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E. G. D. Cohen

Publications and source records attributed to E. G. D. Cohen.

At least 73 records · Page 4Linked to original sources
arXiv1999

Gibbs entropy and irreversible thermodynamics

Recently a number of approaches has been developed to connect the microscopic dynamics of particle systems to the macroscopic properties of systems in nonequilibrium stationary states, via the theory of dynamical systems. This way a direct connection between dynamics and Irreversible Thermodynamics has been claimed to have been found. However, the main quantity used in these studies is a (coarse-grained) Gibbs entropy, which to us does not seem suitable, in its present form, to characterize nonequilibrium states. Various simplified models have also been devised to give explicit examples of how the coarse-grained approach may succeed in giving a full description of the Irreversible Thermodynamics. We analyze some of these models pointing out a number of difficulties which, in our opinion, need to be overcome in order to establish a physically relevant connection between these models and Irreversible Thermodynamics.

cond-mat.stat-mech↗
arXiv1999

Propagation and organization in lattice random media

We show that a signal can propagate in a particular direction through a model random medium regardless of the precise state of the medium. As a prototype, we consider a point particle moving on a one-dimensional lattice whose sites are occupied by scatterers with the following properties: (i) the state of each site is defined by its spin (up or down); (ii) the particle arriving at a site is scattered forward (backward) if the spin is up (down); (iii) the state of the site is modified by the passage of the particle, i.e. the spin of the site where a scattering has taken place, flips ($\uparrow \Leftrightarrow \downarrow $). We consider one dimensional and triangular lattices, for which we give a microscopic description of the dynamics, prove the propagation of a particle through the scatterers, and compute analytically its statistical properties. In particular we prove that, in one dimension, the average propagation velocity is $ = 1/(3-2q)$, with $q$ the probability that a site has a spin $\uparrow$, and, in the triangular lattice, the average propagation velocity is independent of the scatterers distribution: $ = 1/8$. In both cases, the origin of the propagation is a blocking mechanism, restricting the motion of the particle in the direction opposite to the ultimate propagation direction, and there is a specific re-organization of the spins after the passage of the particle. A detailed mathematical analysis of this phenomenon is, to the best of our knowledge, presented here for the first time.

cond-mat.stat-mech↗
arXiv1999

Topological origin of the phase transition in a mean-field model

We argue that the phase transition in the mean-field XY model is related to a particular change in the topology of its configuration space. The nature of this topological transition can be discussed on the basis of elementary Morse theory using the potential energy per particle V as a Morse function. The value of V where such a topological transition occurs equals the thermodynamic value of V at the phase transition and the number of (Morse) critical points grows very fast with the number of particles N. Furthermore, as in statistical mechanics, also in topology the way the thermodynamic limit is taken is crucial.

cond-mat.stat-mech↗
arXiv1999

Note on Two Theorems in Nonequilibrium Statistical Mechanics

An attempt is made to clarify the difference between a theorem derived by Evans and Searles in 1994 on the statistics of trajectories in phase space and a theorem proved by the authors in 1995 on the statistics of fluctuations on phase space trajectory segments in a nonequilibrium stationary state.

cond-mat↗
arXiv1998

Magnetic Properties of a Bose-Einstein Condensate

Three hyperfine states of Bose-condensed sodium atoms, recently optically trapped, can be described as a spin-1 Bose gas. We study the behaviour of this system in a magnetic field, and construct the phase diagram, where the temperature of the Bose condensation $T_{BEC}$ increases with magnetic field. In particular the system is ferromagnetic below $T_{BEC}$ and the magnetization is proportional to the condensate fraction in a vanishing magnetic field. Second derivatives of the magnetisation with regard to temperature or magnetic field are discontinuous along the phase boundary.

cond-mat.stat-mech↗
arXiv1998

The Fermi-Pasta-Ulam problem revisited: stochasticity thresholds in nonlinear Hamiltonian systems

