Bruce Boghosian | Tufts University (original) (raw)
Papers by Bruce Boghosian
We describe a three-dimensional hydrodynamic lattice-gas model of amphiphilic fluids. This model ... more We describe a three-dimensional hydrodynamic lattice-gas model of amphiphilic fluids. This model of the non-equilibrium properties of oil-water-surfactant systems, which is a non-trivial extension of an earlier two-dimensional realisation due to Boghosian, Coveney and Emerton [Boghosian, Coveney, and Emerton 1996, Proc. Roy. Soc. A 452, 1221-1250], can be studied effectively only when it is implemented using high-performance computing and visualisation techniques. We describe essential aspects of the model's theoretical basis and computer implementation, and report on the phenomenological properties of the model which confirm that it correctly captures binary oil-water and surfactant-water behaviour, as well as the complex phase behaviour of ternary amphiphilic fluids.
Fourier acceleration has been successfully applied to the simulation of lattice field theories fo... more Fourier acceleration has been successfully applied to the simulation of lattice field theories for more than a decade. In this paper, we extend the method to the dynamics of discrete particles moving in continuum. Although our method is based on a mapping of the particles' dynamics to a regular grid so that discrete Fourier transforms may be taken, it should be emphasized that the introduction of the grid is a purely algorithmic device and that no smoothing, coarse-graining or mean-field approximations are made. The method thus can be applied to the equations of motion of molecular dynamics (MD), or its Langevin or Brownian variants. For example, in Langevin MD simulations our acceleration technique permits a straightforward spectral decomposition of forces so that the long-wavelength modes are integrated with a longer time step, thereby reducing the time required to reach equilibrium or to decorrelate the system in equilibrium. Speedup factors of up to 30 are observed relative ...
Tsallis has proposed a generalization of Boltzmann-Gibbs thermostatistics by introducing a family... more Tsallis has proposed a generalization of Boltzmann-Gibbs thermostatistics by introducing a family of generalized nonextensive entropy functionals with a single parameter q. These reduce to the extensive Boltzmann-Gibbs form for q=1, but a remarkable number of statistical and thermodynamic properties have been shown to be q-invariant -- that is, valid for any q. In this paper, we address the question of whether or not the value of q for a given viscous, incompressible fluid can be ascertained solely by measurement of the fluid's hydrodynamic properties. We find that the hydrodynamic equations expressing conservation of mass and momentum are q-invariant, but that for conservation of energy is not. Moreover, we find that ratios of transport coefficients may also be q-dependent. These dependences may therefore be exploited to measure q experimentally.
A lattice gas model for Schloegl's second chemical reaction is described and analyzed. Becaus... more A lattice gas model for Schloegl's second chemical reaction is described and analyzed. Because the lattice gas does not obey a semi-detailed-balance condition, the equilibria are non-Gibbsian. In spite of this, a self-consistent set of equations for the exact homogeneous equilibria are described, using a generalized cluster-expansion scheme. These equations are solved in the two-particle BBGKY approximation, and the results are compared to numerical experiment. It is found that this approximation describes the equilibria far more accurately than the Boltzmann approximation. It is also found, however, that spurious solutions to the equilibrium equations appear which can only be removed by including effects due to three-particle correlations.
A complete formulation is given of an exact kinetic theory for lattice gases. This kinetic theory... more A complete formulation is given of an exact kinetic theory for lattice gases. This kinetic theory makes possible the calculation of corrections to the usual Boltzmann / Chapman-Enskog analysis of lattice gases due to the buildup of correlations. It is shown that renormalized transport coefficients can be calculated perturbatively by summing terms in an infinite series. A diagrammatic notation for the terms in this series is given, in analogy with the diagrammatic expansions of continuum kinetic theory and quantum field theory. A closed-form expression for the coefficients associated with the vertices of these diagrams is given. This method is applied to several standard lattice gases, and the results are shown to correctly predict experimentally observed deviations from the Boltzmann analysis.
