Research Interests and Selected Publications

I Statistical Physics and Nonlinear Dynamics of Gravity using One-Dimensional Models: Evolution, Relaxation, Diffusion, Phase Transitions, Fractal Properties and Cosmology.

In thermal physics courses we learn that if the energy of a system is increased, the temperature goes up. In other words the heat capacity is positive. This is regarded as a basic requirement for stability in thermodynamics. Now think about a star: as it ages, it radiates energy, contracts, and its temperature rises. Thus it loses energy but its temperature still increases. The heat capacity is negative! This is remarkable and an apparent contradiction of the laws of thermodynamics. After all, the star is stable, and approximately in local equilibrium. The negative heat capacity is a central feature of most gravitational systems and operates in every type of gravitational contraction, from galaxy formation to the supernova. While these events obviously occur on vastly different time scales, the essential thermodynamics is similar.

The thermodynamics and statistical physics we study in courses is mostly concerned with the interactions between atoms and molecules. These are dominated by short ranged forces. The intuition we develop from this experience simply doesn't transfer to systems in which the interaction between elements is, like the gravitational force, almost purely attractive and of long range. In order to gain a better understanding of the statistical and dynamical features of gravitating systems, in our research group we have been investigating the behavior of idealized systems that astrophysicists have employed to test various conjectures about galaxy and cluster evolution. One example is the system of parallel mass sheets that are constrained to move perpendicular to their surface. This system avoids the singularity and the possibility of escape found in real stellar systems.Another is the system of concentric, spherical, mass shells having purely radial motion. In this model both the force singularity and escape can be controlled by introducing, respectively, internal and external reflecting barriers, or a centrifugal barrier induced by rotation.

By simplifying the interaction, the evolution of the system can be accurately simulated for much longer times than is possible for more realistic models. In earlier studies of the parallel sheet system we demonstrated that,following an initial "violent" relaxation, the system enters a long-lasting metastable phase which, presumably, gradually slides towards equilibrium. [1-6] Previously it was believed by the astrophysics community that evolution was much more rapid. To determine whether or not gravitational dynamics is intrinsically chaotic, we also investigated the ergodic properties of this system. We found stable regions in the phase space for systems with ten members (a twenty dimensional phase space) [7], and evidence of “sticky”regions, in which the system becomes trapped for long periods, in much larger systems. [8]

Astrophysicists model galactic dynamics with the coupled Vlasov-Poisson equations. This is an approximation, which replaces individual mass points (stars) by a continuous mass fluid. It is rigorous in the limit where the number of particles becomes infinite, but the total mass and energy are fixed. It is known that the collisionless evolution predicted by the Vlasov equation does not have increasing entropy and, therefore, does not approach a unique equilibrium distribution. To explain how the effects of discreteness influence the evolution of the parallel sheet system with finite population, we have constructed a diffusion model which treats the acceleration and velocity of a test particle as a two dimensional stochastic process [9]. The source of randomness is the variable time between sheet crossings. This is the strongest stochastic assumption possible in this system. We have shown that the model agrees with simulations for short times [10], and that it predicts that relaxation processes scale as the system population [11]. By including feedback, which accounts for the alteration in crossing distribution induced by the diffusion process, we hope to determine if the model is capable of predicting the evolution for longer times, or if a weaker stochastic assumption is operating in the dynamics.

For a video demonstrating the relaxation process click here. In the simulation, each paticle's position and velocity are represented by a point. The horizontal direction represents position and the vertical direction represents velocity.

It is well known that a large fraction of stars form in open clusters.However, the connection between this fact and the evolution of the parent galaxy is largely unexplored. Using the planar sheet model we investigated the dependence of the time for initial (violent) relaxation on cluster size and found a strong correlation [12]. In collaboration with T. Tsuchiya (Osaka University)we have been examining how clusters influence each other in three-dimensional systems.

Although attempted earlier by at least two other groups, mass segregation, the separation of heavy and light elements, has never been observed in the planar sheet system. Recently we were able to demonstrate mass segregation in a version of the system having two different populations. [13] As far as we know, this is the first time true macroscopic relaxation has been observed in the parallel sheet system.

