Exact Equilibrium Research Articles

Aggregation and disaggregation through insensitivity in stochastic Petri nets

In this paper we apply some results from the theory of insensitivity to stochastic Petri nets. In doing so exact equilibrium distributions are derived for transition merging, marking amalgamation and aggregation and disaggregation for suitable classes of stochastic Petri nets.

Exact solutions for the coagulation-fragmentation equation

The authors present new exact equilibrium and time-dependent solutions to the two main versions of the coagulation-fragmentation equation. Equilibria are first found for the pure coagulation and pure fragmentation equations; equilibrium solutions for the full coagulation-fragmentation equation are then constructed when the kernels governing the reaction rates are chosen to be in a particular form. Time-dependent solutions are also derived for the pure fragmentation equation.

The exact classical vibrational-rotational partition function for the Woolley potential: calculations of the equilibrium constants for the formation of Ar-Ar and Mg-Mg

Exact analytical expressions are derived for the classical vibrational-rotational partition function and for the number of vibrational-rotational energy levels of the one-constant Woolley potential, which can be viewed as a deformed Lennard-Jones (12, 6) potential. These expressions are then used first to fit the Woolley potential to accurate analytic potentials of Ar2 and Mg2 by matching the total number of vibrational-rotational energy levels and, secondly, to calculate the corresponding classical equilibrium constants for the formation of these two molecules. The results are in excellent agreement with the exact classical and quantum-mechanical equilibrium constants calculated from the accurate analytic potentials by Dardi and Dahler (1990) with a maximum error of 4% for Ar2 and 2% for Mg2. In particular, they are in much better agreement than those calculated from various Lennard-Jones (12, 6) potentials.

Ambipolar diffusion, cloud cores, and star formation: Two-dimensional, cylindrically symmetric contraction. II - Results and a length scale for protostellar cores. III - A further parameter study and magnetically controlled accretion rate

Ambipolar diffusion initiates the formation and collapse of cores in self-gravitating, thermally supercritical model clouds which would be in exact equilibrium states if the magnetic field were frozen in the matter. The contraction is followed to an increase of the central density by a factor of 10 6 (e.g., from 3 × 10 3 to 3 × 10 9 cm -3 ). The results are interpreted in terms of thermal, magnetic, and Alfven length scales.

Modeling the effects of mutations on the denatured states of proteins.

We develop a model for the reversible denaturation of proteins and for the effects of single-site mutations on the denatured states. The model is based on short chains of sequences of H (hydrophobic) and P (other) monomers configured as self-avoiding walks on the two-dimensional square lattice. The N (native) state is defined as the unique conformation of lowest contact energy, whereas the D (denatured) state is defined as the collection of all other conformations. With this model we are able to determine the exact partition function, and thus the exact native-denatured equilibrium for various solvent conditions, using the computer to exhaustively enumerate every possible configuration. Previous studies confirm that this model shows many aspects of protein-like behavior. The present study attempts to model how the denatured state (1) depends on the amino acid sequence, and (2) is changed by single-site mutations. The model accounts for two puzzling experimental results: (1) the replacement of a polar residue by a hydrophobic amino acid on the surface of a protein can destabilize a native protein, and (2) the "denaturant slope," m = partial delta G/partial c (where c is the concentration of denaturant--urea, guanidine hydrochloride), can sometimes change by as much as 30% due to a single mutation. The principal conclusion of the present study is that, under strong folding conditions, the denatured conformations that are in equilibrium with the native state are not open random configurations. Instead, they are an ensemble of highly compact conformations with a distribution that depends on the residue sequence and that can be substantially altered by single mutations. Most importantly, we conclude that mutations can exert their dominant effects on protein stability by changing the entropy of folding.