We present a general methodology for constructing lattice Boltzmann models of hydrodynamics with ... more We present a general methodology for constructing lattice Boltzmann models of hydrodynamics with certain desired features of statistical physics and kinetic theory. We show how a methodology of linear programming theory, known as Fourier-Motzkin elimination, provides an important tool for visualizing the state space of lattice Boltzmann algorithms that conserve a given set of moments of the distribution function. We show how such models can be endowed with a Lyapunov functional, analogous to Boltzmann's H, resulting in unconditional numerical stability. Using the Chapman-Enskog analysis and numerical simulation, we demonstrate that such entropically stabilized lattice Boltzmann algorithms, while fully explicit and perfectly conservative, may achieve remarkably low values for transport coefficients, such as viscosity. Indeed, the lowest such attainable values are limited only by considerations of accuracy, rather than stability. The method thus holds promise for high-Reynolds num...
We investigate the growth kinetics of binary immiscible fluids and emulsions in two dimensions us... more We investigate the growth kinetics of binary immiscible fluids and emulsions in two dimensions using a hydrodynamic lattice-gas model. We perform off-critical quenches in the binary fluid case and find that the domain size within the minority phase grows algebraically with time in accordance with theoretical predictions. In the late time regime we find a growth exponent n = 0.45 over a wide range of concentrations, in good agreement with other simluations. In the early time regime we find no universal growth exponent but a strong dependence on the concentration of the minority phase. In the ternary amphiphilic fluid case the kinetics of self assembly of the droplet phase are studied for the first time. At low surfactant concentrations, we find that, after an early algebraic growth, a nucleation regime dominates the late-time kinetics, which is enhanced by an increasing concentration of surfactant. With a further increase in the concentration of surfactant, we see a crossover to loga...
In this work, we study numerically the convergence of the scalar D2Q9 lattice Boltzmann scheme wi... more In this work, we study numerically the convergence of the scalar D2Q9 lattice Boltzmann scheme with multiple relaxation times when the time step is proportional to the space step and tends to zero. We do this by a combination of theory and numerical experiment. The classical formal analysis when all the relaxation parameters are fixed and the time step tends to zero shows that the numerical solution converges to solutions of the heat equation, with a constraint connecting the diffusivity, the space step and the coefficient of relaxation of the momentum. If the diffusivity is fixed and the space step tends to zero, the relaxation parameter for the momentum is very small, causing a discrepency between the previous analysis and the numerical results. We propose a new analysis of the method for this specific situation of evanescent relaxation, based on the dispersion equation of the lattice Boltzmann scheme. A new asymptotic partial differential equation, the damped acoustic system, is ...
A Lattice Boltzmann formulation for relativistic fluids is presented and numerically verified thr... more A Lattice Boltzmann formulation for relativistic fluids is presented and numerically verified through quantitative comparison with recent hydrodynamic simulations of relativistic shock-wave propagation in viscous quark-gluon plasmas. This formulation opens up the possibility of exporting the main advantages of Lattice Boltzmann methods to the relativistic context, which seems particularly useful for the simulation of relativistic fluids in complicated geometries.
We simulate the dynamics of phase assembly in binary immiscible fluids and ternary microemulsions... more We simulate the dynamics of phase assembly in binary immiscible fluids and ternary microemulsions using a three-dimensional hydrodynamic lattice gas approach. For critical spinodal decomposition we perform the scaling analysis in reduced variables introduced by Jury et al. and Kendon et al. We find a late-stage scaling exponent consistent with the inertial regime. However, as observed with the previous lattice-gas model of Appert et al. our data does not fall in the same range of reduced length and time as that of Kendon et al. For off-critical binary spinodal decomposition we observe a reduction of the effective exponent with the volume fraction of the minority phase. However, the n=1/3 Lifshitz-Slyzov-Wagner droplet coalescence exponent is not observed. Adding a sufficient number of surfactant particles to a critical quench of binary immiscible fluids produces a ternary bicontinuous microemulsion. We observe a change in scaling behaviour from algebraic to logarithmic growth for am...
We show that a phenomenological hydrodynamic lattice-gas model of two-phase flow, developed by Ro... more We show that a phenomenological hydrodynamic lattice-gas model of two-phase flow, developed by Rothman and Keller in 1988 and used extensively for numerical simulations since then, can be derived from an underlying model of particle interactions. From this result, we elucidate the nature of the hydrodynamic limit of the Rothman-Keller model.