Equipartition, defined as the approach of the time averaged energy of each system particle to its predicted microcanonical average on the energy hypersurface, has long been regarded as a basic signature of ergodic behavior. In a recent paper we showed that, in fact, equipartition does not necessarily imply ergodicity! Rather, it is a consequence of certain exchange symmetries in the system orbits. This was accomplished by demonstrating that equipartition was satisfied for a well known, non-ergodic, dynamical system [14]. In a series of papers, Tsuchiya, Gouda, and Koneko asserted that equipartition, and some features in its time spectrum, were associated with the macroscopic relaxation of the one dimensional planar sheet model. Using the above arguments coupled with the dynamical simulation of systems with a small number of elements, which were demonstrated to be “non-ergodic” [7], we showed that their conclusions were not justified by this approach [14].

There is observational evidence that black holes exist in some galactic centers. In order for a singularity to develop, large amounts of mass have to concentrate near the center. The formation of this “black hole precursor” is nonrelativistic, and may be driven by a gravitational phase transition. Michael Kiessling raised the possibility of such a transition in 1989 from mean field arguments. An open question is whether the transition is accessible dynamically and on what time scale it occurs. Using mean field (Vlasov) theory we have worked out the equilibrium theory of the phase transition for the system of irrotational concentric shells. We then carried out dynamical simulations in which we recovered all of the central features of gravitational phase transitions and confirmed the theoretical predictions [15-17]. This work was the first to dynamically demonstrate a purely gravitational phase transition.

From other studies intuition suggested that the sum of the squares of angular momentum, say L2, varies very slowly with time in a virialized gravitational system. We asked Toshio Tsuchiya, our colleague in Japan, to test this hypothesis using GRAPE computer architecture, a dedicated system for studying the evolution of large numbers of point masses moving under their mutual gravitational attraction. Toshio was able to confirm our hypothesis to within the numerical accuracy of GRAPE. We then developed the mean field theory of spherical systems of both point masses and rotating shells by determining the stationary solutions of the coupled Vlasov-Poisson equations with extremal entropy subject to the constraints of constant energy and L2. We were able to prove that a concentric shell system in which all angular momentum are constrained to a common plane shares an equivalent thermodynamics with the more conventional three dimensional point mass system in the mean field (Vlasov) limit [18]. In addition we were able to demonstrate from a stability analysis that an entropy ceiling is lacking so that the system shares the possibility of gravitational catastrophe proved by Antonov for 3D systems with the usual constraint of constant energy. In common with Antonov and others, we were able to demonstrate the existence of locally stable (or metastable) solutions that, presumably, could have a long lifetime before transitioning to the collapsing mode.

In another test of the role of the singularity, we explored the mean field theory of a 3D system in which the angular momentum per unit mass has a fixed magnitude. Here the centrifugal barrier is successful in screening the singularity in the Newtonian interaction. For this system we were able to rigorously prove that there is an entropy ceiling in the microcanonical ensemble (MCE) and a corresponding minimum in the free energy for the canonical ensemble (CE) [19]. For each ensemble we derived coupled, nonlinear, differential equations for the local density. Solving them numerically we were able to demonstrate that phase transitions occur in each ensemble [19]. We also found that each ensemble had a critical point and, as in other non-extensive systems, we found the phenomena of ensemble inequivalence; i.e. in contrast with normal systems there are major differences separating the properties of each ensemble. For example, a negative heat capacity can only occur in MCE, and the critical point occurs at a different temperature and value of L2 in each ensemble. We also studied the evolution of the system using dynamical simulation in both the MCE [20] and CE [26]. We were able to demonstrate the existence of the transition in MCE modulo finite size scaling.In CE we were able to see the transition occur dynamically from the metastable to the stable phase, and the reverse for sufficiently small systems. We found that we could see the occurrence of the transition dynamically with only 8 shells!

More recently we have used dynamical simulation to study the concentric shell system where the constraints on the angular momentum are relaxed [25]. Whenever a pair of shells crossed, we allowed them to exchange angular momentum while constraining the total angular momentum and L2. We demonstrated that gravothermal catastrophe, i.e. the formation of a small, dense core and a hot halo, would occur in finite time under suitable circumstances. We also demonstrated that the existence of long-lived metastable states is still possible. This may explain the segregation of globular clusters into two groups, those that are apparently collapsed, and those having a smooth density profile.

