Nelson makes quite a lot out of the fact that thermal fluctuations in a fluid are uncorrelated with their past behaviour. This is true for a simple Newtonian fluid where the current behaviour depends only on the thermal statistics. Such a fluid is "forgetful"...
It is possible to treat Brownian motion in a fluid which has some form of memory. These are called viscoelastic fluids because their flow is determined by how much energy they store (elasticity) compared to how much they dissipate (viscosity). You don't have to look far to find viscoelastic fluids: there is a variety of them in your eyes (e.g. hyaluronic acid, vitreous humor) and up your nose (mucus!). As it happens, viscoelastic fluids abound in biology, especially when one considers all plasma membranes have some manner of viscoelastic properties and that the inside of a cell is a complex architecture of protein fibres on many different length scales.
How do we treat diffusion in such fluids? We use a generalised Langevin equation! This is essentially Newton's law written in a disguised fashion. Firstly, set the total force equal to ma, as usual. We can then write the net force as a sum of a thermal term, f, a potential term, -grad(U), and a dissipation term. The dissipation is represented as an integral (over all previous times) of the microscopic memory function \zeta(t-t') with the velocity at all previous times (essentially a convolution of the particle velocity and the fluid's behaviour).
Physically, the memory function describes how similar the fluid is to how it was a time t' ago i.e. it is an autocorrelation of the thermal force function. In simple fluids it is just a Dirac delta because the fluid only behaves like it is now... now!
It turns out that this concept can be extended to rotational diffusion too (centre of mass and orientation undergo a random walk) and the maths remains essentially analogous.
Anyone interested should check out some lecture slides presented by Thomas Mason (he is definitely one of the big names in this area of research!).
This is very interesting. I suspect that the size of the particle plays a big role. For example the diffusion of a glucose molecule within a cell would probably have a Dirac delta auto-correlation function while a leukocyte diffusing through mucus would be much more complicated.
ReplyDeleteSize is important because is effects the form of the drag coefficient (contained within the memory function... for really complicated fluids the memory function is actually a tensor too).
ReplyDeleteIt turns out that the autocorrelation function (ACF) for a particle in a simple viscous fluid at low Reynolds numbers is an exponential decay. Elastic media might be expected to have a decaying periodic ACF because the fluid tends to `move back to how it was' over short time scales.
The delta function really applied to the autocorrelation of the thermal force (rather than the position of the particle). You are correct though: the viscoelasticity of mucus and leukocytes means that the thermal force acting on them takes a more complicated statistical form than `simple diffusion' would indicate.