We generalize the hydrodynamic lattice gas model to include arbitrary numbers of particles moving... more We generalize the hydrodynamic lattice gas model to include arbitrary numbers of particles moving in each lattice direction. For this generalization we derive the equilibrium distribution function and the hydrodynamic equations, including the equation of state and the prefactor of the inertial term that arises from the breaking of galilean invariance in these models. We show that this prefactor can be set to unity in the generalized model, therby effectively restoring galilean invariance. Moreover, we derive an expression for the kinematic viscosity, and show that it tends to decrease with the maximum number of particles allowed in each direction, so that higher Reynolds numbers may be achieved. Finally, we derive expressions for the statistical noise and the Boltzmann entropy of these models. 1 Lattice Gases Lattice gas automata (LGA) are a class of dynamical systems in which particles move on a lattice in discrete time steps. If the collisions between the particles conserve mass a...
— A joint call for proposals was issued in late 2004- early 2005 by the US’s NSF and UK’s EPSRC f... more — A joint call for proposals was issued in late 2004- early 2005 by the US’s NSF and UK’s EPSRC for applications that aim to utilize the combined computational resources of the US and UK. In response to the call, three computational science groups from UCL, Tufts and Brown Universities teamed up with a middleware team from NIU/Argonne to meet the challenge. Successful demonstra-tions of these applications at SC05 established that not only can applications make effective use of resources that span computational grids but also underscored the need to run at the even larger scales of grids-of-grids. Although the groups had three distinct codes and aims, the projects had the underlying common feature that they were comprised of large-scale distributed applications which required high-end networking and advanced middleware in
Oil-water-surfactant systems can create a wide array of complex structures. We present simulation... more Oil-water-surfactant systems can create a wide array of complex structures. We present simulation studies of structure size growth rates in a two-dimensional lattice gas model of microemulsions. A series of temperature quenches from above the critical mixing temperature was simulated for several concentrations of surfactant in an oil-water-surfactant mixture. Spontaneous micellization was observerd for several quench depths in all surfactant
SIAM J. Appl. Math., 2021
In this work, we modify the Affine Wealth Model of wealth distributions to examine the effects of... more In this work, we modify the Affine Wealth Model of wealth distributions to examine the effects of nonconstant redistribution on the very wealthy. Previous studies of this model, restricted to flat redistribution schemes, have demonstrated the presence of a phase transition to a partially wealth-condensed state, or "partial oligarchy", at the critical value of an order parameter. These studies have also indicated the presence of an exponential tail in wealth distribution precisely at criticality. Away from criticality, the tail was observed to be Gaussian. In this work, we generalize the flat redistribution within the Affine Wealth Model to allow for an essentially arbitrary redistribution policy. We show that the exponential tail observed near criticality in prior work is in fact a special case of a much broader class of critical, slower-than-Gaussian decays that depend sensitively on the corresponding asymptotic behavior of the progressive redistribution model used. We th...
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Recent work on agent-based models of wealth distribution has yielded nonlinear, non-local Fokker–... more Recent work on agent-based models of wealth distribution has yielded nonlinear, non-local Fokker–Planck equations whose steady-state solutions describe empirical wealth distributions with remarkable accuracy using only a few free parameters. Because these equations are often used to solve the ‘inverse problem’ of determining the free parameters given empirical wealth data, there is much impetus to find fast and accurate methods of solving the ‘forward problem’ of finding the steady state corresponding to given parameters. In this work, we derive and calibrate a lattice Boltzmann equation for this purpose. This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.
Pattern Formation and Lattice gas Automata, 1995
A method is described for calculating corrections to the Boltzmann/Chapman-Enskog analysis of lat... more A method is described for calculating corrections to the Boltzmann/Chapman-Enskog analysis of lattice gases due to the buildup of correlations. It is shown that renormalized transport coefficients can be calculated perturbatively by summing terms in an infinite series. A diagrammatic notation for the terms in this series is given, in analogy with the Feynman diagrams of quantum field theory. This theory is applied to an example lattice gas and shown to correctly predict experimental deviation from the Boltzmann prediction.