An important question in cosmology is whether the universe is homogeneous and isotropic on large scales. While this is the fundamental hypothesis of cosmologists, as technology improves observations show the existence of increasingly large structure. One school of thought pioneered by Mandelbrot is that the observable universe is not uniform on any scale. One-dimensional models provide excellent tools for studying the nonlinear dynamics of the expanding universe. While the predictions will differ from the more realistic higher dimensional models, there are many parallels between them. Because we can study a system with numerical accuracy using orders of magnitude more particles per dimension than is possible in 3D, we can confidently make reproducible calculations of fractal properties. By including the Hubble expansion in the parallel sheet system, in collaboration with J. L. Rouet, my French colleague at the Universite d’Orleans, we were able to demonstrate the multi-fractal properties of cluster formation. [21] In common with astronomical observations of galaxy positions, we showed that a bifractal structure is obtained, i.e. both nearly empty regions (voids) and dense regions (clusters) appear to have distinct fractal dimension. This is work in progress [22, 29, 30-35].

In a return to the two mass species system we studied the relaxation towards equilibrium for different populations and mass ratios. Our goal was to flesh out the earlier work with a single mass ratio [13]. In all cases we found the occurrence of equipartition and mass segregation. Using mean field theory we also computed the theoretical mass distribution and compared the predictions with dynamical simulations. Although we let the systems evolve for 108 characteristic system-crossing times, they had not equilibrated. However, the deviations of time averages from the equilibrium predictions were only on the order of a few percent. One interesting feature of the simulations is that convergence to equilibrium improved as a powerlaw in time, indicating that the system has no characteristic time scale for reaching equilibrium. This “scale free” behavior was not anticipated and differs from the usual view discussed in the astrophysics literature.