Single porous crystal membranes: mixtures in steady flow

A treatment, which uses the Langmuir adsorption isotherm as the simplest representation of the type 1 contours typical of most isotherms in porous crystals, has been developed, from the case of steady-state diffusive flow of a single gas through a porous crystal to that of flow of a mixture containing any number of components. Quantitative expressions have been obtained for the departures from sorption equilibrium for each component not only across the gas/solid interfaces but also for adsorption on the external surfaces. These departures must arise whenever the flow density, J, is finite. They are found by taking into account the mass balance equations at and across the gas/solid interfaces at entry to and departure from the crystal membrane.When Jid is the flow density which would arise for the usual but incorrect assumption of exact sorption equilibrium at and across the gas/crystal interfaces, Jid > J and the treatment leads to a simple expression for the ratio Jid/J for a single pure gas flowing alone. For the mixture of gases each component affects the flow of each of the other components and the same is true of the separation factor for any pair of gases in the mixture. In accord with this the irreversible thermodynamic formulation of mixture flow has been developed in a way that takes account of non-equilibrium at and across interfaces.

Modifications of fibre pull-out analysis

CHUN-HWAY HSUEH Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA The interracial properties of fibre-reinforced compo- sites can be evaluated for the results of single-fibre pull-out tests. The shear lag model [1-10] has been used extensively to analyse the stress transfer be- tween the fibre and the matrix, and to interpret the result of a single-fibre pull-out test. To obtain analytical solutions for a single-fibre pull-out, approximations of the analysis are required. For example, the stress transfer between the fibre and the matrix is modelled by a simple relationship between the axial stress gradient in the fibre and the interracial shear stress [1-10]. The radial depend- ences of the axial stresses in both the fibre and the matrix are generally ignored [1-8]. Since the radius of the matrix is generally much greater than that of the fibre in the single-fibre pull-out test, the radial dependence of the axial stress in the matrix is included in Hsueh's analyses [9, 10]. However, to obtain this radial dependence, an approximate shear stress distribution in the matrix is adopted. This shear stress nearly satisfies the free surface condition when the radius of the matrix is much greater than that of the fibre [9, 10] which, in turn, provokes questioning of the eligibility [8] of Hsueh's analyses. Also, the correlations between the exact equilibrium equation and the equations dictating both the stress transfer between the fibre and the matrix, and the shear stress distribution in the matrix have not been addressed. As a complement to the previous study [9], the purposes of the present study were two-fold. First, the correlations between the exact equilibrium equation and the approximate equilibrium equations adopted in the shear lag model were addressed. Secondly, the shear stress adopted in the previous studies [9] was modified to satisfy the free surface condition, and the corresponding differential equation governing the axial stress distribution in the fibre is presented. The results based on this modification are then compared with previous re- sults [9]. The idealized specimen for a single-fibr e pull-out and the corresponding shear lag model is shown in Fig. 1. A fibre with a radius of a is located at the centre of a coaxial cylindrical shell of matrix with outer radius b, which is much greater than a. The radial and the axial co-ordinates are r and z, respectively. An axial stress, or0, is applied to pull out the fibre, and the matrix is fixed at the bottom surface. For an axially symmetric geometry, which has the cylindrical polar co-ordinates r, 0 and z, the

Diffusion through a membrane with pressure-induced structure

A microscopic model of a thin membrane with pressure-induced structure has been considered. The exact equilibrium equation of state of the membrane has been found and with the help of it the isobars and compressibility have been determined. On the basis of the equilibrium membrane properties a stochastic process retrieving the properties after long time has been constructed. Next, the transport of charged colloid particles across the membrane subject to a constant pressure has been considered in the presence of an external electric field and the gradients of the membrane electric potential.

Option Pricing With V. G. Martingale Components1

European call options are priced when the uncertainty driving the stock price follows the V. G. stochastic process (Madan and Seneta 1990). the incomplete markets equilibrium change of measure is approximated and identified using the log return mean, variance, and kurtosis. an exact equilibrium interpretation is also provided, allowing inference about relative risk aversion coefficients from option prices. Relative to Black‐Scholes, V. G. option values are higher, particularly so for out of the money options with long maturity on stocks with high means, low variances, and high kurtosis.

Statistical mechanics of a nonlinear relaxation process. II. Dynamical properties.

The dynamical properties of a semiclassical model of spin relaxation are carefully studied with the help of computer simulation. It is shown that, as an effect of the reaction field, the relaxation time as a function of temperature ranges from an Arrhenius-like behavior at weak temperatures to a linear power-law behavior at high temperatures. It is also shown that a fairly accurate theoretical prediction of the relaxation time can be obtained in both the low- and high-temperature limit. In the latter regime a simple factorization prescription, although leading to small deviations from the exact equilibrium value for the spin components, is shown to provide the most accurate predictions.