The relativistic electromagnetic projection operators discovered by Fradkin are used to obtain a ... more The relativistic electromagnetic projection operators discovered by Fradkin are used to obtain a covariant decomposition of the motion of a relativistic charged particle into parallel motion and perpendicular gyration. The Lagrangian Lie transform method of Littlejohn is used to achieve a transformation to guiding-center coordinates in which the rapid oscillatory motion is removed. The natural guiding-center Poisson bracket structure and
We describe a three-dimensional hydrodynamic lattice-gas model of amphiphilic fluids. This model ... more We describe a three-dimensional hydrodynamic lattice-gas model of amphiphilic fluids. This model of the non-equilibrium properties of oil-water-surfactant systems, which is a non-trivial extension of an earlier two-dimensional realisation due to Boghosian, Coveney and Emerton [Boghosian, Coveney, and Emerton 1996, Proc. Roy. Soc. A 452, 1221-1250], can be studied effectively only when it is implemented using high-performance computing and visualisation techniques. We describe essential aspects of the model's theoretical basis and computer implementation, and report on the phenomenological properties of the model which confirm that it correctly captures binary oil-water and surfactant-water behaviour, as well as the complex phase behaviour of ternary amphiphilic fluids.
Fourier acceleration has been successfully applied to the simulation of lattice field theories fo... more Fourier acceleration has been successfully applied to the simulation of lattice field theories for more than a decade. In this paper, we extend the method to the dynamics of discrete particles moving in continuum. Although our method is based on a mapping of the particles' dynamics to a regular grid so that discrete Fourier transforms may be taken, it should be emphasized that the introduction of the grid is a purely algorithmic device and that no smoothing, coarse-graining or mean-field approximations are made. The method thus can be applied to the equations of motion of molecular dynamics (MD), or its Langevin or Brownian variants. For example, in Langevin MD simulations our acceleration technique permits a straightforward spectral decomposition of forces so that the long-wavelength modes are integrated with a longer time step, thereby reducing the time required to reach equilibrium or to decorrelate the system in equilibrium. Speedup factors of up to 30 are observed relative ...
Tsallis has proposed a generalization of Boltzmann-Gibbs thermostatistics by introducing a family... more Tsallis has proposed a generalization of Boltzmann-Gibbs thermostatistics by introducing a family of generalized nonextensive entropy functionals with a single parameter q. These reduce to the extensive Boltzmann-Gibbs form for q=1, but a remarkable number of statistical and thermodynamic properties have been shown to be q-invariant -- that is, valid for any q. In this paper, we address the question of whether or not the value of q for a given viscous, incompressible fluid can be ascertained solely by measurement of the fluid's hydrodynamic properties. We find that the hydrodynamic equations expressing conservation of mass and momentum are q-invariant, but that for conservation of energy is not. Moreover, we find that ratios of transport coefficients may also be q-dependent. These dependences may therefore be exploited to measure q experimentally.
A lattice gas model for Schloegl's second chemical reaction is described and analyzed. Becaus... more A lattice gas model for Schloegl's second chemical reaction is described and analyzed. Because the lattice gas does not obey a semi-detailed-balance condition, the equilibria are non-Gibbsian. In spite of this, a self-consistent set of equations for the exact homogeneous equilibria are described, using a generalized cluster-expansion scheme. These equations are solved in the two-particle BBGKY approximation, and the results are compared to numerical experiment. It is found that this approximation describes the equilibria far more accurately than the Boltzmann approximation. It is also found, however, that spurious solutions to the equilibrium equations appear which can only be removed by including effects due to three-particle correlations.
A complete formulation is given of an exact kinetic theory for lattice gases. This kinetic theory... more A complete formulation is given of an exact kinetic theory for lattice gases. This kinetic theory makes possible the calculation of corrections to the usual Boltzmann / Chapman-Enskog analysis of lattice gases due to the buildup of correlations. It is shown that renormalized transport coefficients can be calculated perturbatively by summing terms in an infinite series. A diagrammatic notation for the terms in this series is given, in analogy with the diagrammatic expansions of continuum kinetic theory and quantum field theory. A closed-form expression for the coefficients associated with the vertices of these diagrams is given. This method is applied to several standard lattice gases, and the results are shown to correctly predict experimentally observed deviations from the Boltzmann analysis.