1.The Relaxation Time of a One Dimensional Self-Gravitating System, H. L. Wright, B. N. Miller and W. E. Stein, Astrophysics and Space Science, 84, 421(1982).
2. Gravity in One Dimension: A Dynamical and Statistical Study, H. L. Wright and Bruce N. Miller, Physics Review A. 29, 1411 (1984).
3. Gravity in One Dimension: Selective Relaxation, C. Reidl and B. N. Miller, The Astrophysical Journal 318, 248 (1987).
4. Gravity in One Dimension: Quasi-Equilibrium, Charles J. Reidl, Jr. and Bruce N. Miller, The Astrophysical Journal 332, 619 (1988).
5. Gravity in One Dimension: Persistence of Correlation, Bruce N. Miller and Charles J. Reidl, The Astrophysical Journal 348, 203 (1990).
6. Gravity in One Dimension: A Correction for Ensemble Averaging, Charles Reidl and Bruce N. Miller, The Astrophysical Journal, 371, 260 (1991).
7. Gravity in One Dimension: Stability of Periodic Orbits, Charles J. Reidl, Jr. and Bruce N. Miller, Physical Review A, 46, 837 (1992).
8. Gravity in One Dimension: The Critical Population, Charles J. Reidl and Bruce N. Miller, Physical Review E 48, 4250 (1993).
9. Gravity in One Dimension: Diffusion in Acceleration, Bruce N. Miller, Journal of Statistical Physics, 63, 291 (1991).
10. Stochastic Dynamics of Gravity in One Dimension, Kenneth R. Yawn, Bruce N. Miller, and Willard Maier, Physical Review E 52, 3390 (1995).
11. Source of Relaxation in a One Dimensional Gravitating System, Bruce N. Miller, Physical Review E 53, R4279 (1996).
12. Gravity in One Dimension: Population Dependence of Early Relaxation, Charles J. Reidl and Bruce N. Miller, Physical Review E 51, 884 (1995).
13.Equipartition and Mass Segregation in a One Dimensional Gravitating System, Ken Yawn and Bruce N. Miller, Physical Review Letters 79, 3561 (1997).
14.Ergodic Properties and Equilibrium of One-Dimensional Self-Gravitating Systems, Kenneth R. Yawn and Bruce N. Miller, Physical Review E, 56, 2429 (1997).
15. Rapid Relaxation in a One Dimensional Gravitating System, V. Paige Youngkins and Bruce N. Miller, Physical Review E 56, R4963 (1997).
16. Phase Transition in a Model Gravitating System, Bruce N. Miller and Paige Youngkins, Phys. Rev. Lett.81,4794 (1998).
17. Gravitational Phase Transitions in a One Dimensional Spherical System, V. Paige Youmgkins and Bruce N. Miller, Physical Review E 62, 4583-4596 (2000).
18. Mean Field Theory of Spherical Gravitating Systems, Peter Klinko and Bruce N. Miller, Physical Review E 62, 5783-5792 (2000).
19. Rotation-induced Phase Transition in a Spherical Gravitating System, Peter Klinko, Bruce N. Miller, and Igor Prokhorenkov, Physical Review E 63, 066131 (2001).
20. Angular Momentum Induced Phase Transition in Spherical Gravitational Systems: N-Body Simulations, Peter Klinko and Bruce N. Miller, Physical Review E, 65, 056127 (2002).
21. Influence of Expansion on Structure Formation, Bruce N. Miller and Jean-Louis Rouet, Physical Review E, 65, 056121 (2002).
22. Fractal Spectrum of a 1D Gravitating Model, Bruce N. Miller and Jean-Louis Rouet, Physica A, 266 (2002).
23. Spherical Gravitating Systems, Bruce N. Miller, Peter Klinko, and Paige Youngkins, Chaos,Solitons and Fractals 13 , 603-616 (2002).
24. Incomplete Relaxation in a Two-mass, One-dimensional Self-gravitating System, Kenneth Yawn and Bruce N. Miller, Physical Review E 68, 056120 (2003).
25. Dynamical Simulation of Gravothermal Catastrophe, Peter Klinko and Bruce N. Miller, Physical Review Letters, 92 (2), 021102(2004).
26. Dynamical Study of a First Order Gravitational Phase Transition, Peter Klinko and Bruce N. Miller, Physics Letters A, 333 (3-4),187 (2004).
27. Exactly Solvable Analogue of a One Dimensional Gravitational System, Bruce N. Miller, Kenneth R. Yawn, and Bill Maier, Physics Letters A 346, 92-98(2005).
28. Vlasov Theory in Newtonian Gravity: Approach to Equilibrium, Phase Transitions, Gravothermal Catastrophe, and the Expanding Universe, Bruce N. Miller, Transport Theory and Statistical Physics 34, 367-389(2005).
29. Development of fractal geometry in a one-dimensional gravitational system, Bruce N. Miller and Jean-Louis Rouet, Comptes Rendus Physique 7, 383-390 (2006).
30. 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).
31. Ewald sums for one dimension,Bruce N. Miller and Jean-Louis Rouet, Physical Review E 82 , 066203 (1-7) (2010).
32. Cosmology in one dimension: fractal geometry, power spectra and correlation,Bruce N. Miller and Jean-Louis Rouet, Journal of Statistical Physics, P12028 (1-26) (2010).
33. Ewald Sums for One Dimension, Bruce N. Miller and Jean-Louis Rouet, Physical Review E 82, 066203(1-7), (2010).
34. Cosmology in one dimension: Vlasov dynamics, Giovanni Manfredi, Jean-Louis Rouet, Bruce Miller and Yui Shiozawa, Physical Review E, 93, 042211 (1-9) (2016).
35. Cosmology in one dimension: A two component model, Yui Shiozawa and Bruce N. Miller, Chaos, Solitons and Fractals 91, 86-91 (2016).

II Low Dimensional Nonlinear Dynamics: New Mechanisms (beyond KAM) for Generating Chaos

The first problem a physics student learns to solve is the problem of a perfectly elastic ball bouncing on a flat floor. As it bounces along it inscribes a succession of parabolic arcs in space. Every nondissipative dynamical problem physicists solve exactly is structurally identical to this one: The number of conserved quantities is equal to the number of degrees of freedom, so the problem can be reduced to quadrature. We now recognize that such “integrable” cases comprise a small subset of dynamical systems. For example, if segments of the floor on which the ball bounces are tilted, the motion is, in general, chaotic and not solvable for long times with simple equations [see 1,2 below]. In 1954 Kolmogorov, Arnold, and Moser (KAM) proved that chaos could develop in smooth dynamical system whenever an integrable system is perturbed. By smooth, we mean that forces change gradually and are therefore differentiable functions of position. This type of chaos has a well-known signature.

