From Nosé–Hoover chain to Nosé–Hoover network: design of non-Hamiltonian equations of motion for molecular-dynamics with multiple thermostats

A systematic algorithm to design multiple thermostat systems in the framework of the Nosé–Hoover type non-Hamiltonian formulation is presented. Using ‘non uniform’ time transformations in a generalised Hamiltonian equation, we develop the non-Hamiltonian equations of motion for multiple thermostat systems having an arbitrary number of thermostats and arbitrary connections between a physical system and thermostats (‘Nosé–Hoover network’). We then present the algorithm to construct the Nosé–Hoover network equations based on a simple diagram only. On the basis of this algorithm, recursively attached Nosé–Hoover thermostats are introduced as an example of the Nosé–Hoover network and its high efficiency in sampling the canonical distribution for an one-dimensional double-well system is illustrated by numerical calculations.     

Asymptotic Behavior of a Network of Oscillators Coupled to Thermostats of Finite Energy

We study the asymptotic behavior of a finite network of oscillators (harmonic or anharmonic) coupled to a number of deterministic Lagrangian thermostats of finite energy. In particular, we consider a chain of oscillators interacting with two thermostats situated at the boundary of the chain. Under appropriate assumptions, we prove that the vector (p, q) of moments and coordinates of the oscillators in the network satisfies (p, q)(t) → (0, q _c ) as t → ∞, where q _c is a critical point of some effective potential, so that the oscillators just stop. Moreover, we argue that the energy transport in the system stops as well without reaching thermal equilibrium. This result is in contrast to the situation when the energies of the thermostats are infinite, studied for a similar system in [14] and subsequent works, where the convergence to a nontrivial limiting regime was established.The proof is based on a method developed in [22], where it was observed that the thermostats produce some effective dissipation despite the Lagrangian nature of the system.     

Thermostatted kinetic equations as models for complex systems in physics and life sciences

Statistical mechanics is a powerful method for understanding equilibrium thermodynamics. An equivalent theoretical framework for nonequilibrium systems has remained elusive. The thermodynamic forces driving the system away from equilibrium introduce energy that must be dissipated if nonequilibrium steady states are to be obtained. Historically, further terms were introduced, collectively called a thermostat, whose original application was to generate constant-temperature equilibrium ensembles. This review surveys kinetic models coupled with time-reversible deterministic thermostats for the modeling of large systems composed both by inert matter particles and living entities. The introduction of deterministic thermostats allows to model the onset of nonequilibrium stationary states that are typical of most real-world complex systems. The first part of the paper is focused on a general presentation of the main physical and mathematical definitions and tools: nonequilibrium phenomena, Gauss least constraint principle and Gaussian thermostats. The second part provides a review of a variety of thermostatted mathematical models in physics and life sciences, including Kac, Boltzmann, Jager–Segel and the thermostatted (continuous and discrete) kinetic for active particles models. Applications refer to semiconductor devices, nanosciences, biological phenomena, vehicular traffic, social and economics systems, crowds and swarms dynamics.     

On the Fluctuation Relation for Nosé-Hoover Boundary Thermostated Systems

We discuss the transient and steady state fluctuation relation for a mechanical system in contact with two deterministic thermostats at different temperatures. The system is a modified Lorentz gas in which the fixed scatterers exchange energy with the gas of particles, and the thermostats are modelled by two Nosé-Hoover thermostats applied at the boundaries of the system. The transient fluctuation relation, which holds only for a precise choice of the initial ensemble, is verified at all times, as expected. Times longer than the mesoscopic scale, needed for local equilibrium to be settled, are required if a different initial ensemble is considered. This shows how the transient fluctuation relation asymptotically leads to the steady state relation when, as explicitly checked in our systems, the condition found in (D.J. Searles, et al., J. Stat. Phys. 128:1337, 2007), for the validity of the steady state fluctuation relation, is verified. For the steady state fluctuations of the phase space contraction rate Λ and of the dissipation function Ω, a similar relaxation regime at shorter averaging times is found. The quantity Ω satisfies with good accuracy the fluctuation relation for times larger than the mesoscopic time scale; the quantity Λ appears to begin a monotonic convergence after such times. This is consistent with the fact that Ω and Λ differ by a total time derivative, and that the tails of the probability distribution function of Λ are Gaussian.     

Thermostatting the atomic spin dynamics from controlled demons

Deterministic dynamics in extended phase space of a constant temperature interacting spin system is formulated. The spin temperature is recovered through the constrained equation of motion and is in agreement with Rugh’s geometrical approach to temperature for classical Heisenberg spin systems. Detailed comparisons are investigated between state-of-the-art stochastic spin dynamics and deterministic dynamics using a chain of thermostats, for which an accelerated convergence structure is found.     

Non-equilibrium dynamics of two-dimensional infinite particle systems with a singular interaction

The infinite system of Newton's equations of motion is considered for two-dimensional classical particles interacting by conservative two-body forces of finite range. Existence and uniqueness of solutions is proved for initial configurations with a logarithmic order of energy fluctuation at infinity. The semigroup of motion is also constructed and its continuity properties are discussed. The repulsive nature of interparticle forces is essentially exploited; the main condition on the interaction potential is that it is either positive or has a singularity at zero interparticle distance, which is as strong as that of an inverse fourth power.     

