The non-Markovian nature of turbulence 3: the Master Equation.
In the previous post we established that the ‘loss’ term in the transport equation depends on the number of particles in the state currently being studied. This followed straightforwardly from our consideration of hard-sphere collisions. Now we want to establish that this is a general consequence of a Markov process, of which the problem of 
hard spheres is a particular example.
We follow the treatment given in pages 162-163 of the book [1] and consider the case of Brownian motion, as this is relevant to the Edwards self-consistent field theory of turbulence. We again consider a multipoint joint probability distribution and now consider a continuous variable 
which takes on specific values 
at time 
, 
at time 
, and in general 
at time 
; thus;
![\[f_n(x_1,t_1; x_2,t_2; \dots x_n,t_n).\]](https://blogs.ed.ac.uk/physics-of-turbulence/wp-content/ql-cache/quicklatex.com-6a5809d0e3d8cf8c2e144a11d048e33d_l3.png "Rendered by QuickLaTeX.com")
We then introduce the conditional probability density:
![\[p(x_1,t_1|x_2,t_2),\]](https://blogs.ed.ac.uk/physics-of-turbulence/wp-content/ql-cache/quicklatex.com-2fec6dab119fad9620fdd95f378aed8d_l3.png "Rendered by QuickLaTeX.com")
which is the probability density that 
at 
, given that had the value 
when 
. It is defined by the identity:
(1) 
From this equation (see [1]), we can obtain a general relationship between the single-particle probabilities at different times as:
(2) 
Next we formally introduce the concept of a Markov process. We now define this in terms of the conditional probabilities. If:
(3) 
then the current step depends only on the immediately preceding step, and not on any other preceding steps. Under these circumstances the process is said to be Markovian.
It follows that the entire hierarchy of probability distributions can be constructed from the single-particle distribution 
and the transition probability 
. The latter quantity can be shown to satisfy the Chapman-Kolmogorov equation:
(4) 
indicating the transitive property of the transition probability.
It is of interest to consider two specific cases.
First, for a chain which has small steps between events, the integral relation (4) can be turned into a differential equation by expanding the time dependences to first order in Taylor series. Putting 
for simplicity, we may obtain:
(5) 
where 
is the rate per unit time at which transitions from state to state take place. This is known as the master equation.
Secondly, if is a continuum variable, we can further derive the Fokker-Planck equation as:
(6) ![\begin{equation*} \frac{\partial f(x,t)}{\partial t}= \frac{\partial [A(x)f(x,t)]}{\partial x} + \frac{\frac{1}{2}\partial^2[B(x)f(x,t)]}{\partial x^2}. \end{equation*}](https://blogs.ed.ac.uk/physics-of-turbulence/wp-content/ql-cache/quicklatex.com-491f1fea2aaacf4401d2172726e0b637_l3.png "Rendered by QuickLaTeX.com")
This equation describes a random walk with diffusivity 
and friction damping 
. A discussion of this equation as applied to Brownian motion may be found on pages 163-164 of [1] but we will not pursue that here.
In the next post we will discuss the Edwards theory of turbulence (and by extension the other pioneering theories of Kraichnan and of Herring) in the context of the present work.
[1] W. D. McComb. The Physics of Fluid Turbulence. Oxford University Press 1990.
