000).

Path-integral Study of Positronium Decay in Xenon,Terrence L. Reese and Bruce N. Miller, Physical Review E 64, 061201 (2001).

Self-trapping at the liquid-vapor critical point: A path-integral study, Bruce N. Miller and Terrence L. Reese, Physical Review E 78, 061123 (1-10) (2008).

Path Integral Monte Carlo on a Lattice: Extended States, Mark O’Callaghan and Bruce N. Miller,Physical Review E 89, 042124 (1-19)(2014).

Path integral Monte Carlo on a lattice. II. Bound states, Mark O’Callaghan and Bruce N.Miller, Physical Review E 94, 012120 (1-14)(2016).

Nonlinear Dynamics

The fundamental assumption of statistical mechanics is that dynamical systems are “chaotic”. For decades physicists assumed that chaos occurs when a stable dynamical system is perturbed. However, simulations using the first vacuum tube computers broke the proverbial bubble by showing that most of the phase space remains stable under small perturbation. A breakthrough came in 1954 when three mathematicians were able to explain the structure of phase space and its separation into stable and chaotic regions. By studying accelerated billiards with discontinuous boundaries, we have found new sources of orbital instability. We developed a simple model, the wedge billiard, which exhibits the complete range of Hamiltonian chaos. The model has been used successfully as a teaching tool and formed the basis for some deep theorems on the ergodic properties of a many body system. Recently it has been studied experimentally using cold atoms. We have extended our approach with a hyperbolic boundary that can be continuously deformed into either a parabola or wedge. This is also a rich system that shows how nearly integrable behavior can be connected by chaotic regimes. Currently we are investigating a quantum mechanical version of the wedge billiard.

Numerical Study of a Billiard in a Gravitational Field, H. Lehtihet and B. N. Miller, Physica D. 21, 93 (1986).

Dynamics of a Pair of Spherical Gravitating Shells, CHAOS 7, 187 (1997).

Dynamics of a Gravitational Billiard with a Hyperbolic Lower Boundary, CHAOS,9, 841(1999).

Dynamic modeling and simulation of a real world billiard, Alexandre E. Hartl, Bruce N. Miller and Andre P. Mazzoleni, Physics Letters A 375, 3682-3686 (2011).

Dynamics of a dissipative, inelastic gravitational billiard, Alexandre E. Hartl, Bruce N. Miller and Andre Mazzoleni, Physical Review E 87, 032901 (1-11), (2013).

Dynamics of Coulombic and gravitational systems, Pankaj Kumar and Bruce N. Miller, Physical Review E, 93 040202(R), (2016).

Gravitational Evolution, Equilibrium and Fractal Geometry

As a star ages, it radiates energy, heats up, and contracts, so its heat capacity is negative. In an ordinary physical system (e.g. a glass of water) this would not be possible.Because the gravitational force is purely attractive and of infinite range, the evolution and thermodynamic stability of astronomical objects such as galaxies and globular clusters is subtle and complex. In our group, we have used idealized models to study the nature of gravitational evolution in depth. Our studies of a system of parallel mass sheets have demonstrated weak chaos, suggesting that stable regions in the phase space trap the system for long times. Relaxation to equilibrium was demonstrated for the first time in a planar system with two distinct populations. The evolution of a system of concentric mass shells approaches equilibrium much more rapidly than its planar counterpart. Mean field theory predicts that a phase transition occurs in this system at sufficiently low energy. This was verified for the first time by our group using dynamical simulation. We have also investigated how rotation influences core collapse. In other work, we have constructed a completely integrable cousin of an N-body gravitational system that, along with the energy, conserves an N dimensional generalization of angular momentum. Recently, by including the Hubble expansion in the evolution, we are studying the multifractal properties of clustering in models of a matter-dominated universe.

Angular Momentum Induced Phase Transition in a Spherical Gravitational System: N-Body Simulations, Phys. Rev. E, 65, 056127 (2002).

Influence of Expansion on Structure Formation, Phys. Rev. E, 65, 056121(2002).

Incomplete Relaxation in a Two-mass, One-dimensional Gravitating System, Phys. Rev. E 68 , Nov 1,2003.

Dynamical Simulation of Gravothermal Catastrophe, Peter Klinko and Bruce N. Miller, Physical Review Letters, 92 (2), 021102 (2004).

Exactly Integrable Analogue of a One Dimensional Gravitational System, Bruce N. Miller, Kenneth R. Yawn and Bill Maier, Physics Letters A 346, 92-98 (2005).

Fractal Geometry in an expanding, one-dimensional, Newtonian universe, Bruce N. Miller, Jean-Louis Rouet, and Emmanuel Le Guirriec, Physical Review E 76 , 036705 (1-14) (2007).

Ewald Sums for One Dimension, Bruce N. Miller and Jean-Louis Rouet, Physical Review E 82, 066203(1-7), (2010).

Cosmology in One Dimension: Fractal Geometry, Power Spectra and Correlation, Bruce N. Miller and Jean-Louis Rouet, Journal of Statistical Mechanics, P12028 (1-26) (2010).

Cosmology in one dimension: Vlasov dynamics, Giovanni Manfredi, Jean-Louis Rouet, Bruce Miller and Yui Shiozawa, Physical Review E, 93, 042211 (1-9) (2016).

Cosmology in one dimension: A two component model, Yui Shiozawa and Bruce N. Miller, Chaos, Solitons and Fractals 91, 86-91 (2016).