Relaxation of non-equilibrium current sheets

The dynamic evolution of a collisionless current sheet which is initially not in equilibrium is investigated by means of a one-dimensional electromagnetic implicit particle simulation code. In general a relaxation of a collisionless system into an exact equilibrium is not possible, although on macroscopic scales an equilibrium can be reached. The macroscopic view is inherently included in observations as well as in a particle simulation. Our simulation results show a filamentation of the distribution function in phase space. Macroscopic quantities such as kinetic energy exhibit an oscillation which is damped on Alfvèn time scales.

Exact equilibrium crystal shapes in two dimensions for free-fermion models.

Using a conceptually novel approach that maps a two-dimensional interface exactly onto a Feynman-Vdovichenko lattice walker, we derive an exact and general solution for the equilibrium crystal shapes (ECS's) of free-fermion models, i.e., models that are solvable via the Feynman-Vdovichenko or (equivalently) Pfaffian methods. The ECS for these models is given by the locus of purely imaginary poles of the determinant of the ``momentum-space'' lattice-path propagator. The ECS may, therefore, simply be read off from the analytical expression for the bulk free energy. From these shapes one can then obtain numerically (but to arbitrary accuracy) the high-temperature direction-dependent correlation length of the dual system. We give several examples of previously unknown Ising ECS's, and we examine in detail the free-fermion case of the eight-vertex model. The free-fermion eight-vertex model includes the modified potassium dihydrogen phosphate (KDP) model, which is not in the Ising universality class. The ECS of the modified KDP model is shown to be the limiting case of the ECS of an antiferromagnetic 2\ifmmode\times\else\texttimes\fi{}1 phase on a triangular lattice in the limit of infinite interactions. The ECS of the modified KDP model is lenticular at finite temperature and has sharp corners. We explain the physics of this lens shape from an elementary calculation.

Finite shearing of a FRR block with a circular hole

By using the idealized theory of fiber-reinforced materials, an exact equilibrium solution is obtained for the finite shearing of a rubber block reinforced by fibers in the shearing direction and weakened by a circular hole. Deformed configurations of the hole and block ends are obtained in explicit forms. It is found that the infinite normal stress occurs along the fiber lines tangent to the hole while the infinite hydrostatic pressure along the normal lines tangent to the hole. The edge stress around the hole obtained in an explicit form can be reduced to that given by the infinitesimal theory of anisotropic elasticity for the infinite ratio of elastic moduli when the block is infinite in length. These results are also presented in the figures.

Nonlinear Burnett coefficients.

We derive exact equilibrium fluctuation expressions for the nonlinear Burnett coefficients for the electrical conductivity of a fluid. The Burnett coefficients define a Taylor-series expansion of the conductivity in powers of the electrical current density. The fluctuation expressions may be calculated from an analysis of the time-dependent fluctuations of the equilibrium system constrained so that the mean electrical current is zero.

Equilibrium crystal shapes and correlation lengths: A general exact result in two dimensions.

We show that the exact equilibrium crystal shape (ECS) for a wide class of two-dimensional Ising models can simply be read off from the analytical expression for the bulk free energy: The ECS is given as the locus of purely imaginary zeros of the ``momentum''-space lattice-path propagator. From these shapes, one may obtain numerically, to arbitrary accuracy, the high-temperature direction-dependent correlation lengths of the dual system. These exact ECS's and corresponding correlation lengths were previously known only for the rectangular, triangular, and honeycomb lattices.