We present a general methodology for constructing lattice Boltzmann models of hydrodynamics with ... more We present a general methodology for constructing lattice Boltzmann models of hydrodynamics with certain desired features of statistical physics and kinetic theory. We show how a methodology of linear programming theory, known as Fourier-Motzkin elimination, provides an important tool for visualizing the state space of lattice Boltzmann algorithms that conserve a given set of moments of the distribution function. We show how such models can be endowed with a Lyapunov functional, analogous to Boltzmann's H, resulting in unconditional numerical stability. Using the Chapman-Enskog analysis and numerical simulation, we demonstrate that such entropically stabilized lattice Boltzmann algorithms, while fully explicit and perfectly conservative, may achieve remarkably low values for transport coefficients, such as viscosity. Indeed, the lowest such attainable values are limited only by considerations of accuracy, rather than stability. The method thus holds promise for high-Reynolds num...
We investigate the growth kinetics of binary immiscible fluids and emulsions in two dimensions us... more We investigate the growth kinetics of binary immiscible fluids and emulsions in two dimensions using a hydrodynamic lattice-gas model. We perform off-critical quenches in the binary fluid case and find that the domain size within the minority phase grows algebraically with time in accordance with theoretical predictions. In the late time regime we find a growth exponent n = 0.45 over a wide range of concentrations, in good agreement with other simluations. In the early time regime we find no universal growth exponent but a strong dependence on the concentration of the minority phase. In the ternary amphiphilic fluid case the kinetics of self assembly of the droplet phase are studied for the first time. At low surfactant concentrations, we find that, after an early algebraic growth, a nucleation regime dominates the late-time kinetics, which is enhanced by an increasing concentration of surfactant. With a further increase in the concentration of surfactant, we see a crossover to loga...
In this work, we study numerically the convergence of the scalar D2Q9 lattice Boltzmann scheme wi... more In this work, we study numerically the convergence of the scalar D2Q9 lattice Boltzmann scheme with multiple relaxation times when the time step is proportional to the space step and tends to zero. We do this by a combination of theory and numerical experiment. The classical formal analysis when all the relaxation parameters are fixed and the time step tends to zero shows that the numerical solution converges to solutions of the heat equation, with a constraint connecting the diffusivity, the space step and the coefficient of relaxation of the momentum. If the diffusivity is fixed and the space step tends to zero, the relaxation parameter for the momentum is very small, causing a discrepency between the previous analysis and the numerical results. We propose a new analysis of the method for this specific situation of evanescent relaxation, based on the dispersion equation of the lattice Boltzmann scheme. A new asymptotic partial differential equation, the damped acoustic system, is ...
A Lattice Boltzmann formulation for relativistic fluids is presented and numerically verified thr... more A Lattice Boltzmann formulation for relativistic fluids is presented and numerically verified through quantitative comparison with recent hydrodynamic simulations of relativistic shock-wave propagation in viscous quark-gluon plasmas. This formulation opens up the possibility of exporting the main advantages of Lattice Boltzmann methods to the relativistic context, which seems particularly useful for the simulation of relativistic fluids in complicated geometries.
We simulate the dynamics of phase assembly in binary immiscible fluids and ternary microemulsions... more We simulate the dynamics of phase assembly in binary immiscible fluids and ternary microemulsions using a three-dimensional hydrodynamic lattice gas approach. For critical spinodal decomposition we perform the scaling analysis in reduced variables introduced by Jury et al. and Kendon et al. We find a late-stage scaling exponent consistent with the inertial regime. However, as observed with the previous lattice-gas model of Appert et al. our data does not fall in the same range of reduced length and time as that of Kendon et al. For off-critical binary spinodal decomposition we observe a reduction of the effective exponent with the volume fraction of the minority phase. However, the n=1/3 Lifshitz-Slyzov-Wagner droplet coalescence exponent is not observed. Adding a sufficient number of surfactant particles to a critical quench of binary immiscible fluids produces a ternary bicontinuous microemulsion. We observe a change in scaling behaviour from algebraic to logarithmic growth for am...
We show that a phenomenological hydrodynamic lattice-gas model of two-phase flow, developed by Ro... more We show that a phenomenological hydrodynamic lattice-gas model of two-phase flow, developed by Rothman and Keller in 1988 and used extensively for numerical simulations since then, can be derived from an underlying model of particle interactions. From this result, we elucidate the nature of the hydrodynamic limit of the Rothman-Keller model.