A Macroscopic System with Undamped Periodic Compressional Oscillations

A class of macroscopic systems is described which have the remarkable feature that they can sustain undamped compressional radial oscillations. They consist of an arbitrary number of particles confined by a harmonic potential and interacting among themselves through conservative forces scaling as the inverse cube of distances. The radial oscillation leads to a variation of the thermodynamic quantities characterizing the system. The system therefore does not approach equilibrium, since the (macroscopic) amplitude of the oscillation does not decrease as time goes to infinity. The oscillation is harmonic and isochronous, that is, its frequency is fixed and independent of the initial condition. These results hold independently of the dimension of the system and are also valid in the quantal context.     

The plastic coupled map lattice: A novel image-processing paradigm

Coupled map lattices (CML) can describe many relaxation and optimization algorithms currently used in image processing. We recently introduced the "plastic-CML" as a paradigm to extract (segment) objects in an image. Here, the image is applied by a set of forces to a metal sheet which is allowed to undergo plastic deformation parallel to the applied forces. In this paper we present an analysis of our "plastic-CML" in one and two dimensions, deriving the nature and stability of its stationary solutions. We also detail how to use the CML in image processing, how to set the system parameters and present examples of it at work. We conclude that the plastic-CML is able to segment images with large amounts of noise and large dynamic range of pixel values, and is suitable for a very large scale integration (VLSI) implementation.     

Anti-phase synchronization and ergodicity in arrays of oscillators coupled by an elastic force

We have proposed a mechanism of interaction between two non-linear dissipative oscillators, leading to exact and robust anti-phase and in-phase synchronization. The system we have analyzed is a model for the Huygens’s two pendulum clocks system, as well as a model for synchronization mediated by an elastic media. Here, we extend these results to arrays, finite or infinite, of conservative pendula coupled by linear elastic forces. We show that, for two interacting pendula, this mechanism leads always to synchronized anti-phase small amplitude oscillations, and it is robust upon variation of the parameters. For three or more interacting pendula, this mechanism leads always to ergodic non-synchronized oscillations. In the continuum limit, the pattern of synchronization is described by a quasi-periodic longitudinal wave.     

Dynamics of a pair of spherical gravitating shells

The dynamical N body problem for a system of mass points interacting solely through gravitational forces is not integrable. The difficulties which arise in constructing accurate numerical codes for simulating the motion over long time scales are legend. Thus, in order to test their theories, astronomers and astrophysicists resort to simpler, one-dimensional models which avoid the problems of binary formation, escape, and the singularity of the inverse square force law. To date, the most frequently employed "test" model consists of a system of parallel mass sheets moving perpendicular to their surface. While this system avoids all of the above problems, the time scale for reaching equilibrium is extremely long and probably arises from the system's weak ergodic properties, which become manifest even in the three sheet system. Here we consider a different one-dimensional gravitating system consisting of nonrotating concentric mass shells. For the case of two shells we investigate the structure of the phase space by studying the stability of periodic trajectories. By employing an event driven algorithm, we are able to directly investigate the influence of the singularity without having to resort to regularization of the force. Although stable structures are present at every energy, we find that the ergodic properties of this system are more robust than its planar counterpart. (c) 1997 American Institute of Physics.     

Ergodic Properties of Random Billiards Driven by Thermostats

We consider a class of mechanical particle systems interacting with thermostats. Particles move freely between collisions with disk-shaped thermostats arranged periodically on the torus. Upon collision, an energy exchange occurs, in which a particle exchanges its tangential component of the velocity for a randomly drawn one from the Gaussian distribution with the variance proportional to the temperature of the thermostat. In the case when all temperatures are equal one can write an explicit formula for the stationary distribution. We consider the general case and show that there exists a unique absolutely continuous stationary distribution. Moreover under rather mild conditions on the initial distribution the corresponding Markov dynamics converges to the equilibrium with exponential rate. One of the main technical difficulties is related to a possible overheating of moving particle. However as we show in the paper non-compactness of the particle velocity can be effectively controlled.     

Perturbed GUE Minor Process and Warren’s Process with Drifts

We consider the minor process of (Hermitian) matrix diffusions with constant diagonal drifts. At any given time, this process is determinantal and we provide an explicit expression for its correlation kernel. This is a measure on the Gelfand–Tsetlin pattern that also appears in a generalization of Warren’s process (Electron. J. Probab. 12:573–590, 2007), in which Brownian motions have level-dependent drifts. Finally, we show that this process arises in a diffusion scaling limit from an interacting particle system in the anisotropic KPZ class in 2+1 dimensions introduced in Borodin and Ferrari (Commun. Math. Phys., 2008). Our results generalize the known results for the zero drift situation.     

