# The Propagator Method to Solve the Fokker—Planck Equation

• Jian-Zhong Gu
• Yin-Sheng Ling

## Abstract

In a lot of physical problems one should solve the evolution equation
$$\frac{{dU\left( t \right)}}{{dt}}\;{\text{ = }}\;A(t)U(t)$$
(1.a)
$$U\left( 0 \right)\;{\text{ = }}\;1$$
(1.b)
If the A(t) belongs to a n dimensional Lie algebra G
$$A\left( t \right)\;{\text{ = }}\;\sum\limits_{i = 1}^m {{a_i}} \left( t \right){T_i}\;\;\;\left( {m \leqslant n} \right)$$
(2)
in which {T 1.T 2??T n} are the generators of the Lie algebra G. then, the evolution operator U(t) can be expressed in the following way[1]
$$U(t)\; = \;\prod\limits_{j = 1}^n {exp} \left\{ {{g_j}\left( t \right)\;{T_j}} \right\}\;\;\;\left( {{g_j}\left( 0 \right)\; = 0.\;\;j = 1 - n} \right)$$
(3)
Substituting (3) into (1), leads to
$$\sum\limits_{j = 1}^n {{g_j}} \left( t \right)\prod\limits_{k = 1}^{j - 1} {exp} \left\{ {{g_k}\left( t \right)\;ad\;{T_k}} \right\}\;{T_j} = \sum\limits_{i = 1}^n {{a_i}} \left( t \right){T_i}$$
(4)
Comparing the coefficients of the two sides before the generators T i, we can obtain a set of differential equations which is satisfied by the unknown functions g j (t).

## References

1. [1]
Fritz wolf, J. Math. Phys. 29 (2): 305 (1988).
2. [2]
Yin-Sheng Ling.Physica Energiae Fortis and Physica Nuclearis. 8: 743 (1992).Google Scholar
3. [3]
H. Hofmann. Nucl. Phys. A 394: 477 (1983).