We generalize the hydrodynamic lattice gas model to include arbitrary numbers of particles moving... more We generalize the hydrodynamic lattice gas model to include arbitrary numbers of particles moving in each lattice direction. For this generalization we derive the equilibrium distribution function and the hydrodynamic equations, including the equation of state and the prefactor of the inertial term that arises from the breaking of galilean invariance in these models. We show that this prefactor can be set to unity in the generalized model, therby effectively restoring galilean invariance. Moreover, we derive an expression for the kinematic viscosity, and show that it tends to decrease with the maximum number of particles allowed in each direction, so that higher Reynolds numbers may be achieved. Finally, we derive expressions for the statistical noise and the Boltzmann entropy of these models. 1 Lattice Gases Lattice gas automata (LGA) are a class of dynamical systems in which particles move on a lattice in discrete time steps. If the collisions between the particles conserve mass a...
— A joint call for proposals was issued in late 2004- early 2005 by the US’s NSF and UK’s EPSRC f... more — A joint call for proposals was issued in late 2004- early 2005 by the US’s NSF and UK’s EPSRC for applications that aim to utilize the combined computational resources of the US and UK. In response to the call, three computational science groups from UCL, Tufts and Brown Universities teamed up with a middleware team from NIU/Argonne to meet the challenge. Successful demonstra-tions of these applications at SC05 established that not only can applications make effective use of resources that span computational grids but also underscored the need to run at the even larger scales of grids-of-grids. Although the groups had three distinct codes and aims, the projects had the underlying common feature that they were comprised of large-scale distributed applications which required high-end networking and advanced middleware in
Oil-water-surfactant systems can create a wide array of complex structures. We present simulation... more Oil-water-surfactant systems can create a wide array of complex structures. We present simulation studies of structure size growth rates in a two-dimensional lattice gas model of microemulsions. A series of temperature quenches from above the critical mixing temperature was simulated for several concentrations of surfactant in an oil-water-surfactant mixture. Spontaneous micellization was observerd for several quench depths in all surfactant
SIAM J. Appl. Math., 2021
In this work, we modify the Affine Wealth Model of wealth distributions to examine the effects of... more In this work, we modify the Affine Wealth Model of wealth distributions to examine the effects of nonconstant redistribution on the very wealthy. Previous studies of this model, restricted to flat redistribution schemes, have demonstrated the presence of a phase transition to a partially wealth-condensed state, or "partial oligarchy", at the critical value of an order parameter. These studies have also indicated the presence of an exponential tail in wealth distribution precisely at criticality. Away from criticality, the tail was observed to be Gaussian. In this work, we generalize the flat redistribution within the Affine Wealth Model to allow for an essentially arbitrary redistribution policy. We show that the exponential tail observed near criticality in prior work is in fact a special case of a much broader class of critical, slower-than-Gaussian decays that depend sensitively on the corresponding asymptotic behavior of the progressive redistribution model used. We th...
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Recent work on agent-based models of wealth distribution has yielded nonlinear, non-local Fokker–... more Recent work on agent-based models of wealth distribution has yielded nonlinear, non-local Fokker–Planck equations whose steady-state solutions describe empirical wealth distributions with remarkable accuracy using only a few free parameters. Because these equations are often used to solve the ‘inverse problem’ of determining the free parameters given empirical wealth data, there is much impetus to find fast and accurate methods of solving the ‘forward problem’ of finding the steady state corresponding to given parameters. In this work, we derive and calibrate a lattice Boltzmann equation for this purpose. This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.
Pattern Formation and Lattice gas Automata, 1995
A method is described for calculating corrections to the Boltzmann/Chapman-Enskog analysis of lat... more A method is described for calculating corrections to the Boltzmann/Chapman-Enskog analysis of lattice gases due to the buildup of correlations. It is shown that renormalized transport coefficients can be calculated perturbatively by summing terms in an infinite series. A diagrammatic notation for the terms in this series is given, in analogy with the Feynman diagrams of quantum field theory. This theory is applied to an example lattice gas and shown to correctly predict experimental deviation from the Boltzmann prediction.
The relativistic electromagnetic projection operators discovered by Fradkin are used to obtain a ... more The relativistic electromagnetic projection operators discovered by Fradkin are used to obtain a covariant decomposition of the motion of a relativistic charged particle into parallel motion and perpendicular gyration. The Lagrangian Lie transform method of Littlejohn is used to achieve a transformation to guiding-center coordinates in which the rapid oscillatory motion is removed. The natural guiding-center Poisson bracket structure and