In our work with idealized gravitational models [1, 2] we have found other, new mechanisms for generating chaotic behavior in conservative dynamical systems with a few degrees of freedom. To understand these phenomena more completely we have constructed different models, which capture this feature. In related work we showed how chaos could arise in a driven one-dimensional system in which the absolute value of the acceleration is constant. More recently we have shown that coexisting stable and unstable regions exist in the system of two concentric gravitational shells described above [4]. Here too we find evidence of phase space geometry that is not explained by K. A. M. theory.

Our study of a ball bouncing between tilted floors, now more commonly known as the wedge billiard, demonstrates that a discontinuity in a system map can also contribute to chaos[1, 2]. In that system, purely integrable, purely chaotic, and coexisting chaotic and stable regions in the phase space exist depending entirely on the angle formed by the two floor segments. It is worth noting that if the floor has a parabolic shape, the system is completely integrable. A generalization of the wedge billiard is the transitional system in which the floor cross-section is hyperbolic: Then a high energy ball sees a wedge, whereas a low energy ball sees a parabola. We have studied the behavior of this system as the energy is increased [5]. From the above, at low energy the phase space is mostly stable. As the energy is raised, the system should approach a mixed or chaotic region depending on the angle between the asymptotes. Surprisingly we have found that the transition domain the system can be either chaotic or highly stable depending on the angle.

1.Numerical Study of a Billiard in a Gravitational Field, H. Lehtihet and B. N. Miller, Physica D. 21, 93 (1986).
2. Stochastic Modeling of a Billiard in a Gravitational Field: Powerlaw Behavior of Lyapunov Exponents, B. N. Miller and K.Ravishankar, Journal of Statistical Physics 53, 1299 (1988).
3. Dynamics in a Discontinuous Field: The Smooth Fermi Piston, G. A. Worrell, A. Matulich, and B. N. Miller, CHAOS, 3, 397 (1993).
4. Ergodic Properties of a Pair of Concentric Gravitating Shells, Bruce N. Miller and V. Paige Youngkins, CHAOS 7, 187 (1997).
5. Dynamics of a Gravitational Billiard with a Hyperbolic Lower Boundary, M. L. Ferguson, B. N. Miller, and M. A. Thompson, CHAOS 9, 841 (1999).
6. 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).
7.Dynamics of a dissipative, inelastic gravitational billiard, Alexandre E. Hartl, Bruce N. Miller and Andre Mazzoleni, Physical Review E 87, 032901 (1-11),(2013).
8.Dynamics of Coulombic and gravitational systems, Pankaj Kumar and Bruce N. Miller, Physical Review E, 93 040202(R), (2016).

III Quantum Localization of a Low Mass Particle: Applications to Positron and Positronium Annihilation in Fluids

In our solid-state physics course we learn that a free, or quasi-free, electron is essentially a plane wave propagating through the system. Conductivity, or mobility, is determined by the amount of scattering from lattice defects and dislocations. Similar behavior is observed in liquids well below the critical temperature. It is not widely recognized that, under the right conditions, a light particle (electron, positron, or positronium atom) can actually trap itself in a bubble or droplet it helps to create and stabilize. The bubbles and droplets are on the order of 20A in diameter. It is hard to imagine that a particle with the electron mass can strongly influence the behavior of atoms: It is as if a Volkswagen beetle were moving a Mack truck! However, self-trapping is purely a quantum effect that occurs because the long wavelength of the light particle effectively allows it to interact with many atoms simultaneously. Manifestations of self-trapping are the increase in decay rate of the positron (it gets more electrons to annihilate with by forming a droplet), the decrease in decay rate of positronium (it forms a bubble, so there are fewer electrons available to the positron), and a reduction of the electron mobility (it has to drag the bubble or droplet along).

