Frank C. Andrews

Simple approach to the equilibrium statistical mechanics of the hard sphere fluid

The simple physical interpretation of the statistical mechanical expression for the reciprocal of the activity of a classical fluid is explored by using it to derive the well−known equation of state for one−dimensional fluids of hard rods. Direct extension of this derivation to three−dimensional hard spheres yields analytical equations for the activity and the pressure of the fluid branch which fit the molecular dynamics data about as well as the Pade approximant or the empirical equation of Carnahan and Starling. They represent a significant improvement over existing statistical theories, e.g., that of Percus and Yevick. The approach also yields equations which qualitatively describe the hard sphere crystalline branch.

A simple approach to the equilibrium statistical mechanics of two‐dimensional fluids

A simple statistical mechanical theory previously developed for three‐dimensional hard spheres is applied to a system of two‐dimensional hard disks to obtain analytical equations for activity and pressure. The first order result, based on only the third virial coefficient, fits the molecular dynamics data for fluid disks significantly better than the seven‐term virial series, but (unlike the case in three dimensions) not so well as the scaled particle theory. The second order result, involving the fourth virial coefficient, is the equal of the Pade approximant (3,4) with seven correct virial coefficients built in, and is significantly better than the scaled particle theory. The simple theory is surprisingly accurate even for the hard disk crystal. A simple theory of the two‐dimensional Lennard‐Jones fluid is obtained by incorporation of attractive wells with a hard core of temperature‐dependent diameter. Comparison of the theory with the 17 high‐density, supercritical pressures obtained by Fehder using mo...

Colligative properties of simple solutions.

Vapor pressure lowering, osmotic pressure, boiling point elevation, and freezing point depression are all related quantitatively to the decrease in �1soln upon the addition of solute in forming a solution. In any equilibrium system, regardless of whether it is in a gravitational field or whether it contains walls, semipermeable membranes, phase transitions, or solutes, all equilibria are maintained locally, in the small region of the equilibrium, by the equality of �1soln. If there are several subsystems in a gravitational field, at any fixed height, �i will have the same value in each subsystem into which substance i can get, and �i + Migh is constant throughout the entire system. In a solution, there is no mechanism by which solvent and solute molecules could sustain different pressures. Both the solvent and solute are always under identical pressures in a region of solution, namely, the pressure of the solution in that region. Since nature does not know which component we call the solvent and which the solute, equations should be symmetric in the two (acknowledging that the nonvolatile component, if any, is commonly chosen to be solute). Simple molecular pictures illustrate what is happening to cause pressure (positive or negative) in liquids, vapor pressure of liquids, and the various colligative properties of solutions. The only effect of solute involved in these properties is that it dilutes the solvent, with the resulting increase in S and decrease in �1soln. Water can be driven passively up a tree to enormous heights by the difference between its chemical potential in the roots and the ambient air. There is nothing mysterious about the molecular bases for any of these phenomena. Biologists can use the well-understood pictures of these phenomena with confidence to study what is happening in the complicated living systems they consider.

A simple equilibrium statistical mechanical theory of dense hard sphere fluid mixtures

The equilibrium statistical mechanical theory of Andrews for a hard core fluid is reformulated generally from simple, rigorous probability theory and is shown to apply to multicomponent systems. A simple expression is rigorously derived for the third virial coefficient of a mixture of nonadditive hard spheres. In the cases of additive potentials or binary systems, the expression is a great simplification of the previously known result. The exact theory of mixtures of not necessarily additive one‐dimensional hard lines is developed simply for uniform mixing. Expressions for the reciprocal of the activity and the pressure in mixtures of additive hard spheres are determined approximately by similar methods. Calculations for binary mixtures with diameter ratios of 1.1 to 1, 1.667 to 1, and 3 to 1 are compared with computer experiments. The theory is significantly better than either the equation of state obtained from the Percus–Yevick equation or from the scaled particle theory. Indeed, it is about as accurat...