The Fermi-Pasta-Ulam $α$-model of harmonic oscillators with cubic anharmonic interactions is studied from a statistical mechanical point of view. Systems of N= 32 to 128 oscillators appear to be large enough to suggest statistical mechanical behavior. A key element has been a comparison of the maximum Lyapounov coefficient $λ_{max}$ of the FPU $α$-model and that of the Toda lattice. For generic initial conditions, $λ_{max}(t)$ is indistinguishable for the two models up to times that increase with decreasing energy (at fixed N). Then suddenly a bifurcation appears, which can be discussed in relation to the breakup of regular, soliton-like structures. After this bifurcation, the $λ_{max}$ of the FPU model appears to approach a constant, while the $λ_{max}$ of the Toda lattice appears to approach zero, consistent with its integrability. This suggests that for generic initial conditions the FPU $α$-model is chaotic and will therefore approach equilibrium and equipartition of energy. There is, however, a threshold energy density $ε_c(N)\sim 1/N^2$, below which trapping occurs; here the dynamics appears to be regular, soliton-like and the approach to equilibrium - if any - takes longer than observable on any available computer. Above this threshold the system appears to behave in accordance with statistical mechanics, exhibiting an approach to equilibrium in physically reasonable times. The initial conditions chosen by Fermi, Pasta and Ulam were not generic and below threshold and would have required possibly an infinite time to reach equilibrium.

chao-dyn↗
arXiv1998

On the Validity of the Conjugate Pairing Rule for Lyapunov Exponents

For Hamiltonian systems subject to an external potential, which in the presence of a thermostat will reach a nonequilibrium stationary state, Dettmann and Morriss proved a strong conjugate pairing rule (SCPR) for pairs of Lyapunov exponents in the case of isokinetic (IK) stationary states which have a given kinetic energy. This SCPR holds for all initial phases of the system, all times t and all numbers of particles N. This proof was generalized by Wojtkovski and Liverani to include hard interparticle potentials. A geometrical reformulation of those results is presented. The present paper proves numerically, using periodic orbits for the Lorentz gas, that SCPR cannot hold for isoenergetic (IE) stationary states, which have a given total internal energy. In that case strong evidence is obtained for CPR to hold for large N and t, where it can be conjectured that the larger N, the smaller t will be. This suffices for statistical mechanics.

chao-dyn↗
arXiv1997

Note on Phase Space Contraction and Entropy Production in Thermostatted Hamiltonian Systems

The phase space contraction and the entropy production rates of Hamiltonian systems in an external field, thermostatted to obtain a stationary state are considered. While for stationary states with a constant kinetic energy the two rates are formally equal for all numbers of particles N, for stationary states with constant total (kinetic and potential) energy this only obtains for large N. However, in both cases a large number of particles is required to obtain equality with the entropy production rate of Irreversible Thermodynamics. Consequences of this for the positivity of the transport coefficients and for the Onsager relations are discussed. Numerical results are presented for the special case of the Lorentz gas.

cond-mat.stat-mech↗
arXiv1997

Orbital measures in non-equilibrium statistical mechanics: the Onsager relations

We assume that the properties of nonequilibrium stationary states of systems of particles can be expressed in terms of weighted orbital measures, i.e. through periodic orbit expansions. This allows us to derive the Onsager relations for systems of $N$ particles subject to a Gaussian thermostat, under the assumption that the entropy production rate is equal to the phase space contraction rate. Moreover, this also allows us to prove that the relevant transport coefficients are not negative. In the appendix we give an argument for the proper way of treating grazing collisions, a source of possible singularities in the dynamics.

chao-dyn↗
arXiv1997

Square root singularity in the viscosity of neutral colloidal suspensions at large frequencies

