arXiv subjects

Bruce M. Boghosian

Publications and source records attributed to Bruce M. Boghosian.

At least 19 recordsLinked to original sources

arXiv2025

A Dynamical Equation for the Lorenz Curve: Dynamics of incomplete moments of probability distributions arising from Fokker-Planck equations

Fokker-Planck equations (forward Kolmogorov equations) evolve probability densities in time from an initial condition. For distributions over the real line, these evolution equations can sometimes be transformed into dynamics over the incomplete zeroth and first moments. We call this perspective the Lorenz dynamics of the system after the Lorenz curve description of distributions of wealth. This offers the benefit of presenting the dynamics over a compact domain. The integral transformation is motivated and then stated for a general class of Fokker-Planck equations. Following this, the transformed equation is solved for the heat equation and some variants thereof. Finally, some equations arising from the application of kinetic theory to idealized economic systems are transformed and analyzed in this new light.

math.AP↗

arXiv2024

Bounding the approach to oligarchy in a variant of the yard-sale model

We present analytical results for the Gini coefficient of economic inequality under the dynamics of a modified Yard-Sale Model of kinetic asset exchange. A variant of the Yard-Sale Model is introduced by modifying the underlying binary transaction of the classical system. It is shown that the Gini coefficient is monotone under the resulting dynamics but the approach to oligarchy, as measured by the Gini index, can be bounded by a first-order differential inequality used in conjunction with the differential Gronwall inequality. This result is in the spirit of entropy -- entropy production inequalities for diffusive PDE. The asymptotics of the modified system, with a redistributive tax, are derived and shown to agree with the original, taxed Yard-Sale Model, which implies the modified system is as suitable for matching real wealth distributions. The Gini -- Gini production inequality is shown to hold for a broader class of models.

cond-mat.stat-mech↗

arXiv2021

The Nonuniversality of Wealth Distribution Tails Near Wealth Condensation Criticality

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 thereby demonstrate that the functional form of the tail of the wealth distribution in a near-critical society is not universal in nature but rather entirely determined by the specifics of public policy decisions. This is significant because most major economies today are observed to be near-critical.

q-fin.GN↗

arXiv2018

The Affine Wealth Model: An agent-based model of asset exchange that allows for negative-wealth agents and its empirical validation

We present a stochastic, agent-based, binary-transaction Asset-Exchange Model (AEM) for wealth distribution that allows for agents with negative wealth. This model retains certain features of prior AEMs such as redistribution and wealth-attained advantage, but it also allows for shifts as well as scalings of the agent density function. We derive the Fokker-Planck equation describing its time evolution and we describe its numerical solution, including a methodology for solving the inverse problem of finding the model parameters that best match empirical data. Using this methodology, we compare the steady-state solutions of the Fokker-Planck equation with data from the United States Survey of Consumer Finances over a time period of 27 years. In doing so, we demonstrate agreement with empirical data of an average error less than 0.16\% over this time period. We present the model parameters for the US wealth distribution data as a function of time under the assumption that the distribution responds to their variation adiabatically. We argue that the time series of model parameters thus obtained provides a valuable new diagnostic tool for analyzing wealth inequality.

q-fin.GN↗

arXiv2017

Efficient lattice Boltzmann models for the Kuramoto-Sivashinsky equation

In this work, we improve the accuracy and stability of the lattice Boltzmann model for the Kuramoto-Sivashinsky equation proposed in \cite{2017_Otomo}. This improvement is achieved by controlling the relaxation time, modifying the equilibrium state, and employing more and higher lattice speeds, in a manner suggested by our analysis of the Taylor-series expansion method. The model's enhanced stability enables us to use larger time increments, thereby more than compensating for the extra computation required by the high lattice speeds. Furthermore, even though the time increments are larger than those of the previous scheme, the same level of accuracy is maintained because of the smaller truncation error of the new scheme. As a result, total performance with the new scheme on the D1Q7 lattice is improved by 92 $\%$ compared to the original scheme on the D1Q5 lattice.