Extensive Form Games in Continuous Time: Pure Strategies

We propose a new framework for games in continuous time that conforms as closely as possible to the conventional discrete-time framework. In this paper, we take the view that continuous time can be viewed as discrete time, but with a grid that infinitely fine. Specifically, we define a class continuous-time strategies with the following property: when restricted to an arbitrary, increasingly fine sequence discrete-time grids, any profile strategies drawn from this class generates a convergent sequence outcomes, whose limit independent the sequence grids. We then define the continuous-time outcome to be this limit. Because our continuous-time model conforms so closely to the conventional discrete-time model, we can readily compare the predictions the two frameworks. Specifically, we ask two questions. First, discrete-time with a very find grid a good proxy for continuous time? Second, does every subgame perfect equilibrium in our model have a discrete-time analog? Our answer to the first question the following upper result: Suppose a sequence discrete-time e-subgame-perfect equilibria increasingly closely approximates (in a special sense) a given continuous-time profile, with - converging to zero along as the period length shrinks. Then the continuous-time profile will be an exact equilibrium for the corresponding continuous-time game. Our second answer a lower hemi-continuity result that holds under weak conditions. Fix a perfect equilibrium for a continuous-time game and a positive ?. Then for any sufficiently fine grid, there will exist an e-subgame perfect equilibrium for the corresponding game played on that grid which is within - of the continuous-time equilibrium. Our model yields sharp predictions in a variety industrial organization applications. We first consider several variants a familiar preemption model. Next, we analyze a stylized model a patent race. Finally, we obtain a striking uniqueness result for a class repeated games.

Theory of spin-relaxation processes

The relaxation properties of a spin system weakly coupled to lattice degrees of freedom are described using an equation of motion for the spin density matrix. This equation is derived using a general weak coupling theory which has been previously developed. To second order in the weak coupling parameter, the results are in agreement with those obtained by Bloch, Wangsness and Redfield but the derivation does not make use of second order perturbation theory for short times. The derivation can be extended beyond second order and ensures that the spin density matrix relaxes to its exact equilibrium form to the appropriate order in the weak coupling parameter.

Nonlinear Burnett coefficients.

We derive exact equilibrium fluctuation expressions for the nonlinear Burnett coefficients for the electrical conductivity of a fluid. The Burnett coefficients define a Taylor-series expansion of the conductivity in powers of the electrical current density. The fluctuation expressions may be calculated from an analysis of the time-dependent fluctuations of the equilibrium system constrained so that the mean electrical current is zero.

Arbitrage and Diversification in a General Equilibrium Asset Economy

This paper presents a theory of equilibrium asset pricing that generalizes the recent work of Connor. The model extends Connor's results to more general sets of asset returns and consumer preferences, introduces production, and provides a framework for analyzing exact and approximate equilibrium asset pricing. The other major contribution of the paper is the introduction of geometric arguments that exploit the properties of induced prefer- ences over assets. This method of analyzing asset pricing provides an intuitively appealing way of analyzing equilibrium asset pricing theories. unifies several well-known asset-pricing models. This paper extends Connor's finite-asset model in a number of directions. These extensions are nontrivial, because they allow the theory to encompass asset-pricing models that were excluded by Connor's assumptions. The second major contribution of the paper is to introduce a method of argument that exploits the geometric properties of induced preferences over assets, and characterizes asset prices using these proper- ties. This technique provides an intuitively appealing way of analyzing asset pricing theories. (For a simple geometric exposition of the main ideas in this paper, see Milne (1987).) 0.2. To place this paper in the context of the literature, it is important to appreciate the development of asset-pricing models over the last two decades. The first-and perhaps the most famous-asset-pricing model is the Capital Asset Pricing Model (CAPM) developed by Sharpe (1964) and Lintner (1965). The model demonstrated a striking result: assuming (i) consumers characterize portfolios in terms of the means and variance of rates of return, (ii) there exists a risk-free asset, and (iii) asset equilibrium; then any asset's expected rate of return could be written as a linear function of the riskless interest rate, and the expected rate of return on the endowment of risky assets (the so-called market portfolio). The original CAPM result was derived from a one period model, but Merton (1973) showed that in a continuous time model where returns could be char- acterized by a diffusion process, the same mean-variance argument could be applied to obtain the CAPM formula. It was well-known that if consumers had von Neumann-Morgenstern prefer- ences, then either quadratic utility and/or multivariate normality of returns were

A closed-form solution for inverse kinematics of robot manipulators with redundancy

A closed-form solution formula for inverse kinematics of manipulators with redundancy is derived using the Lagrangian multiplier method. The proposed method is proved to provide the exact equilibrium state for the resolved-motion method. The repeatability problem in the resolved-motion method does not exist in the proposed method. The method is demonstrated to give more accurate trajectories than the resolved-motion method.