A statistical theory of traffic flow on highways—I. Steady-state flow in low-density limit

A SIMPLE BUT REALISTIC MODEL OF TRAFFIC FLOW ON HIGHWAYS IS DEVELOPED FOR ANALYSIS BY AN INCREASINGLY SOPHISTICATED THEORY. THE MODEL UTILIZES A DISTRIBUTION OF DESIRED SPEEDS BY THE CARS AND CERTAIN DETAILED INFORMATION ON THE PASSING BEHAVIOR OF THE CARS. THIS PAPER TREATS STEADY- STATE FLOW ON UNIFORM ROADWAYS IN THE LIMIT OF DENSITIES SO LOW THAT NO QUEUING NEED BE CONSIDERED. CORRECTIONS TO THE DESIRED LANE BEHAVIOR CAUSED BY PASSING ARE FORMULATED EXPLICITLY, AND EXAMPLES ARE WORKED OUT FOR BOTH CONTINUOUS AND DISCRETE VELOCITY SPACES. /AUTHOR/

A simple approach to the equilibrium statistical mechanics of Lennard‐Jones fluids

A simple statistical mechanical theory previously developed for hard spheres and for two‐dimensional fluids is applied to three‐dimensional particles interacting through a Lennard‐Jones 6–12 potential. The results are compared with the molecular dynamics experiments of Verlet, with surprisingly good agreement. Employing the potential parameters obtained for argon from second virial coefficient data results in pressure isotherms which fit the experimental data of Michels with greater accuracy than the choice of potential warrants.

Gravitational Effects on Concentrations and Partial Pressures in Solutions: A Thermodynamic Analysis

Thermodynamic analysis establishes the equilibrium relationships between the concentrations and partial pressures of the components of liquid and gaseous solutions in the presence of a gravitational field. The conditions of equilibrium between a column of gas and gas-saturated water and the conditions of equilibrium governing a model of the distribution of radioactive heat sources in surface rocks are deduced from the theory.

A statistical theory of traffic flow on highways—III. Distributions of desired speeds

Abstract The development of statistical theories of traffic flow on highways has often started with a model in which each driver retains a fixed desired speed. This has been taken to mean that the distribution of desired speeds for the cars on the road at a given time was independent of density. However, it is shown that this model really implies that the desired speed distribution for the cars crossing any fixed point on the road in a certain period of time will be density-independent. The relation between the two distributions is computed, and the effect of the distinction on a theory of the density dependence of traffic behavior is discussed. The important input for a statistical theory of traffic flow seems to be a description of the desired speeds of the cars using the road in a period of time and a model of passing and queuing behavior. The number of cars crossing a fixed point in unit time appears to be more basic in such a theory than the density of cars.

A statistical theory of traffic flow on highways—II. Three-car interactions and the onset of queuing☆

THE MODEL OF TRAFFIC FLOW ON HIGHWAYS INTRODUCTED IN A PREVIOUS PAPER IS ANALYZED IN THE FORM OF AN EXPANSION IN DENSITY FOR STEADY-STATE FLOW ON UNIFORM ROADWAYS. CORRECTIONS ARE DEVELOPED TO ACCOUNT FOR 3-CAR INTERACTIONS, WHICH INVOLVE 2-CAR QUEUES. THE INTERESTING PROBABILITIES ARE DERIVED TO THIS ORDER AND SAMPLES ARE WORKED OUT SHOWING THE INITIATION OF DECREASED AVERAGE SPEED DUE TO QUEUING. /AUTHOR/

A statistical theory of traffic flow on highways—IV. Semi-empirical steady state theory

Abstract A simple semi-empirical model of traffic flow on highways, similar to one developed by Miller, is introduced. This consists of the distribution of desired speeds for the cars entering the road and of the passing frequencies from queues going various speeds as functions of the number of cars crossing a fixed point in unit time. Equations are derived giving the time evolution of all relevant probabilities. These equations are solved formally through an expansion in powers of the number of cars crossing a fixed point in unit time. It then becomes only a straightforward algebraic problem to compute any quantity of interest as a function of the number of cars crossing a fixed point in unit time for traffic conditions in which only queues of finite lengths are important.