physics.comp-ph↗

arXiv2016

The Growth of Oligarchy in a Yard-Sale Model of Asset Exchange: A Logistic Equation for Wealth Condensation

The addition of wealth-attained advantage (WAA) to the Yard-Sale Model (YSM) of asset exchange has been demonstrated to induce wealth condensation. In a model of WAA for which the bias is a continuous function of the wealth difference of the transacting agents, the condensation was shown to arise from a second-order phase transition to a coexistence regime. In this paper, we present the first analytic time-dependent results for this model, by showing that the condensed wealth obeys a logistic equation in time.

q-fin.GN↗

arXiv2016

Oligarchy as a Phase Transition: The effect of wealth-attained advantage in a Fokker-Planck description of asset exchange

In earlier work, we derived a nonlinear, nonlocal Fokker-Planck equation for the Yard-Sale Model of asset exchange. In the absence of redistribution, we showed that the Gini coefficient is a Lyapunov functional for this model, tending to one in the time-asymptotic limit, corresponding to maximal inequality. When a one-parameter model of redistribution is introduced, we showed that the model admits a steady state similar to Pareto's Law. In this work, we analyze the form of this distribution in greater detail, both analytically and numerically. We find that, while Pareto's Law is approximately valid for low redistribution, it gives way to something like Gibrat's Law at higher redistribution. We also prove that, while this Pareto or Gibrat behavior persists over many orders of magnitude, it ultimately gives way to gaussian decay at extremely large wealth. Following the work of Moukarzel et al., we introduce a bias in favor of the wealthier agent. We derive the corresponding modification to the Fokker-Planck equation, and we show this leads to wealth condensation when the bias exceeds a critical value. Earlier work took the bias to be a discontinuous function of the wealth differential between the two transacting agents, and reported a first-order phase transition to absolute oligarchy. By contrast, in this work we take the bias to be a continuous function of the wealth differential, and consequently we observe a second-order phase transition with a region of coexistence between the oligarch and a distribution of non-oligarchs. We additionally show that the onset of wealth condensation has a reciprocal effect on the character of the non-oligarchical part of the distribution. Specifically, we show that the above-mentioned gaussian decay at extremely large wealth is valid both above and below criticality, but degenerates to exponential decay precisely at criticality.

physics.soc-ph↗

arXiv2014

An $H$ theorem for Boltzmann's equation for the Yard-Sale Model of asset exchange

In recent work, Boltzmann and Fokker-Planck equations were derived for the "Yard-Sale Model" of asset exchange. For the version of the model without redistribution, it was conjectured, based on numerical evidence, that the time-asymptotic state of the model was oligarchy -- complete concentration of wealth by a single individual. In this work, we prove that conjecture by demonstrating that the Gini coefficient, a measure of inequality commonly used by economists, is an $H$ function of both the Boltzmann and Fokker-Planck equations for the model.

q-fin.GN↗

arXiv2014

Fokker-Planck Description of Wealth Dynamics and the Origin of Pareto's Law

The so-called "Yard-Sale Model" of wealth distribution posits that wealth is transferred between economic agents as a result of transactions whose size is proportional to the wealth of the less wealthy agent. In recent work [B.M. Boghosian, "Kinetics of Wealth and the Pareto Law," {\it Phys. Rev. E} {\bf 89} (2014) 042804], it was shown that this results in a Fokker-Planck equation governing the distribution of wealth. With the addition of a mechanism for wealth redistribution, it was further shown that this model results in stationary wealth distributions that are very similar in form to Pareto's well known law. In this paper, a much simpler derivation of that Fokker-Planck equation is presented.

