In the intermediate time regime the evolution of the jump process has to be considered more carefully. It is assumed that the occupation probabilities W_k of finding a site k among n possible sites are stochastic variables of a stationary Markov process whose time dependence can be described by a master equation:

(1) The quantity Π is the transition matrix whose elements Π_kj are the probabilities per unit time for a jump from site j to site k. Since Π_kjW_j is the rate of population of site k by site j, the quantity Π_kkW_k has to be the rate with which site k is depleteted by jumps to other sites:

(2) which is equivalent to the statement, that the sum over each column of Π has to vanish.

The jump process is independent from the NMR experiment, especially the origin in time, t=0. The NMR experiment is always conducted in a stationary situation as far as the stochstic process is concerned. The W_k(t) therefore can be substituted by their equilibrium values (here always denoted as populations) which are determined by the following relation:

(3)

Weblab supports a transition matrix with the following restrictions:
a) Only jumps between next neighbours are considered
b) The principle of detailed balance is valid, which means: for all j and k.

With these assumptions, one has just 2n non-zero transition probabilities for all cases considered here. Detailed balance has to be fullfilled, which implies a further condition, so that you are actually only dealing with 2n-1 independent quantities.

You have to tell Weblab, which of the possible transition matrix elements should get the value that you have specified as jump rate Ω. In the case n=2, for example, there are two options, Π₁₂ and Π₂₁. If you select transition 1 ← 2 then Π₁₂ is set to Ω and Π₂₁ is set to b · Ω.

For a better comparability with the treatment in the fast limit you also can enter populations instead of transition matrix elements. In this case Weblab will calculate a transition matrix in agreement with eq. (3) and restrictions a) and b). In case of more then 2 sites there is no one-to-one corresponence and more solutions are possible. In this case Weblab sets all non-zero elements in each row equal.