Microcanonical ensemble study of a classical fluid of hard rods under gravity

The study of a system of hard rods in a box of finite length in the presence of a uniform gravitational field is made by means of the microcanonical ensemble. Explicit expressions are derived for the phase volume and the density of states, the primary functions of this ensemble. Related statistical quantities are reported, such as the entropy, the temperature, the heat capacity and the forces exerted on the fluid by the bottom and top walls. The microcanonical number density and higher order molecular distribution functions are also derived. Received: 7 April 1998 / Received in final form: 20 July 1998 / Accepted: 28 July 1998     

Microcanonical ensemble extensive thermodynamics of Tsallis statistics

The microscopic foundation of the generalized equilibrium statistical mechanics based on the Tsallis entropy is given by using the Gibbs idea of statistical ensembles of the classical and quantum mechanics. The equilibrium distribution functions are derived by the thermodynamic method based upon the use of the fundamental equation of thermodynamics and the statistical definition of the functions of the state of the system. It is shown that if the entropic index ξ=1/(q−1) in the microcanonical ensemble is an extensive variable of the state of the system, then in the thermodynamic limit the principle of additivity and the zero law of thermodynamics are satisfied. In particular, the Tsallis entropy of the system is extensive and the temperature is intensive. Thus, the Tsallis statistics completely satisfies all the postulates of the equilibrium thermodynamics. Moreover, evaluation of the thermodynamic identities in the microcanonical ensemble is provided by the Euler theorem. The principle of additivity and the Euler theorem are explicitly proved by using the illustration of the classical microcanonical ideal gas in the thermodynamic limit.     

Microcanonical Monte Carlo study of one dimensional self-gravitating lattice gas models

In this study we present a microcanonical Monte Carlo investigation of one dimensional (1 ? d) self-gravitating toy models. We study the effect of hard-core potentials and compare to the results obtained with softening parameters and also the effect of the topology on these systems. In order to study the effect of the topology in the system we introduce a model with the symmetry of motion in a line instead of a circle, which we denominate as 1 /r model. The hard-core particle potential introduces the effect of the size of particles and, consequently, the effect of the density of the system that is redefined in terms of the packing fraction of the system. The latter plays a role similar to the softening parameter ? in the softened particles’ case. In the case of low packing fractions both models with hard-core particles show a behavior that keeps the intrinsic properties of the three dimensional gravitational systems such as negative heat capacity. For higher values of the packing fraction the ring model behaves as the potential for the standard cosine Hamiltonian Mean Field model while for the 1 /r model it is similar to the one-dimensional systems. In the present paper we intend to show that a further simplification level is possible by introducing the lattice-gas counterpart of such models, where a drastic simplification of the microscopic state is obtained by considering a local average of the exact N-body dynamics.     

Microcanonical study of the planar spin model

The planar spin model is simulated by microcanonical and Monte Carlo algorithms. The average action, topological charge density and spin-spin correlation functions are measured on 15 × 15 and 30 × 30 lattices at temperatures surrounding the critical point. In general very good agreement between the methods is found. Accurate runs (5.10⁵ sweeps) expose the finite volume differences between the canonical and microcanonical ensembles. A small shift (of order 1/volume) of the temperature brings the two sets of measurements into excellent agreement.     

Microcanonical analysis of a finite three-dimensional Ising system

All singularities which occur at phase transitions are smeared out in finite size systems. It is the purpose of this paper to show that this smearing is a characteristic of the canonical ensemble. It can be greatly reduced if the microcanonical ensemble is used. In order to demonstrate this the density of states of a three-dimensional 10×10×10 Ising system is determined numerically and is analysed in terms of a model entropy. The critical exponents and turn out to be not too far from the values which have been obtained by critical scaling.Dedicated to Prof. H. Wagner on the occasion of his 60th birthday     

Ergodic Properties of Microcanonical Observables

The problem of the existence of a strong stochasticity threshold in the FPU- model is reconsidered, using suitable microcanonical observables of thermodynamic nature, like the temperature and the specific heat. Explicit expressions for these observables are obtained by exploiting rigorous methods of differential geometry. Measurements of the corresponding temporal autocorrelation functions locate the threshold at a finite value of the energy density, which is independent of the number of degrees of freedom.     