Heat transfer between two degrees of freedom

A particularly simple chaotic nonequilibrium open system with two Cartesian degrees of freedom, characterized by two distinct temperatures T(x) and T(y), is introduced. The two temperatures are maintained by Nose-Hoover canonical-ensemble thermostats. Both the equilibrium (no net heat transfer) and nonequilibrium (dissipative) Lyapunov spectra are characterized for this simple system.     

Dynamic Simulation of Random Packing of Polydispersive Fine Particles

In this paper, we perform molecular dynamic (MD) simulations to study the two-dimensional packing process of both monosized and random size particles with radii ranging from 1.0 to 7.0 μm. The initial positions as well as the radii of five thousand fine particles were defined inside a rectangular box by using a random number generator. Both the translational and rotational movements of each particle were considered in the simulations. In order to deal with interacting fine particles, we take into account both the contact forces and the long-range dispersive forces. We account for normal and static/sliding tangential friction forces between particles and between particle and wall by means of a linear model approach, while the long-range dispersive forces are computed by using a Lennard-Jones-like potential. The packing processes were studied assuming different long-range interaction strengths. We carry out statistical calculations of the different quantities studied such as packing density, mean coordination number, kinetic energy, and radial distribution function as the system evolves over time. We find that the long-range dispersive forces can strongly influence the packing process dynamics as they might form large particle clusters, depending on the intensity of the long-range interaction strength.     

Statistics of particle trajectories at short time intervals reveal fN-scale colloidal forces

We describe and implement a technique for extracting forces from the relaxation of an overdamped thermal system with normal modes. At sufficiently short time intervals, the evolution of a normal mode is well described by a one-dimensional Smoluchowski equation with constant drift velocity v, and diffusion coefficent D. By virtue of fluctuation dissipation, these transport coefficients are simply related to conservative forces, F, acting on the normal mode: F=kBTv/D. This relationship implicitly accounts for hydrodynamic interactions, requires no mechanical calibration, makes no assumptions about the form of conservative forces, and requires no prior knowledge of material properties. We apply this method to measure the electrostatic interactions of polymer microspheres suspended in nonpolar microemulsions.     

Fourier’s Law for a Microscopic Model of Heat Conduction

We consider a chain of N harmonic oscillators perturbed by a conservative stochastic dynamics and coupled at the boundaries to two gaussian thermostats at different temperatures. The stochastic perturbation is given by a diffusion process that exchange momentum between nearest neighbor oscillators conserving the total kinetic energy. The resulting total dynamics is a degenerate hypoelliptic diffusion with a smooth stationary state. We prove that the stationary state, in the limit as N→ ∞, satisfies Fourier’s law and the linear profile for the energy average     

Wave-Corpuscle Mechanics for Electric Charges

It is well known that the concept of a point charge interacting with the electromagnetic (EM) field has a problem. To address that problem we introduce the concept of wave-corpuscle to describe spinless elementary charges interacting with the classical EM field. Every charge interacts only with the EM field and is described by a complex valued wave function over the 4-dimensional space time continuum. A system of many charges interacting with the EM field is defined by a local, gauge and Lorentz invariant Lagrangian with a key ingredient—a nonlinear self-interaction term providing for a cohesive force assigned to every charge. An ideal wave-corpuscle is an exact solution to the Euler-Lagrange equations describing both free and accelerated motions. It carries explicitly features of a point charge and the de Broglie wave. Our analysis shows that a system of well separated charges moving with nonrelativistic velocities are represented accurately as wave-corpuscles governed by the Newton equations of motion for point charges interacting with the Lorentz forces. In this regime the nonlinearities are “stealthy” and don’t show explicitly anywhere, but they provide for the binding forces that keep localized every individual charge. The theory can also be applied to closely interacting charges as in hydrogen atom where it produces discrete energy spectrum.     

In-flight and collisional dissipation as a mechanism to suppress Fermi acceleration in a breathing Lorentz gas

Some dynamical properties for a time dependent Lorentz gas considering both the dissipative and non dissipative dynamics are studied. The model is described by using a four-dimensional nonlinear mapping. For the conservative dynamics, scaling laws are obtained for the behavior of the average velocity for an ensemble of non interacting particles and the unlimited energy growth is confirmed. For the dissipative case, four different kinds of damping forces are considered namely: (i) restitution coefficient which makes the particle experiences a loss of energy upon collisions; and in-flight dissipation given by (ii) F=-ηV(2); (iii) F=-ηV(μ) with μ≠1 and μ≠2 and; (iv) F=-ηV, where η is the dissipation parameter. Extensive numerical simulations were made and our results confirm that the unlimited energy growth, observed for the conservative dynamics, is suppressed for the dissipative case. The behaviour of the average velocity is described using scaling arguments and classes of universalities are defined.     

Long-time asymptotics of the long-range Emch-Radin model

The long-time asymptotic behaviour is studied for a long-range variant of the Emch-Radin model of interacting spins. We derive upper and lower bounds on the expectation values of a class of observables. We prove analytically that the time scale at which the system relaxes to equilibrium diverges with the system size N, displaying quasistationary nonequilibrium behaviour. This finding implies that, for large enough N, equilibration will not be observed in an experiment of finite duration.