Thermostating by deterministic scattering: construction of nonequilibrium steady states

We present a novel approach for constructing nonequilibrium steady states. It is based on a deterministic and time-reversible mechanism for dissipating energy from a subsystem into a thermal reservoir. The key idea is to thermalize a moving particle by appropriately modeling its microscopic collision rules with a boundary mimicking a thermal reservoir with arbitrarily many degrees of freedom. We demonstrate our method for the periodic Lorentz gas with an external electric field. By applying our thermostat we do not find an ergodic breakdown with increasing field strength.     

Thermodynamics based on the principle of least abbreviated action: Entropy production in a network of coupled oscillators

We present some novel thermodynamic ideas based on the Maupertuis principle. By considering Hamiltonians written in terms of appropriate action-angle variables we show that thermal states can be characterized by the action variables and by their evolution in time when the system is nonintegrable. We propose dynamical definitions for the equilibrium temperature and entropy as well as an expression for the nonequilibrium entropy valid for isolated systems with many degrees of freedom. This entropy is shown to increase in the relaxation to equilibrium of macroscopic systems with short-range interactions, which constitutes a dynamical justification of the Second Law of Thermodynamics. Several examples are worked out to show that this formalism yields the right microcanonical (equilibrium) quantities. The relevance of this approach to nonequilibrium situations is illustrated with an application to a network of coupled oscillators (Kuramoto model). We provide an expression for the entropy production in this system finding that its positive value is directly related to dissipation at the steady state in attaining order through synchronization.     

Nonequilibrium quasi-classical effective meson gas

We treat a gas of interacting relativistic effective mesons (similar to those produced in a heavy-ion collision), regarded as a nonequilibrium statistical system. The large number of degrees of freedom gives rise to statistical features. We suppose large occupation numbers, temperature somewhat below typical critical temperatures and the quasi-classical regime. At the initial time t0, the gas is off-equilibrium. We analyze the time evolution of the quasi-classical effective meson gas for t>t₀ by using a new moment method. Long-time approximations, which could yield the approach to global thermal equilibrium, are presented.     

Interacting Bosons at Finite Temperature: How Bogolubov Visited a Black Hole and Came Home Again

The structure of the thermal equilibrium state of a weakly interacting Bose gas is of current interest. We calculate the density matrix of that state in two ways. The most effective method, in terms of yielding a simple, explicit answer, is to construct a generating function within the traditional framework of quantum statistical mechanics. The alternative method, arguably more interesting, is to construct the thermal state as a vector state in an artificial system with twice as many degrees of freedom. It is well known that this construction has an actual physical realization in the quantum thermodynamics of black holes, where the added degrees of freedom correspond to the second sheet of the Kruskal manifold and the thermal vector state is a state of the Unruh or the Hartle–Hawking type. What is unusual about the present work is that the Bogolubov transformation used to construct the thermal state combines in a rather symmetrical way with Bogolubov's original transformation of the same form, used to implement the interaction of the nonideal gas in linear approximation. In addition to providing a density matrix, the method makes it possible to calculate efficiently certain expectation values directly in terms of the thermal vector state of the doubled system.     

Nonequilibrium quasi-classical effective meson gas: Thermalization

We consider a gas of interacting relativistic effective mesons (qualitatively, like those produced in a heavy-ion collision), regarded as an out-of-equilibrium statistical system. We suppose large occupation numbers, temperature somewhat below typical critical temperatures and the quasi-classical regime. At some initial time t₀, let the gas be in a nonequilibrium state, with spatial inhomogeneities. The time evolution of the gas for t > t ₀ is studied by a moment method, and appropriate long-time approximations, which could yield the approach to global thermal equilibrium, are discussed.     

Dynamical chaos in the Lorentz lattice gas

This paper provides an introduction to the applications of dynamical systems theory to nonequilibrium statistical mechanics, in particular to a study of nonequilibrium phenomena in Lorentz lattice gases with stochastic collision rules. Using simple arguments, based upon discussions in the mathematical literature, we show that such lattice gases belong to the category of dynamical systems with positive Lyapunov exponents. This is accomplished by showing how such systems can be expressed in terms of continuous phase space variables. Expressions for the Lyapunov exponent of a one-dimensional Lorentz lattice gas with periodic boundaries are derived. Other quantities of interest for the theory of irreversible processes are discussed.     

Equilibration,Generalized Equipartition,and Diffusion in Dynamical Lorentz Gases

We demonstrate approach to thermal equilibrium in the fully Hamiltonian evolution of a dynamical Lorentz gas, by which we mean an ensemble of particles moving through a d-dimensional array of fixed soft scatterers that each possess an internal harmonic or anharmonic degree of freedom to which moving particles locally couple. We analytically predict, and numerically confirm, that the momentum distribution of the moving particles approaches a Maxwell-Boltzmann distribution at a certain temperature T, provided that they are initially fast and the scatterers are in a sufficiently energetic but otherwise arbitrary stationary state of their free dynamics—they need not be in a state of thermal equilibrium. The temperature T to which the particles equilibrate obeys a generalized equipartition relation, in which the associated thermal energy k _B T is equal to an appropriately defined average of the scatterers’ kinetic energy. In the equilibrated state, particle motion is diffusive.     

