Classical Many-body Systems Research Articles

Molecular dynamics simulations of spin and pure liquids with preservation of all the conservation laws.

A methodology is developed to integrate numerically the equations of motion for classical many-body systems in molecular dynamics simulations. Its distinguishable feature is the possibility to preserve, independently on the size of the time step, all the conservation laws inherent in the description without breaking the time reversibility. As a result, an implicit second-order algorithm is derived and applied to pure liquids, as well as spin liquids, for which the dynamics is characterized by the conservation of total energy, linear and angular momenta, as well as magnetization and individual spin lengths. It is demonstrated on the basis of Lennard-Jones and Heisenberg fluid models that when such quantities as energy and magnetization must be conserved perfectly, the algorithm turns out to be more efficient than popular decomposition integrators and standard predictor-corrector schemes.

Iso-g(2) Processes in Equilibrium Statistical Mechanics

The pair correlation function g(2)(r) in a classical many-body system depends in a nontrivial way both on the number density ρ and on the pair interactions ν(r), and a long-standing goal of statistical mechanics has been to predict these effects quantitatively. The present investigation focuses on a restricted circumstance whereby simultaneous isothermal changes in ρ and ν(r) have exactly canceling effects on g(2). By appealing to the isothermal compressibility relation, we establish that an upper limit for density increase exists for this “iso-g(2)” process, and at this limit in three dimensions the correspondingly modified pair interaction develops a long-ranged Coulombic character. Using both the standard hypernetted chain and Percus−Yevick approximations, we have examined the iso-g(2) process for rigid rods in one dimension that starts at zero density, and maintains the simple step-function pair correlation during density increase, a process that necessarily terminates at a covering fraction of one-ha...

Fluctuation Theorem for Heat Flow

Thermal conduction in a classical many-body system which is in contact with two isothermal reservoirs maintained at different temperatures is considered. The probability that when observed for a finite time, the heat flux of a finite system flows in the reverse direction to that required by the Second Law of Thermodynamics is calculated from first principles. Analytical expressions are given for the probability of observing Second Law violating fluctuations in this system. These expressions constitute an application of the fluctuation theorem to the problem of thermal conduction. The expressions are tested using nonequilibrium molecular dynamics simulations of heat flow between thermostated walls.

Effective field theory of slowly moving `extreme black holes'

We consider the non-relativistic effective field theory of`extreme black holes' in the Einstein-Maxwell-dilaton theory withan arbitrary dilaton coupling. We investigatethe finite-temperature behaviourof agas of `extreme black holes' using the effective theory.The total energy of the classical many-body system is also derived.

Regular Keplerian motions in classical many-body systems

A clear and simple physical approach to the explanation of exact particular solutions of the classical many-body problem is suggested. When the motion of individual bodies coupled by mutual gravitational forces in a many-body system occurs along conic sections, each body can be treated as moving not under the pull of the other moving bodies, but rather under a stationary central inverse-square gravitational field. These solutions describing possible amazingly simple (Keplerian) many-body motions are illustrated by computer simulations. Some pedagogical and philosophical aspects of the problem are discussed.

Comment on "Solution of classical stochastic one-dimensional many-body Systems"

In a recent Letter, Bares and Mobilia proposed the method to find solutions of the stochastic evolution operator $H=H_0 + {\gamma\over L} H_1$ with a non-trivial quartic term $H_1$. They claim, ``Because of the conservation of probability, an analog of the Wick theorem applies and all multipoint correlation functions can be computed.'' Using the Wick theorem, they expressed the density correlation functions as solutions of a closed set of integro-differential equations. In this Comment, however, we show that applicability of Wick theorem is restricted to the case $\gamma = 0$ only.

Solution of Classical Stochastic One-Dimensional Many-Body Systems

We propose a simple method that allows, in one dimension, to solve exactly a wide class of classical stochastic many-body systems far from equilibrium. For the sake of illustration and without loss of generality, we focus on a model that describes the asymmetric diffusion of hard core particles in the presence of an external source and instantaneous annihilation. Starting from a Master equation formulation of the problem we show that the density and multi-point correlation functions obey a closed set of integro-differential equations which in turn can be solved numerically and/or analytically

Nonlinear Response for Time-Dependent External Fields: Shear Flow and Color Conductivity

We present a generalization of the nonlinear response theory for autonomous systems which can be applied to classical many-body systems in large time-dependent external fields. Our formalism represents the first practical application of response theory to such problems, and provides a method of evaluating averages of phase functions that is more efficient than direct computer simulation. Our expressions for the nonlinear time-dependent response are tested against nonequilibrium molecular dynamics computer simulation of two simple nonautonomous systems. The relation of our results to known special cases (time-dependent linear response and time-independent nonlinear response) is discussed.