The asymptotic frequency $ω$, dependence of the dynamic viscosity of neutral hard sphere colloidal suspensions is shown to be of the form $η_0 A(ϕ) (ωτ_P)^{-1/2}$, where $A(ϕ)$ has been determined as a function of the volume fraction $ϕ$, for all concentrations in the fluid range, $η_0$ is the solvent viscosity and $τ_P$ the Péclet time. For a soft potential it is shown that, to leading order steepness, the asymptotic behavior is the same as that for the hard sphere potential and a condition for the cross-over behavior to $1/ωτ_P$ is given. Our result for the hard sphere potential generalizes a result of Cichocki and Felderhof obtained at low concentrations and agrees well with the experiments of van der Werff et al, if the usual Stokes-Einstein diffusion coefficient $D_0$ in the Smoluchowski operator is consistently replaced by the short-time self diffusion coefficient $D_s(ϕ)$ for non-dilute colloidal suspensions.

cond-mat.soft↗
arXiv1996

Scaling of Particle Trajectories on a Lattice I: Critical Behavior

The scaling behavior of the closed trajectories of a moving particle generated by randomly placed rotators or mirrors on a square or triangular lattice is studied numerically. For most concentrations of the scatterers the trajectories close exponentially fast. For special critical concentrations infinitely extended trajectories can occur which exhibit a scaling behavior similar to that of the perimeters of percolation clusters. In addition to the two critical exponents $τ=15/7$ and $d_{f}=7/4$ found before, the critical exponent $σ=3/7$, which is associated with the scaling function for trajectory size away from criticality, also appears. This exponent determines structural scaling properties of closed trajectories of finite size when they approach infinity, at criticality. New scaling behavior was found for the square lattice partially occupied by rotators, indicating a different universality class from that of percolation clusters. An argument for the scaling behavior found along the critical lines is presented.

cond-mat↗
arXiv1996

Scaling of Particle Trajectories on a Lattice II: The Critical Region

The scaling behavior of the closed trajectories of a moving particle generated by randomly placed rotators or mirrors on a square or triangular lattice in the critical region are investigated. We study numerically two scaling functions: $f(x)$ related to the trajectory length distribution $n_S$ and $h(x)$ related to the trajectory size $R_S$ (gyration radius) as introduced by Stauffer for the percolation problem, where $S$ is the length of a closed trajectory. The scaling function $f(x)$ is in most cases found to be symmetric double Gaussians with the same characteristic size exponent $σ=0.43\approx 3/7$ as was found at criticality. In contrast to previous assumptions of an exponential dependence of $n_S$ on $S$, the Gaussian functions lead to a stretched exponential dependence of $n_S$ on $S$, $n_S\sim e^{-S^{6/7}}$. However, for the rotator model on the partially occupied square lattice, an alternative scaling function near criticality is found, leading to a new exponent $σ'=1.6\pm0.3$ and a super exponential dependence of $n_S$ on $S$. The appearance of the same exponent $σ=3/7$ describing the behavior at and near the critical point is discussed. Our numerical simulations show that $h(x)$ is essentially a constant, which depends on the type of lattice and on the concentration of the scatterers.

cond-mat↗
arXiv1996

Boltzmann and Statistical Mechanics

Lecture given at the International Meeting ``Boltzmann's Legacy - 150 Years after his Birth'', organized by the Accademia Nazionale dei Lincei, 25 - 28 May 1994, in Rome, to be published in: ``Atti dell"Accademia Nazionale dei Lincei'', 1997.