Field-dependent conductivity and diffusion in a two-dimensional Lorentz gas

The conductivity and diffusion of a color-charged two-dimensional thermostatted Lorentz gas in a color field is studied by a variety of methods. In this gas, point particles move through a regular triangular array of soft scatterers, where, in the presence of a field, a nonequilibrium stationary state is reached by coupling to a Gaussian thermostat. The zero-field conductivity and diffusion coefficient are computed with equilibrium molecular dynamics dynamics from the Green-Kubo formula and the Einstein relation. Their values are consistent and approach those obtained by Machta and Zwanzig in the limit of hard (disk) scatterers. The field-dependent conductivity is obtained from its constitutive relation, from the coupling constant to the thermostat, and by using the recently derived conjugate pairing rule of Evans, Cohen, and Morriss, from the two maximal Lyapunov exponents of the Lorentz gas in the stationary state. All these methods give consistent results. Finally, elements of the field-dependent diffusion tensor have been computed. At zero field, they are consistent with the zero-field conductivity, but they vanish beyond a critical field strength, suggesting a dynamical phase transition at the critical field; the conductivity appears to remain finite, approaching a constant value for large field strengths.     

Nonequilibrium quantum meson gas: Dimensional reduction

A nonequilibrium quantum gas of interacting relativistic effective mesons, ressembling qualitatively those produced in a heavy-ion collision, is described by a scalar quantum field in (1 + 3) -dimensional Minkowski space. For high temperature and large temporal and spatial scales, we justify that classical statistical mechanics including quantum renormalization effects describe approximately the gas: nonequilibrium dimensional reduction (NEDR). As a source of hints, we treat the gas at equilibrium in real-time formalism and obtain simplifications for high temperature and large spatial scales, thereby extending a useful equilibrium dimensional reduction known for the imaginary-time formalism. By assumption, the nonequilibrium initial state of the gas, not far from thermal equilibrium, includes interactions and inhomogeneities. We use nonequilibrium real-time generating functionals and correlators at nonzero temperature. In the NEDR regime, our arguments yield: 1) renormalized correlators simplify, 2) the perturbative series for those simplified correlators can be resummed into a new nonequilibrium generating functional, Z’ _{r, dr} , which is super-renormalizable and includes renormalization effects (large position-dependent thermal self-energies and effective couplings). Z’ _{r, dr} could enable to study nonperturbatively changes in the phase structures of the field, by proceeding from the nonequilibrium quantum regime to the NEDR one.     

Two problems concerning hot hadronic matter and high energy collisions (Equilibrium formation,plasma deflagration)

This paper addresses two questions concerning the hydrodynamical approach to high energy collisions producing large multiplicities of hadrons. The first one concerns the difficulty of understanding in terms of successive parton interactions the formation of local thermal equilibrium for the small and short-lived blobs of excited hadronic matter created in such collisions. We argue that the number of successive parton interactions is not the only relevant factor for equilibrium formation, another factor being the early randomization present in all experiments which observe a subsystem of the complete final state and average over many unobserved degrees of freedom. This conjecture helps to understand the high degree of universality of hadronic jets and the fact that quite different dynamical models manage to describe the same data. The second problem concerns the hadronization of a blob of quark-gluon plasma as could be produced in a very high energy collision. Assuming the transition of plasma to hadron gas to have high latent heat, we show for small chemical potential that the plasma can deflagrate and convert a fraction of its latent heat into collective flow of the hadron gas. In such deflagrations very little entropy is produced, but the flow velocity of the hadron gas with respect to the plasma can be more than half of the velocity of light.     

Lorentz gas shear viscosity via nonequilibrium molecular dynamics and Boltzmann's equation

When nonequilibrium molecular dynamics is used to impose isothermal shear on a two-body periodic system of hard disks or spheres, the equations of motion reduce to those describing a Lorentz gas under shear. In this shearing Lorentz gas a single particle moves, isothermally, through a spatially periodic shearing crystal of infinitely massive scatterers. The curvilinear trajectories are calculated analytically and used to measure the dilute Lorentz gas viscosity at several strain rates. Simulations and solutions of Boltzmann's equation exhibit shear thinning resembling that found inN-body nonequilibrium simulations. For the three-dimensional Lorentz gas we obtained an exact expression for the viscosity which is valid at all strain rates. In two dimensions this is not possible due to the anisotropy of the scattering.     

Energy transfer in nonequilibrium space-charge-dominated beams

Space-charge modes similar to those observed in recent experiments appear in simulations of nonequilibrium charged particle beams with anisotropy. The modes couple degrees of freedom, causing energy transfer and equipartitioning without halo formation in just a few betatron wavelengths. The rate depends on a single free parameter quantifying the space-charge intensity of the final state. Traditional stability analyses are shown not to apply to high-intensity laboratory beams originating with a large perturbation from equilibrium.     