Negative thermal expansion in the Gaussian core model

The pressure equation of state of the classical many-body system with Gaussian pair interactions has been examined over a wide density range. Both analytical and simulational methods have been involved. A simply-connected region of negative thermal expansion has emerged from the study. It includes portions of the BCC crystal and fluid sections of the equilibrium phase diagram.

Nonlinear Response for Time-dependent External Fields

We generalize nonlinear response theory for autonomous systems so that it describes the response of classical many-body systems to large time-dependent external fields. Our formalism represents the first practical application of response theory to such problems. Our expressions for the nonlinear time-dependent response are tested against nonequilibrium molecular dynamics computer simulation. The relation of our results to known special cases (time-dependent linear response and time-independent nonlinear response) is discussed.

Exact time evolution of the density of a classical many-body system: The open one-dimensional gravitational gas.

We introduce an exact equation for the time evolution of a classical many-body system, and apply it to the one-dimensional gravitational gas. An irreversible approach to a final density distribution is found for a large class of initial momentum distributions, allowing us to introduce the idea of ``semiergodicity.'' \textcopyright{} 1996 The American Physical Society.

Chaotic scattering theory, thermodynamic formalism, and transport coefficients.

The foundations of the chaotic scattering theory for transport and reaction-rate coefficients for classical many-body systems are considered here in some detail. The thermodynamic formalism of Sinai, Ruelle, and Bowen [D. Ruelle, Thermodynamic Formalism (Addison-Wesley, Reading, MA, 1978)] is employed to obtain an expression for the escape rate for a phase-space trajectory of a system to leave a finite region of phase space for the first time. This expression relates the escape rate to the difference between the sum of the positive Lyapunov exponents and the Kolmogorov-Sinai entropy for the fractal set of phase-space trajectories that are trapped forever in the finite region. This relation is well known for systems of a few degrees of freedom and is extended here to systems with many degrees of freedom. The formalism is applied to smooth hyperbolic systems, to cellular-automata lattice gases, and to hard-sphere systems. In the last case, the geometric constructions of Sinai and co-workers [Russ. Math. Surv. 25, 137 (1970); 42, 181 (1987)] for billiard systems are used to describe the relevant chaotic scattering phenomena. Some applications of this formalism to nonhyperbolic systems are also discussed.

A Classical Approach in Predictive Statistical Mechanics: A Generalized Boltzmann Formalism

We consider a mechano-statistical formalism for the description of nonequilibrium classical Hamiltonian many-body systems. It is described how such formalism, the so called nonequilibrium statistical operator method at the classical mechanical level is obtained through the use of a variational principle that recovers known approaches as particular cases. The method is shown to be encompassed in the context of Jaynes' Predictive Statistical Mechanics. On the basis of this formalism it is shown how to obtain a nonlinear generalized transport theory of large scope. This theory is applied to derive the equations of evolution for the single- and two-particle distribution functions to obtain generalized transport equations including relaxation effects to all orders in the interaction strength, valid, in principle, for arbitrary nonequilibrium dissipative states. The classical Boltzmann equation and Boltzmann's H-theorem are recovered within restrictive approximations. Finally we discuss the connection with phenomenological irreversible thermodynamics, for which the method provides microscopic foundations in what is termed Informational Statistical Thermodynamics. The questions of entropy production and a generalized H-theorem are considered and conceptual aspects of the method are discussed.

A special‐purpose computer for molecular dynamics: GRAPE‐2A

Molecular dynamics simulations have been extensively used in research of proteins. Since these simulations are quite computer intensive, their acceleration is of main interest of the research. In molecular dynamics simulations, almost all computing time is consumed in calculating the forces between particles, e.g., Coulomb and van der Waals forces. We have designed and built GRAPE-2A (GRAvity PipE 2A), a special-purpose computer for use in simulations of classical many-body systems. GRAPE-2A calculates forces exerted on a particle from the other particles. GRAPE-2A can calculate force of an arbitrary functional form of a central force. The host computer, which is connected to GRAPE-2A through the VME bus, performs other calculations such as time integration. The peak speed of GRAPE-2A is 180 Mflops. We can also stimulate systems with periodic boundary conditions by the Ewald method, using GRAPE-2A and another special-purpose computer, WINE (Wave space INtegrator for the Ewald method).

