The Markov stochastic
transition matrix (Mi,j) is by definition a mathematical probability matrix that works in some limited mathematical domains.
The question arises as to why it cannot solve physical problems (PDE of heat diffusion/conduction, PDE of Poisson and Laplace, derivation of integration and differentiation formulas, etc.) which are simply solved by the Transition-matrix B (B i, j) of the Cairo Technique?
We assume that the Markov stochastic transition matrix is simply unphysical:
i- Mi,j is not necessarily symmetric and
ignores the detailed rules of balance and reciprocity.
ii-There is no place for the source/sink term S and the boundary conditions BC which are in a way a source/sink term.
Nothing's missing; what the statements made illustrate is misunderstanding and the statements (i) and (ii) are, simply, wrong.
It would be a good idea to actually study the relevant material, before making such statements, that are, trivially, wrong. A good place to start might be any textbook or lecture notes on Monte Carlo methods, e.g. https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=0bfe9e3db30605fe2d4d26e1a288a5e2997e7225
There are several potential limitations or sources of error that may be present in a Markov stochastic transition matrix, which could lead to missing information or inaccurate predictions. Some examples include:
Overall, while the Markov stochastic transition matrix is a useful tool for modeling transitions between states, it is important to be aware of its limitations and potential sources of error, in order to interpret its results appropriately.
Markov was part of the great tradition of mathematics in Russia and never missed his goal.
He designed and passed on a non-physical stochastic chain and never claimed it could solve physical problems.
We present the following brief comparison between classical stochastic Markov chains and recent B-matrix chains for the statistical transition imposed via the Cairo technique in the year 2020,
Markov defined his original nxn square transition matrix M(i,j) via two assumptions:
(1) the transition probabilities Mi,j are equal to or greater than zero.
(2) the sum of the transition probabilities Sigma Mi,j=1, for all the columns j=1 to n, (called the vertical matrix) OR the sum of the transition probabilities Sigma Mi,j=1, for all the rows i =1 to n , (called horizontal matrix)
Obviously the Markov stochastic chains can be converted into B-matrix chains and vice versa.On the other hand, the well-founded B-matrix and B-transition matrix chains rely on four assumptions. The first two assumptions are those of Markov with a slight modification of the second to allow the insertion of boundary conditions BC.The other two assumptions are physical, namely,(3) the matrix B is symmetric in space like nature itself, i.e.Bij = Bji.condition (3) represents detailed equilibrium and reciprocity.(4) the input elements of the main diagonal (RHS) are all equal to a constant input RO.Where RO is an element of [0,1] and represents the residual or remanence of the energy density at a point in spacetime.It is clear that RO must be zero when solving the Poisson and Laplace PDE and somewhere between zero and one when solving the heat diffusion/conduction PDE.
The question arises as to why Markov stochastic chains cannot solve physical problems (PDE of heat diffusion/conduction, PDE of Poisson and Laplace, derivation of integration and differentiation formulas, etc.) which are simply solved by the Transition-matrix B (B i, j) of the Cairo Technique.
what is missing in the Markov stochastic chains?
The answer is that the physical conditions (3) and (4) are missing.
Decisive difference is that in Markov chains which does not account for space x, the time interval dt is arbitrary prechosen and not necessarily relevant to space while in B-matrix chains time is merged or woven in a unitary 4D x-t space which means that the time interval dt is imperial.
It worth mention that the B-matrix chains are relativistic invariant as it should be.
[Google Search, Professor Mourad el Idrissi]Q: What are the limits of Markov analysis?If the time interval is too short, then Markov models are inappropriate because the individual moves are not random, but rather deterministically related in time. This example suggests that Markov models are generally inappropriate over sufficiently short time intervals.
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