Cumulative author index of Volumes 351 to 360

The dynamic evolution of granular gases is fundamentally different from molecular gases due to the energy loss during collisions. Nevertheless techniques of kinetic theory are useful in a regime, when the granular particles are moving rapidly and the gas is sufficiently dilute. In these lecture notes we analyse in detail the collision of two rough particles which is inelastic due to incomplete normal and tangential restitution as well as Coulomb friction. Based on the Walton model a time evolution operator for the many particle system is introduced, a formalism which is well suited for simple approximations. We discuss free cooling of granular particles with particular emphasis on the exchange of energy between rotational and translational degrees of freedom.     

From quantal multiple scattering theory to a classical transport equation—A soluble model for nucleon-nucleus and nucleus-nucleus collisions at high energy

Several ad hoc models describe inelastic nucleon-nucleus and nucleus-nucleus collisions at high energy. Why do they work and how are they related? We investigate this question by studying an exactly soluble model. It is based on Glauber's multiple scattering approach. The following results are derived and discussed: (1) The inclusive cross section for the observation of one nucleon is the space integral of a Wigner transform. (2) The Wigner transform obeys a classical transport equation. (3) The equation is equivalent to the Boltzmann equation at high energy. (4) The interpenetration of two nuclei is viewed as a diffusion phenomenon governed by a Fokker-Planck equation. (5) Hydrodynamic equations are shown to yield approximate solutions to the transport equation. (6) A kind of thermal equilibrium is quickly reached in a nuclear collision process. (7) The equilibrium equation of state corresponds to an ideal gas with two degrees of freedom.     

Equipartition of rotational and translational energy in a dense granular gas

Experiments quantifying the rotational and translational motion of particles in a dense, driven, 2D granular gas floating on an air table reveal that kinetic energy is divided equally between the two translational and one rotational degrees of freedom. This equipartition persists when the particle properties, confining pressure, packing density, or spatial ordering are changed. While the translational velocity distributions are the same for both large and small particles, the angular velocity distributions scale with the particle radius. The probability distributions of all particle velocities have approximately exponential tails. Additionally, we find that the system can be described with a granular Boyle's law with a van?der?Waals-like equation of state. These results demonstrate ways in which conventional statistical mechanics can unexpectedly apply to nonequilibrium systems.     

Exchange Fluctuations in a Nonequilibrium Lorentz Gas

It has been shown that a correlation mechanism that is based on the exchange interaction and destroys the relation between distribution functions and response (Price relation) occurs in a nonequilibrium Lorentz gas (particles interact only with the thermostat). The physical nature of this phenomenon is that the scattering of particles of the gas in the same state on a single particle of the thermostat creates a flux of correlated pairs, which depends on the form of a nonequilibrium distribution function, making impossible the existence of a universal relation between distribution functions and response.     

Collision of a structurally composite particle with a barrier

We suggest a simple model of a structurally composite particle with internal degrees of freedom and study the simplest kinematical and dynamical properties of such a particle. The collision of a structurally composite particle with one internal degree of freedom with a barrier is analyzed in detail. We show that both total cooling and “heating” of the internal degree of freedom can be observed during the reflection of such a particle. We calculate such basic parameters of the collision as its duration, the number of collisions in the interaction time, and the velocities of the envelope and the internal particle after the collision. Properties characteristic of chaotic scattering are shown to appear when a structurally composite particle collides with a barrier.     

Quantum liquids of particles with generalized statistics

We propose a phenomenological approach to quantum liquids of particles obeying generalized statistics of a fermionic type, in the spirit of the Landau Fermi liquid theory. The approach is developed for fractional exclusion statistics. We discuss both equilibrium (specific heat, compressibility, and Pauli spin susceptibility) and nonequilibrium (current and thermal conductivities, thermopower) properties. Low-temperature quantities have the same temperature dependences as for the Fermi liquid, with the coefficients depending on the statistics parameter. The novel quantum liquids provide an explicit realization of systems with a non-Fermi liquid Lorentz ratio in two and more dimensions. Consistency of the theory is verified by deriving the compressibility and f-sum rules.     

Applications of periodic orbit theory toN-particle systems

In recent years a number of new techniques have become available in nonequilibrium statistical mechanics, all derived from dynamical system theory, especially from the thermodynamic formalism of Ruelle. We focus here on periodic orbit theory, and we compare it with a novel approach proposed by Evans, Cohen, and Morriss, and developed further by Gallavotti and Cohen. We argue that the two approaches based on such theories are equivalent for systems of many particles if the underlying dynamics is similar to that of Anosov systems, and that such equivalence should remain in more general situations. We extend our previous explanation of irreversibility in the thermostatted Lorentz gas toN-particle diffusion and shearing systems.