Hard spheres systems

This implies, with the indicated choice of hard sphere diameter d, that the compressibilities of tlie reference system and the equivalent of the hard sphere system are the same. 

Alder B J and Wainwright T E 1957 Phase transition for a hard sphere system J. Chem. Phys. 27 1208-9 Alder B J and Wainwright T E 1962 Phase transition in elastic disks Phys. Rev. 127 359-61... 

T B J and T E Wainwright 1957. Phase Transition for a Hard-sphere System. Journal of Chemical Physics 27 1208-1209. 

IV. Geometrically Based Integral Equation Hierarchy for Hard Sphere Systems 151... 

Although Eqs. (33), (34), and especially (35), are useful they have a problem. They all predict that the hard sphere system is a fluid until = 1. This is beyond close packing and quite impossible. In fact, hard spheres undergo a first order phase transition to a solid phase at around pd 0.9. This has been estabhshed by simulations [3-5]. To a point, the BGY approximation has the advantage here. As is seen in Fig. 1, the BGY equation does predict that dp dp)j = 0 at high densities. However, the location of the transition is quite wrong. Another problem with the PY theory is that it can lead to negative values of g(r). This is a result of the linearization of y(r) - 1 that... 

IV. GEOMETRICALLY-BASED INTEGRAL EQUATION HIERARCHY FOR HARD SPHERE SYSTEMS... 

Alternative integral equations for the cavity functions of hard spheres can be derived [61,62] using geometrical and physical arguments. Theories and results for hard sphere systems based on geometric approaches include the scaled particle theory [63,64], and related theories [65,66], and approaches based on zero-separation theorems [67,68]. These geometric theories have been reviewed by Stell [69]. 

This expression was shown to be valid for hard sphere systems of arbitrary density. Moreover, the free-path distribution scaled by A = (uto) is nearly density-independent and almost the same as in the limit of zero density [74]. 

This is an indication of the collective nature of the effect. Although collisions between hard spheres are instantaneous the model itself is not binary. Very careful analysis of the free-path distribution has been undertaken in an excellent old work [74], It showed quite definite although small deviations from Poissonian statistics not only in solids, but also in a liquid hard-sphere system. The mean free-path X is used as a scaling length to make a dimensionless free-path distribution, Xp, as a function of a free-path length r/X. In the zero-density limit this is an ideal exponential function (Ap)o- In a one-dimensional system this is an exact result, i.e., Xp/(Xp)0 = 1 at any density. In two dimensions the dense-fluid scaled free-path distributions agree quite well with each other, but not so well with the zero-density scaled distribution, which is represented by a horizontal line (Fig. 1.21(a)). The maximum deviation is about... 

In a hard-sphere system, the trajectories of particles are determined by momentum conserving binary collisions. The interactions between particles are assumed to be pair-wise additive and instantaneous. In the simulation, the collisions are processed one by one according to the order in which the events occur. For not too dense systems, the hard-sphere models are considerably faster than the soft-sphere models. Note that the occurrence of multiple collisions at the same instant cannot be taken into account. 

Fig. 20. Excess compressibility yIS for a system of inelastic hard spheres, as function of the coefficient of normal restitution, for one solid fraction (as = 0.05). The excess compressibility has been normalized by the excess compressibility y is of the elastic hard spheres system. Other simulation parameters are as in Fig. 19.

Figure 8 Compressibility factor P/fiksT versus density p = pa3 of the hard-sphere system as calculated from both free-volume information (Eq. [8]) and the collision rate measured in molecular dynamics simulations. The empirically successful Camahan-Starling84 equation of state for the hard-sphere fluid is also shown for comparison. (Adapted from Ref. 71).

Structural Precursor to Freezing in the Hard-Disk and Hard-Sphere Systems. 

This is the isothermal bulk modulus. Thus we can use our simulation data in Figure 5.1 and calculate a modulus for a hard sphere system. Equations (5.14) to (5.16) form an interesting hierarchy of equations ... 

This is strong evidence for assuming that dispersions of ideal hard spheres would be expected to show a transition in the viscous behaviour between

short time selfdiffusion coefficient Z)s. This still shows a significant value after the order-disorder transition. The problem faced by the rheologist in interpreting hard sphere systems is that at high concentrations there is... 

Figure 3 depicts the spectmm of Lyapunov exponents in a hard-sphere system. The area below the positive Lyapunov exponent gives the value of the Kolmogorov-Sinai entropy per unit time. The positive Lyapunov exponents show that the typical trajectories are dynamically unstable. There are as many phase-space directions in which a perturbation can amplify as there are positive Lyapunov exponents. All these unstable directions are mapped onto corresponding stable directions by the time-reversal symmetry. However, the unstable phase-space directions are physically distinct from the stable ones. Therefore, systems with positive Lyapunov exponents are especially propitious for the spontaneous breaking of the time-reversal symmetry, as shown below. 

Mixtures of equisized charged spheres were also treated by the MSA. Such a system is then uniquely characterized by the ratio of the critical temperatures of the pure components. Harvey [235] found that a continuous critical curve from the dipolar solvent to the molten salt is maintained until the critical temperature of the ionic component exceeds that of the dipolar component by a factor of about 3.6. This ratio is much higher than theoretically predicted for nonionic model fluids. We recall that for NaCl the critical line is still continuous at a critical temperature ratio of about 5. Thus, the MSA of the charged-hard-sphere-dipolar-hard-sphere system captures, at least in part, some unusual features of real salt-water systems with regard to their critical curves. 

It has been discussed in the previous section that the long-time part in the memory function gives rise to the slow long-time tail in the dynamic structure factor. In the case of a hard-sphere system the short-time part is considered to be delta-correlated in time. In a Lennard-Jones system a Gaussian approximation is assumed for the short-time part. Near the glass transition the short-time part in a Lennard-Jones system can also be approximated by a delta correlation, since the time scale of decay of Tn(q, t) is very large compared to the Gaussian time scale. Thus the binary term can be written as... 

Small valence systems exhibit a peculiar phase diagram when the valence is significantly smaller than 12, the average number of nearest neighbors in the liquid state of a hard sphere system. Reducing the valence makes it possible to obtain... 

With the switch geometry chosen as for the LJ systems discussed above, the difference between the free energies of fee and hep hard-sphere systems can be determined precisely and transparently [48, 56]. Some of the results are included in Table I. 

Here, we report some basic results that are necessary for further developments in this presentation. The merging process of a test particle is based on the concept of cavity function (first adopted to interpret the pair correlation function of a hard-sphere system [75]), and on the potential distribution theorem (PDT) used to determine the excess chemical potential of uniform and nonuniform fluids [73, 74]. The obtaining of the PDT is done with the test-particle method for nonuniform systems assuming that the presence of a test particle is equivalent to placing the fluid in an external field [36]. 

The fluid phase for a hard-sphere system is stable up to rj = 0.49, at which point a solid phase with rj = 0.55 is predicted to coexist in equilibrium with the fluid phase. Carnahan and Starling [30] have proposed the following simple and accurate equation of state ... 

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