Irreducible memory functions

The first such relation involving the irreducible memory functions is based on a physically intuitive notion Brownian motion and diffusion are two intimately related concepts we might say that collective diffusion is the macroscopic superposition of the Brownian motion of many individual colloidal particles. It is then natural to expect that collective diffusion should be related in a simple manner to self-diffusion. In the original proposal of the SCGLE theory [18], such connections were made at the level of the memory functions. Two main possibilities were then considered, referred to as the additive and the multiplicative Vineyard-like approximations. The first approximates the difference [C(k, z) - O Kk, z)], and the second the ratio [C k, z)IO k, z)], of the memory functions, by their exact short-time limits, using the fact that the exact short-time values, C P(fe, t) and (35)SEXP( 0, of these memory functions are known in terms of equilibrium structural properties [18]. The label SEXP refers to the single exponential time dependence of these memory functions. 

Either of these Vineyard-like approximations, along with an additional closure relation, will allow the exact results for A (t), F(k, t), and F %k, t) to constitute a closed set of equations. The closure relation consists of an independent approximate determination of the self irreducible memory function O Kk, t). One inmitive notion behind the proposed closure relation is the expectation that the -dependent self-diffusion properties, such as F k, t) itself or its memory function O k, t), should... 

The exact memory function expressions for F(k, t) and F Hk, t) in Equations 1.23 and 1.24 were extended to colloidal mixtures in reference [20]. Written in matrix form and in Laplace space, these exact expressions for the matrices F(]c, t) and F k, t) (when convenient, their A -dependence will be explicitly exhibited) in terms of the corresponding irreducible memory function matrices C(k, t) and 0 k, t) read... 

The main conclusion of the previous comparisons is that, except for very short times, in reality there is no practical reason for preferring the multiplicative version of the SCGLE theory over its additive counterpart, particularly if we are interested in intermediate and long times. Since the additive approximation is numerically simpler to implement, we shall no longer refer to the multiplicative approximation. Still, one of the remaining practical difficulties of the SCGLE theory is the involvement of the SEXP irreducible memory functions z) and z) the need to previously... 

In the case where the correlation function <3> (f) has the form of Eq. (148), with p fitting the condition 2 < p < 3, the generalized diffusion equation is irreducibly non-Markovian, thereby precluding any procedure to establish a Markov condition, which would be foreign to its nature. The source of this fundamental difficulty is that the density method converts the infinite memory of a non-Poisson renewal process into a different type of memory. The former type of memory is compatible with the occurrence of critical events resetting to zero the systems memory. The second type of memory, on the contrary, implies that the single trajectories, if they exist, are determined by their initial conditions. 

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