1.3.1 Spatial Correlations
While a gas in the dilute limit is well described by point particles, the finite size of the particles influences the properties of a fluid at higher densities. In particular, the particles can no longer be regarded as spatially uncorrelated.
Spatial correlations between the particles can be partially quantified in terms of the pair correlation function, n2g(r−r0) = Pi6=jhδ(ri−r)δ(rj−r0)i, which quantifies the deviation of the two particle density from its uncorrelated value, n2. For hard sphere fluids, the contact value, χ ≡ g(2a), of the pair correlation function is of particular importance, as it completely determines the equation of state, p/nT = 1 + 4ϕχ(ϕ), where p is the pressure [69]. The most widely used approximate expression for the contact value of the pair correlation function was derived by Carnahan and Starling [113],
χCS(ϕ) = 1 +ϕ/2
(1−ϕ)3. (1.9)
Its validity is discussed, e.g., in Ref. [114]. In particular, χCS shows no sign of the crystallization transition and consequently, it can only be valid sufficiently far below the freezing density,ϕf. The mean free path,`0, is directly related to contact value χ [115],
a/`0 = 3√
2ϕχ, (1.10)
as is the collision frequency, ω0, in the Enskog approximation [48], ωE = 12ϕχ
a s
T
πm. (1.11)
Experimentally, the structure factor, Sq= 1 +nFT[g−1](q), is easier to measure than the pair correlation function. Upon introducing the Fourier transformed
1.3 Dense Fluids
densities,ρq =FT[ρ], whereρ =ρ(r) is the density field in real space, the structure factor can alternatively be expressed asSq=hρqρ−qi=|ρq|2[69]. A particularly useful theoretical approach to calculate the static structure factor in the fluid phase was introduced by Percus and Yevick [116]. On the one hand, it allows for deriving explicit expressions in the case of hard spheres [117–119]. On the other hand it can be made quantitatively quite accurate by introducing an effective density [120].
Van Noije et al. [74] studied static correlation functions of the hydrodynamic variables density,n, momentum current,mj=nmu, and of the granular temperature, T, in a two dimensional driven granular system. Their simulation results showed growing correlation functions for small wave numbersq. This had also been observed by Peng and Ohta [73] before and was recently analyzed further by Headet al. [121].
Via a granular fluctuating hydrodynamics theory, van Noijeet al. were able to relate this increase on large length scales to the onset of aq−2 divergence as q→0.
New Results
The theoretical analysis in chapter 3 closely follows van Noije et al. [74]. While I found the technical reasoning mostly correct, I will conclude that the small q divergence is an immediate consequence of the violation of momentum conservation by the driving which they deemed inconsequential. This is supported by the finding that upon using a driving mechanism that conserves momentum locally, the smallq divergence vanishes.
Static structure factors I measured in large scale simulations of systems in three dimensions will support these considerations. So far, no specific structure factor theory for (driven) inelastic hard spheres exists. Thus, these measurements will eventually be needed as input for the mode coupling theories developed in the subsequent chapters of this thesis. Independently, I will use these measurements in combination with the hydrodynamic theory to determine the speed of sound, sound damping constant and the shear viscosity.
1.3.2 Long-Time Tails
Let us start by looking at the diffusion of tracer particles in a host fluid. As we are not interested in mixtures, let’s assume that the tracer particles are physically identical to the host particles; they are only labeled differently. The concentration,c, of these particles will obey a continuity equation,∂tc=−∇j, wherej is the particle current. Within the context of linear response theory, it is natural to assume that this current will be linearly related to the concentration gradient,j=−D∇c. The constant of proportionality,D, is called the coefficient of diffusion or the diffusivity.
From that we get the diffusion equation,∂tc=D∆c. Green [122] and Kubo [123]
found that transport coefficients such as the coefficient of diffusion are related to
1 Collective Effects in Dense Fluids
the correlation functions of the corresponding or conjugate fluxes,D=R0∞ψ(t)dt, where, ψ(t) =hv(0)v(t)i, is the velocity autocorrelation function (VACF). This is one manifestation of a fluctuation dissipation relation which expresses a transport coefficient in terms of the correlation function of spontaneous fluctuations.
If one simply assumes that the velocity of a tagged particle is disturbed by collisions with its surrounding particles, one would assume that the VACF is of the form, ψ(t) = exp(−t/τc), whereτcis the time scale of collisions. This implies D∝τc−1. As mentioned in the introduction, an algebraic,t−d/2, rather than an exponential decay is found in reality. While a precise derivation of this relation needs considerable technical machinery, one can give a simple quantitative argument [69]. The more rigorous theories will be discussed in chapter 5 when I will investigate theVACFfor a dense granular fluid.
Imagine that because of the repeated collisions of the tagged particle with the surrounding particles of the liquid its momentum, ps, gets redistributed among those particles. For simplicity let us assume that the momentum gets uniformly distributed within a sphere of volume V(t)∝R(t)d. Because momentum transfer occurs diffusively, the size of this sphere grows asR(t)∝√
tand thus the momentum of the tagged particle at time twill be ps(t) ∝t−d/2 with the same result for the VACF,ψ(t)∝ hps(0)ps(t)i ∝t−d/2.
The fact that the Kubo integral, R0∞ψ(τ)dτ, does not exist in two dimensions, implies that the linear diffusion law shown above does not hold in planar fluids [64].
Fortunately, it does hold in three dimensions but here the correlations impede a virial expansion of the coefficient of diffusion [65]. The latter becomes a non-analytic function of the density,
D(ϕ)/D0 = 1 +D1ϕ+D20ϕ2lnϕ+D2ϕ2+. . . , (1.12) including logarithmic terms [124]. For hard spheres, the low density limit is given by the Enskog diffusivity,
D0 = a 8ϕχ
q
πT /m. (1.13)
New Results
In order to explain the simulation results on the long time tails and back scattering by Fiege et al. [67] (see Fig. 1.4), I will derive a mode coupling theory for the tagged particle velocity in chapter 5. This allows me to describe the coupling of the tagged particle to the collective density and current modes,ρq, andjq, respectively. I will show that the coupling to the transverse current jqT is responsible for the long time tails, exactly like in elastic hard sphere fluid.