Mean field theory, in which a single quantum particle experiences a continuous potential resulting from the average distribution of nearby, classical, atoms, does a good job of modeling self-trapping at about 5K in helium. However, in gases with a higher critical temperature, such as argon, we showed some time ago that it is not very successful [1-3]. At these temperatures fluctuations apparently play an important role and mean field theory breaks down. To demonstrate that mean field theory “over-predicts” localization we proved that it predicts that all quantum states of a light particle are self-trapped in a one-dimensional system [4].

In order to study the effect of fluctuations on the quantum states of a positron or positronium atom, we employed the Feynman-Kac discrete path integral and showed how it could be extended to determine the annihilation rate of a positron [5]. This powerful technique allows the calculation of thermal averages by replacing the single quantum particle with a long “necklace” of classical oscillators. Special Monte Carlo methods can be used to sample the system with a fast computer. With this method we were able to replicate the essential features of positron decay in fluid xenon and demonstrate that the anomalous behavior observed in experiments at high density results from the percolation of the positron in channels of reduced potential energy.[6]. We also showed that positronium could become weakly trapped in xenon, but that fluctuations played a dominant role in determining the decay rate [7]. For each system we found that the magnitude of fluctuations of the positron decay rate was on the order of the thermal average [8]!

By marrying the path integral with the density functional theory of classical fluids, in the middle of the last decade Chandler, Singh, and Richardson developed an analytic theory for modeling the electron-fluid system. Called the RISM (interaction site mechanism)-polaron theory, it borrows ideas from Feynman’s polaron approximation to the path integral and modern theories of the two-body correlation function in fluids. This is the only analytic theory that includes the effect of quantum fluctuations on predicted values. We were the first to test this theory using realistic interaction potentials. In a series of papers we applied the method to the prediction of the behavior of a thermalized positron [9, 10], electron [11], and positronium atom [12].Because of the differences in the essential character of the quantum particle-atom interaction for each of these systems, a new algorithm had to be developed in each case to solve the complex coupled integro-differential equations. By employing both the Percus-Yevic and Hypernetted chain closures of the Ornstein-Zernicke equation, we were able to study the degree of localization of the quantum particle over a wide range of fluid temperature and density. For the case of the positron and positronium atom, we were able to compare the predictions with the numerically accurate Monte Carlo predictions we had carried out earlier [6-8]. We concluded that RISM-polaron theory strongly under-estimates the degree of localization below the critical density, but has real promise in the high density regime. Chandler has suggested that the method could be improved by developing different Ornstein-Zernicke closures that are more suitable for the quantum-fluid interaction. This remains an open question.

A number of experiments in positron and positronium annihilation focus on the temperature dependence of the decay rate in the low density regime, and two research groups are conducting new measurements. To support this work, we developed a virial expansion for the decay rate in which each temperature dependent coefficient can be expressed by a modified path integral [13]. We have used this method to carry out the first predictions of the temperature dependence of the positron and positronium annihilation rate at low density [14]. In other work it was demonstrated that the positron is able to bind a small cluster of highly polarizable atoms [15]. It will be interesting to learn how this clustering influences the decay rate. This can be explored using higher level terms in the virial expansion, but the numerical convergence of our current algorithm for these terms is extremely slow.

A difficulty associated with the path integral method is the large amount of computer time required to obtain convergence. This limits the range of parameter space (e.g. the density and temperature of the host fluid) that can be effectively studied. To circumvent this problem we have constructed a model in which both the atoms and the light particle are restricted to lattice sites. With respect to the light quantum particle the result is similar to a tight binding model. Following Feynman’s original approach, we proved that thermal averages could be obtained by representing the quantum particle by an N step random walk on the lattice with variable step size. [16] We have used this method with good success for a one-dimensional system and plan to generalize to two dimensions, where a phase transition occurs naturally. By quenching the lattice we plan to study how Anderson (disorder induced) localization competes with self-trapping. Which mechanism dominates is an open question in positron physics.