Microcanonical simulation of Ising systems

Numerical simulations of the microcanonical ensemble for Ising systems are described. We explain how to write very fast algorithms for such simulations, relate correlations measured in the microcanonical ensemble to those in the canonical ensemble and discuss criteria for convergence and ergodicity.     

Microcanonical Analysis on a System with Long-Range Interactions

We study a long-range interacting spin chain placed in a staggered magnetic field using microcanonical approach and obtain the global phase diagram. We find that this model exhibits both first order phase transition and second order phase transition separated by a tricritical point, and temperature jump can be observed in the first order phase transition.     

Heat transfer in locally heated liquid films moving under the action of gravity and gas flow

Heat transfer in a viscous liquid film moving under the action of gravity and a gas flow on a substrate with a locally heated rectangular area is investigated. The heat exchange coefficient is given on the liquid-gas surface; the heat flux to the liquid is given on the heated area; the substrate surface outside the heated area is heat-insulated. An analytical solution in a form of a convergent series is obtained for the liquid temperature distribution in the film. The influence of the dimensionless criteria on the obtained solution is analyzed. The effect of heat flux inhomogeneity on the temperature distribution is considered.     

Deposition of particles under gravity in a kinetic lattice model

We study the growth through particle deposition of the surface of a discrete two-dimensional system, in which the motion of particles is affected by infinite gravity and the Kob-Andersen kinetic rule. Computer simulation results are found to be consistent with previous results in literature, showing that this particular case belongs to the same universality class as Ballistic Deposition, the Eden model, one step solid-on-solid (SOS) deposition and Kardar-Parisi-Zhang (KPZ) characterized by scaling exponents α=0.5, β=1/3=0.33 and z=α/β=1.5.     

Microcanonical analyses of peptide aggregation processes

We propose the use of microcanonical analyses for numerical studies of peptide aggregation transitions. Performing multicanonical Monte Carlo simulations of a simple hydrophobic-polar continuum model for interacting heteropolymers of finite length, we find that the microcanonical entropy behaves convex in the transition region, leading to a negative microcanonical specific heat. As this effect is also seen in first-order-like transitions of other finite systems, our results provide clear evidence for recent hints that the characterization of phase separation in first-order-like transitions of finite systems profits from this microcanonical view.     

Consistency of Microcanonical and Canonical Finite-Size Scaling

Typically, in order to obtain finite-size scaling laws for quantities in the microcanonical ensemble, an assumption is taken as a starting point. In this paper, consistency of such a Microcanonical Finite-Size Scaling Assumption with its commonly accepted canonical counterpart is shown, which puts Microcanonical Finite-Size Scaling on a firmer footing.     

Geometry of a two-dimensional quantum gravity: Numerical study

A two-dimensional quantum gravity is simulated by means of the dynamical triangulation model. The size of the lattice was up to hundred thousand triangles. Massively parallel simulations and recursive sampling were implemented independently and produced similar results. Wherever the analytical predictions existed, our results confirmed them. The cascade process of baby universes formulation à la Coleman-Hawking scenario in a two-dimensional case has been observed. We observed that there is a simple universal inclusive probability for a baby universe to appear. This anomalous branching of surfaces led to a rapid growth of the integral curvature inside a circle. The volume of a disk in the internal metric has been proven to grow faster than any power of radius. The scaling prediction for the mean square extent given by the Liouville theory has been confirmed. However, the naive expectation for the average Liouville lagrangian ∫(φ)² is about 1 order of magnitude different from the results. This apparently points out to some flaws in the current definition of a Liouville model.     

Non-linear forced vibrations of a beam carrying concentrated mass under gravity

The non-linear dynamic behavior of a simply supported beam, with ends restrained to remain a fixed distance apart, carrying a concentrated mass and subjected to a harmonic exciting force at an arbitrary point under the influence of gravity is analysed. By using the one mode approximation and applying Galerkin's method, the governing equation of motion is reduced to the well known Duffing type equation. The harmonic balance method is applied to solve the equation and the dynamic response of a concentrated mass is derived. The effects of the weight, the location, and the vibratory amplitude of the concentrated mass on the natural frequency are also discussed.