q-fin.GN↗

arXiv2013

The Kinetics of Wealth and the Origin of the Pareto Law

An important class of economic models involve agents whose wealth changes due to transactions with other agents. Several authors have pointed out an analogy with kinetic theory, which describes molecules whose momentum and energy changes due to interactions with other molecules. We pursue this analogy and derive a Boltzmann equation for the time evolution of the wealth distribution of a population of agents for the so-called Yard-Sale Model of wealth exchange. We examine the solutions to this equation by a combination of analytical and numerical methods, and investigate its long-time limit. We study an important limit of this equation for small transaction sizes, and derive a partial integrodifferential equation governing the evolution of the wealth distribution in a closed economy. We then describe how this model may be extended to include features such as inflation, production and taxation. In particular, we show that the model with taxation is capable of explaining the basic features of the Pareto law, namely a lower cutoff to the wealth density at small values of wealth, and approximate power-law behavior at large values of wealth.

physics.soc-ph↗

arXiv2010

A Robust Numerical Method for Integration of Point-Vortex Trajectories in Two Dimensions

The venerable 2D point-vortex model plays an important role as a simplified version of many disparate physical systems, including superfluids, Bose-Einstein condensates, certain plasma configurations, and inviscid turbulence. This system is also a veritable mathematical playground, touching upon many different disciplines from topology to dynamic systems theory. Point-vortex dynamics are described by a relatively simple system of nonlinear ODEs which can easily be integrated numerically using an appropriate adaptive time stepping method. As the separation between a pair of vortices relative to all other inter-vortex length scales decreases, however, the computational time required diverges. Accuracy is usually the most discouraging casualty when trying to account for such vortex motion, though the varying energy of this ostensibly Hamiltonian system is a potentially more serious problem. We solve these problems by a series of coordinate transformations: We first transform to action-angle coordinates, which, to lowest order, treat the close pair as a single vortex amongst all others with an internal degree of freedom. We next, and most importantly, apply Lie transform perturbation theory to remove the higher-order correction terms in succession. The overall transformation drastically increases the numerical efficiency and ensures that the total energy remains constant to high accuracy.

nlin.CD↗

arXiv2008

Exact Hydrodynamics of the Lattice BGK Equation

We apply the projection operator formalism to the problem of determining the asymptotic behavior of the lattice BGK equation in the hydrodynamic limit. As an alternative to the more usual Chapman-Enskog expansion, this approach offers many benefits. Most remarkably, it produces absolutely exact, though non-Markovian, hydrodynamic difference equations as an intermediate step. These are accurate to all orders in Knudsen number and hence contain all of the physics of the Burnett equations and beyond. If appropriate, these equations may then be Taylor expanded to second order in Knudsen number to obtain the usual hydrodynamic equations that result from the Chapman-Enskog analysis. The method offers the potential to derive hydrodynamic difference equations for complex fluids with sharp gradients, such as immiscible and amphiphilic flow, for which the assumptions underlying the Chapman-Enskog approach are generally invalid.

nlin.CG↗

arXiv2008

An order-preserving property of additive invariant for Takesue-type reversible cellular automata

We show that, for a fairly large class of reversible, one-dimensional cellular automata, the set of additive invariants exhibits an algebraic structure. More precisely, if $f$ and $g$ are one-dimensional, reversible cellular automata of the kind considered by Takesue, we show that there is a binary operation on these automata $\vee$ such that $ψ(f)\subseteq ψ(f\vee g)$, where $ψ(f)$ denotes the set of additive invariants of $f$ and $\subseteq$ denotes the inclusion relation between real subspaces.

nlin.CG↗

arXiv2005

Type-II Quantum Algorithms

We review and analyze the hybrid quantum-classical NMR computing methodology referred to as Type-II quantum computing. We show that all such algorithms considered so far within this paradigm are equivalent to some classical lattice-Boltzmann scheme. We derive a sufficient and necessary constraint on the unitary operator representing the quantum mechanical part of the computation which ensures that the model reproduces the Boltzmann approximation of a lattice-gas model satisfying semi-detailed balance. Models which do not satisfy this constraint represent new lattice-Boltzmann schemes which cannot be formulated as the average over some underlying lattice gas. We close the paper with some discussion of the strengths, weaknesses and possible future direction of Type-II quantum computing.

