occasional meanderings in physics' brave new world

Name:
Location: New Zealand

Marni D. Sheppeard

## Friday, September 25, 2009

### M Theory Lesson 297

A Markov chain for a three state system has a $3 \times 3$ transition matrix, with entry $A_{ij}$ the probability that the system transitions from state $i$ to state $j$.

If the system is time symmetric, the matrix is both symmetric and magic, in that all rows and columns will sum to $1$. But for time asymmetric systems, it is only necessary that the columns sum to $1$. Matrix multiplication stands for one step in the Markov process. A typical long time convergence will result in a matrix of the form:

a a a
b b b
c c c

In fact, if all entries of $A^{k}$ (for some $k$) are strictly positive, then this always happens, and the limiting vector is called the steady state vector for the system.