Classical solitons for a one-dimensional many-body system with inverse-square interaction.

We investigate solitons in the one-dimensional classical many-body system of particles interacting by an inverse-square pair potential. We use a combination of numerical results for finite systems, and exact results for the continuum limit, to deduce properties of solitons in the thermodynamic limit. Two remarkable conjectures are supported by these results, and the correspondence with the exact results for the quantum system is established.

Unpredictability of symmetry breaking in a phase transition.

We study the sensitive dependence on initial conditions of the final states of systems undergoing a symmetry-breaking phase transition. If the knowledge of the initial state is incomplete the macroscopic final state is unpredictable. This is demonstrated by computer simulations for two simple classical many-body systems. Next, this unpredictability is discussed within the more general framework of quantum field theories. For infinite systems a mixing property is exposed that prevents long-time predictions on the basis of local measurements. Finally, we remark on the more philosophical question whether the symmetry breaking of a highly symmetric theory after the big bang can be predicted or has to be considered as an historical accident.

A new Monte Carlo power method for the eigenvalue problem of transfer matrices

We propose a new Monte Carlo method for calculating eigenvalues of transfer matrices leading to free energies and to correlation lengths of classical and quantum many-body systems. Generally, this method can be applied to the calculation of the maximum eigenvalue of a nonnegative matrix  such that all the matrix elements of Âk are strictly positive for an integerk. This method is based on a new representation of the maximum eigenvalue of the matrix  as the thermal average of a certain observable of a many-body system. Therefore one can easily calculate the maximum eigenvalue of a transfer matrix leading to the free energy in the standard Monte Carlo simulations, such as the Metropolis algorithm. As test cases, we calculate the free energies of the square-lattice Ising model and of the spin-1/2XY Heisenberg chain. We also prove two useful theorems on the ergodicity in quantum Monte Carlo algorithms, or more generally, on the ergodicity of Monte Carlo algorithms using our new representation of the maximum eigenvalue of the matrixÂ.

Parametric correlations and diffusion in quantum spectra.

The evolution of the quantum spectrum as the function of a control parameter can be mapped onto the molecular dynamics of a classical many-body system. This approach is used to study the parametric motion of energy levels in classically mostly regular and in mostly chaotic regimes. The parametric energy-level dynamics is described in terms of the velocity autocorrelation function and the friction kernel of a disordered many-body system. We show that the parametric dynamics is sensitively dependent on the underlying classical dynamics, and therefore provide new measures of quantum chaology

Self-consistent mean-field analysis of classical many-body systems at finite temperature

We present a methodology for calculating the thermodynamic and configurational properties of a classical system of N interacting particles. For solids we show how the method can be used to determine the structural properties in the equilibrium state. As an application we analyze the formation of misfit dislocations in a nickel-gold bicrystal with a (100) interface. We show that the dislocation exists in a state of minimum Helmholtz free energy. The equilibrium structure and energy are also determined through Monte Carlo calculations and they show good agreement with the results obtained from our method.

Infinite-temperature dynamics of the equivalent-neighbor XYZ model.

The dynamics of the classical XYZ model with uniform interaction is investigated by the recursion method and, in part, by exact analysis. The time evolution is anharmonic for arbitrary N (number of spins); only the cases N=2 and \ensuremath{\infty} are completely integrable. For the special (uniaxially symmetric) equivalent-neighbor XXZ model, the nonlinearities in the equations of motion disappear in the limit N\ensuremath{\rightarrow}\ensuremath{\infty}, and the spin autocorrelation functions are determined exactly for infinite temperature: The function 〈${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$(t)${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$〉 exhibits a Gaussian decay to a nonzero constant, and the function 〈${\mathit{S}}_{\mathit{i}}^{\mathit{x}}$(t)${\mathit{S}}_{\mathit{i}}^{\mathit{x}}$〉 decays to zero, algebraically or like a Gaussian, depending on the amount of uniaxial anisotropy. For the general XYZ case, the T=\ensuremath{\infty} dynamical behavior includes four different universality classes, categorized according to the decay law of the spectral densities at high frequencies. That decay law governs the growth rate of the sequence of recurrents that determine the relaxation function in the continued-fraction representation. The four universality classes may serve as prototypes for a classification of the dynamics of classical and quantum many-body systems in general.