We believe that nature's response to any time-dependent event is purely statistical.
Therefore, the stationary classical statistical distribution such as Gaussian and the quantum statistical distribution such as Plancks radiation law can be easily derived from the B matrix statistical chains.
Additionally, these chains can show how any form of initial state vector eventually reaches the stationary statistical distribution over time t as t approaches infinity.
IYH Dear Ismail Abbas M. Abbas
Yes, this is correct. In constructing a Markov chain with matrix B (transition matrices, commonly denoted as B), the key is to design transition probabilities that satisfy the detailed balance condition with respect to the desired stationary distribution.
For a Gaussian distribution in the continuous variable (or its discrete analogue), one may set up a chain with local moves and symmetric proposals, using an acceptance rule (such as that in Metropolis–Hastings) to enforce π(x) ∝ exp(–x²/(2σ²)).
For Planck’s law, the system is modeled by a discrete set of quantized energies. By setting the detailed balance condition to recover π(n) ∝ exp(–β n hν) (a geometric distribution), and by calculating the average energy of the mode, one can recover the Bose–Einstein form that directly leads to the Planck spectrum when combined with the mode density factor.
Kindly find a handout for you below with step‐by‐step derivations for how one might “engineer” transition matrices so that their stationary distributions are (a) a Gaussian and (b) a distribution leading to Planck’s law.
Method Constructing Markov Chains for Target Distributions: Gaussia...
This is only a preliminary response aimed first at shedding light on the question and its answer and secondly to thank our friend from Cornell University in the United States.
His answer is exact in that I don't think we can add anything more.
*The only two transition matrices we know today are the mathematical Markov transition matrix M which has nothing to do with physics or even mathematics.
The second is the statistical transition matrix B introduced recently by the author (in 2020) which is capable of solving all time-dependent situations of classical and quantum physics as well as mathematical problems.
The main improvement of matrix B is that it is doubly symmetric and its main diagonal entry RO ∈ [0,1] is fully controllable depending on the nature of the problem.
To be continued.
1-A statistical solution to the Laplace and Poisson PDE, ResearchGate,IJISRT,2020.
Before answering the question in detail, let us first show that the 4D discrete space xt of matrix B is a complete measured space.
If the distance between two points in this space is defined as the SQRT of (X2-X1)^2+(Y2-Y1)^2+(Z2-Z1)^2+(t2-t1)^2 then the distance from a point to itself is equal to 0 while the distance between two distinct points is equal to 1 (when the x-t space is normalized).
To show that this space is complete, let us return to pragmatic philosophy:
1- When the space we propose solves all the problems of classical and quantum physics in addition to mathematics, then this space is fair and complete and its adversaries are wrong.
2- When the spaces proposed by Minkowski, Einstein, Schrodinger, Hilbert. . etc. do not solve all the problems of classical and quantum physics in addition to mathematics, then these spaces are neither complete nor correct.
To be completed.@Daniyel Yaacov Bilar
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