cond-mat↗
arXiv1996

Viscosity of Colloidal Suspensions

Simple expressions are given for the Newtonian viscosity $η_N(ϕ)$ as well as the viscoelastic behavior of the viscosity $η(ϕ,ω)$ of neutral monodisperse hard sphere colloidal suspensions as a function of volume fraction $ϕ$ and frequency $ω$ over the entire fluid range, i.e., for volume fractions $0 < ϕ< 0.55$. These expressions are based on an approximate theory which considers the viscosity as composed as the sum of two relevant physical processes: $η(ϕ,ω) = η_{\infty}(ϕ) + η_{cd}(ϕ,ω)$, where $η_{\infty}(ϕ) = η_0 χ(ϕ)$ is the infinite frequency (or very short time) viscosity, with $η_0$ the solvent viscosity, $χ(ϕ)$ the equilibrium hard sphere radial distribution function at contact, and $η_{cd}(ϕ,ω)$ the contribution due to the diffusion of the colloidal particles out of cages formed by their neighbors, on the Péclet time scale $τ_P$, the dominant physical process in concentrated colloidal suspensions. The Newtonian viscosity $η_N(ϕ) = η(ϕ,ω= 0)$ agrees very well with the extensive experiments of Van der Werff et al and others. Also, the asymptotic behavior for large $ω$ is of the form $η_{\infty}(ϕ) + A(ϕ)(ωτ_P)^{-1/2}$, in agreement with these experiments, but the theoretical coefficient $A(ϕ)$ differs by a constant factor $2/χ(ϕ)$ from the exact coefficient, computed from the Green-Kubo formula for $η(ϕ,ω)$. This still enables us to predict for practical purposes the visco-elastic behavior of monodisperse spherical colloidal suspensions for all volume fractions by a simple time rescaling.

chao-dyn↗
arXiv1996

Lyapunov Exponents from Kinetic Theory for a Dilute, Field-driven Lorentz Gas

Positive and negative Lyapunov exponents for a dilute, random, two-dimensional Lorentz gas in an applied field, $\vec{E}$, in a steady state at constant energy are computed to order $E^{2}$. The results are: $λ_{\pm}=λ_{\pm}^{0}-a_{\pm}(qE/mv)^{2}t_{0}$ where $λ_{\pm}^{0}$ are the exponents for the field-free Lorentz gas, $a_{+}=11/48, a_{-}=7/48$, $t_{0}$ is the mean free time between collisions, $q$ is the charge, $m$ the mass and $v$ is the speed of the particle. The calculation is based on an extended Boltzmann equation in which a radius of curvature, characterizing the separation of two nearby trajectories, is one of the variables in the distribution function. The analytical results are in excellent agreement with computer simulations. These simulations provide additional evidence for logarithmic terms in the density expansion of the diffusion coefficient.

chao-dyn↗
arXiv1995

Dynamical ensembles in stationary states

We propose as a generalization of an idea of Ruelle to describe turbulent fluid flow a chaotic hypothesis for reversible dissipative many particle systems in nonequilibrium stationary states in general. This implies an extension of the zeroth law of thermodynamics to non equilibrium states and it leads to the identification of a unique distribution $\m$ describing the asymptotic properties of the time evolution of the system for initial data randomly chosen with respect to a uniform distribution on phase space. For conservative systems in thermal equilibrium the chaotic hypothesis implies the ergodic hypothesis. We outline a procedure to obtain the distribution $\m$: it leads to a new unifying point of view for the phase space behavior of dissipative and conservative systems. The chaotic hypothesis is confirmed in a non trivial, parameter--free, way by a recent computer experiment on the entropy production fluctuations in a shearing fluid far from equilibrium. Similar applications to other models are proposed, in particular to a model for the Kolmogorov--Obuchov theory for turbulent flow.

chao-dyn↗
arXiv1995

Diffusion in Lorentz Lattice Gas Cellular Automata: the honeycomb and quasi-lattices compared with the square and triangular lattices

We study numerically the nature of the diffusion process on a honeycomb and a quasi-lattice, where a point particle, moving along the bonds of the lattice, scatters from randomly placed scatterers on the lattice sites according to strictly deterministic rules. For the honeycomb lattice fully occupied by fixed rotators two (symmetric) isolated critical points appear to be present, with the same hyperscaling relation as for the square and the triangular lattices. No such points appear to exist for the quasi-lattice. A comprehensive comparison is made with the behavior on the previously studied square and triangular lattices. A great variety of diffusive behavior is found ranging from propagation, super-diffusion, normal, quasi-normal, anomalous to absence of diffusion. The influence of the scattering rules as well as of the lattice structure on the diffusive behavior of a point particle moving on the all lattices studied so far, is summarized.

comp-gas↗