quant-ph↗

arXiv2005

From Dirac to Diffusion: Decoherence in Quantum Lattice Gases

We describe a model for the interaction of the internal (spin) degree of freedom of a quantum lattice-gas particle with an environmental bath. We impose the constraints that the particle-bath interaction be fixed, while the state of the bath is random, and that the effect of the particle-bath interaction be parity invariant. The condition of parity invariance defines a subgroup of the unitary group of actions on the spin degree of freedom and the bath. We derive a general constraint on the Lie algebra of the unitary group which defines this subgroup, and hence guarantees parity invariance of the particle-bath interaction. We show that generalizing the quantum lattice gas in this way produces a model having both classical and quantum discrete random walks as different limits. We present preliminary simulation results illustrating the intermediate behavior in the presence of weak quantum noise.

quant-ph↗

arXiv2005

Lattice-gas simulations of dynamical geometry in one dimension

We present numerical results obtained using a lattice-gas model with dynamical geometry defined by Hasslacher and Meyer (Int. J. Mod. Phys. C. 9 1597 (1998)). The (irreversible) macroscopic behaviour of the geometry (size) of the lattice is discussed in terms of a simple scaling theory and obtained numerically. The emergence of irreversible behaviour from the reversible microscopic lattice-gas rules is discussed in terms of the constraint that the macroscopic evolution be reproducible. The average size of the lattice exhibits power law growth with exponent 1/2 at late times. The deviation of the macroscopic behaviour from reproducibility for particular initial conditions (``rogue states'') is investigated as a function of system size. The number of such ``rogue states'' is observed to decrease with increasing system size. Two mean-field analyses of the macroscopic behaviour are also presented.

cond-mat.stat-mech↗

arXiv2003

Covariant Lagrangian Methods of Relativistic Plasma Theory

We obtain a covariant decomposition of the motion of a relativistic charged particle into parallel motion and perpendicular gyration, and transform to guiding-center coordinates using Lie transforms. The natural guiding-center Poisson bracket structure and Hamiltonian are derived. The guiding-center equations of motion are presented to one order higher than the usual drifts, and the correction to the gyromomentum is given. We then allow for eikonal wave perturbations to the Lagrangian action. We develop a manifestly gauge-invariant and covariant oscillation-center theory to arbitrarily high order, and thereby derive the relativistic ponderomotive Hamiltonian. We sum the guiding-center action over a distribution and add the Maxwell action to obtain the total action of a guiding-center plasma. Upon variation, this yields self-consistent covariant relativistic kinetic and field equations; from these we identify the guiding-center current density and the guiding-center magnetization. Noether's theorem then yields covariant conservation laws for the momentum-energy and the angular momentum of a relativistic guiding-center plasma; from these we identify the guiding-center stress-energy tensor and the guiding-center spin angular momentum tensor. Repeating this calculation for guiding/oscillation centers, we find that variation yields self-consistent relativistic kinetic and field equations for the plasma in the wave field, including the wave dispersion relation; from these we identify the wave magnetization and susceptibility. Noether's theorem then yields conservation laws for the guiding-center plasma in the presence of a wave field, including the wave contribution to the stress-energy and spin angular momentum tensors.

physics.plasm-ph↗

arXiv2003

On the dependence of the Navier Stokes equations on the distribution of moleular velocities

In this work we introduce a completely general Chapman Enskog procedure in which we divide the local distribution into an isotropic distribution with anisotropic corrections. We obtain a recursion relation on all integrals of the distribution function required in the derivation of the moment equations. We obtain the hydrodynamic equations in terms only of the first few moments of the isotropic part of an arbitrary local distribution function. The incompressible limit of the equations is completely independent of the form of the isotropic part of the distribution, whereas the energy equation in the compressible case contains an additional contribution to the heat flux. This additional term was also found by Boghosian and by Potiguar and Costa in the derivation of the Navier Stokes equations for Tsallis thermostatistics, and is the only additional term allowed by the Curie principle.

cond-mat.stat-mech↗