The approach to the traffic-flow problem based on an integral differential equation of the Boltzmann type which has been considered by one of us (I P ) in a recent paper is further developed. The possibility of passing is explicitly introduced into the equation for the velocity distribution function. As in the previous paper, it is shown that at sufficiently high concentration a collective flow process must take place. In order to study more specifically the effects of one car on another, we define reduced n-car distribution functions giving the probability of finding a cluster of n cars all having the same velocity. We derive an equation for the evolution of this distribution function. Study of it yields some information as to the way traffic changes from relatively free flow to completely hindered, “condensed” flow.
BOOK REVIEWS a mathematically unsophisticated audience, and so mathematical arguments for the most part proceed formally or are kept, on an intuitive level. On occasion, however, the unavoidable amount of imprecision which such an approach entails is compounded through carelessness. For example, the adjective "continuous" is used in a number of objectionable ways, not the least of which is the definition of a "continuous signal" as "any function of a continuous variable." In a somewhat different category is the incorrect assertion made in a footnote on p. 48 to the effect that, a necessary condition for a differential equation to uniquely specify a system is that the system lie nonanticipative. Despite these critical comments, a reader armed with the appropriate mathematical caveats will find much of value in this book, which is likely to be successful as an introductory text on modern system theory.
Vapor pressure lowering, osmotic pressure, boiling point elevation, and freezing point depression are all related quantitatively to the decrease in micro(1)(soln) 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 micro(1)(soln). If there are several subsystems in a gravitational field, at any fixed height, microi will have the same value in each subsystem into which substance i can get, and microi + M(i)gh 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 micro(1)(soln). 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 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 Padé 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 molecular dynamics shows deviations averaging only 3 1/2%. Agreement with the high-density, low-temperature data obtained by Tsien and Valleau using Monte Carlo techniques is not so good. Calculations for the Lennard-Jones 6–12 potential are compared with calculations for the 6–12–3 potential, proposed as more suited than the 6–12 for adsorbed gases.
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 Padé 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.
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