We have continued to explore the use of the path integral to study low mass particles in fluids. As computers improve in speed and memory we are able to obtain improved convergence for larger systems. We have revisited the system of a low mass particle interacting with a host atomic fluid for the case where the atom-atom interaction is given by the Lennard-Jones potential and the particle-fluid atom interaction is a hard sphere [17]. We used this system to model positronium in Xenon, and compared our results for the pick-off decay rate with experimental measurements. Recently we have taken advantage of new, highly accurate, computations of the liquid-vapor critical point of a Lennard-Jones fluid to study self-trapping in that thermodynamic state. Due to the divergent compressibility, the system is most vulnerable to self-trapping here. We studied the localization properties as a function of hard sphere diameter and found that as this quantity increased, the stability of the localized state was enhanced. We also found that the fluid density in the neighborhood of the trapped particle varied slowly compared with trapping in the liquid state [18]. We plan to apply the path integral method to the computation of the angular distribution of annihilation photons from self-trapped para-positronium in the near future.

1. Macroscopic Theory of Localization in a Classical Fluid, Bruce N. Miller, Physical Review A. 30, 3343 (1984).
2. Self-Trapping of a Light Particle in a Dense Fluid: A Mesoscopic Model, Bruce N. Miller and Terrence Reese, Physical Review Part A 39, 4735 (1989).
3. Self-Trapping of a Light Particle in a Dense Fluid: Application of Scaled Density Functional Theory to the Decay of Ortho-Positronium, Terrence Reese and Bruce N. Miller, Physical Review A, 42, 6068 (1990).
4. Exact Solution of One-Dimensional Self-Trapping Systems in the Mean Field Approximation;Yzhong Fan and Bruce Miller; Journal of Physics A 22, 3849-3858 (1989).
5. Localization in Fluids: A Comparison of Competing Theories and their Application to Positron Annihilation, Bruce N. Miller and Yzhong Fan, Physical Review A 42, 2228 (1990).
6. Positron Annihilation in Xenon: The Path-integral Approach, G. A. Worrell and Bruce N. Miller, Physical Review A, 46, 3380 (1992).
7. Positronium in Xenon: The Path Integral Approach,Terrence Reese and Bruce N. Miller, Physical Review E, 47, 2581 (1993).
8. Positron Lifetime Distributions in Fluids, Bruce N. Miller, Terrence Reese, and Gregory A.Worrell, Physical Review E, 47, 4083 (1993).
9. Localization in a Lennard-Jones Fluid: The Path-Integral Approach, Yzhong Fan and Bruce N. Miller, Journal of Chemical Physics, 93, 4322 (1990).
10. Polaron Theory of Positron Annihilation in Xenon, Jiqiang Chen and Bruce N. Miller, Physical Review E, 48, 3667 (1993).
11. Polaron Theory of an Excess Electron in Xenon, Jiqiang Chen and Bruce N. Miller, Journal of Chemical Physics 100, 3013 (1994).
12. Polaron Theory of Positronium Localization and Annihilation in Xenon, Jiqiang Chen and Bruce N. Miller, Physical Review B 49, 15 615(1994).
13. Virial Expansion of a Quantum Particle in a Classical Gas: Application to the Orthopositronium Decay Rate, Gregory A. Worrell, Bruce N. Miller, and Terrence L. Reese, Physical Review A 53, 2101 (1996).
14. Virial Expansion of the Positron and Positronium Decay Rate in a Dilute Gas: The Linear Contribution, Bruce N. Miller, Terrence L. Reese, and Gregory A. Worrell, Canadian Journal of Physics 78, 548 (1996).
15. Interaction Between Slow Positrons and Atoms, V. A.Dzuba, V. V. Flambaum, W. A. King, B. N. Miller, and O. P. Sushkov, Physica Scripta, T46, 248 (1993).
16. Monte Carlo Study of Localization on a One Dimensional Lattice, Henry Guo and Bruce N. Miller, Journal of Statistical Physics 98, 347 (2000).
17. Path-integral Study of Positronium Decay in Xenon, Terrence L. Reese and Bruce N. Miller, Physical Review E 64, 061201 (2001).
18. Self-Trapping at the Liquid-Vapor Critical Point, B. N. Miller and T. L. Reese, Modern Physics Letters B 20, 169(2006).
19. 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).
20. Path Integral Monte Carlo on a Lattice: Extended States, Mark O’Callaghan and Bruce N. Miller,Physical Review E 89, 042124 (1-19)(2014).
21. 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).

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