Lie Algebraic Discussions for Time-Inhomogeneous Linear Birth–Death Processes with Immigration

Abstract

Analytical solutions for time-inhomogeneous linear birth–death processes with immigration are derived. While time-inhomogeneous linear birth–death processes without immigration have been studied by using a generating function approach, the processes with immigration are here analyzed by Lie algebraic discussions. As a result, a restriction for time-inhomogeneity of the birth–death process is understood from the viewpoint of the finiteness of the dimensionality of the Lie algebra.

Appendix: The Wei–Norman Method

For readers’ convenience, the Wei–Norman method is briefly explained in this appendix. The Wei–Norman method is one of the Lie algebraic method to solve linear differential equations with varying coefficients. As for the related algebraic method, the so-called Magnus expansion, see the review paper in [26]. For the details of the Wei–Norman method, see the original papers [15, 16].

Let $\mathcal {L}$ be a finite-dimensional Lie algebra generated by $H_1, \dots , H_L$ under the commutator product. Note that the following procedures are applicable only if the Lie algebra has a finite dimension.

For later use, we define an adjoint operator, $\mathrm {ad}$, which is a linear operator on $\mathcal {L}$ and

$$\begin{aligned}&(\mathrm {ad} H_i) H_j \equiv [H_i, H_j] = H_i H_j - H_j H_i,\end{aligned}$$

(45)

$$\begin{aligned}&(\mathrm {ad} H_i)^2 H_j = [H_i, [H_i, H_j] ], \end{aligned}$$

(46)

and so on.

Define a time-evolution operator $U(t)$, which satisfies

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t} U(t) = H(t) U(t) \end{aligned}$$

(47)

and $U(0) = I$, where $I$ is the identity operator. In addition, the operator $H(t)$ is assumed to be written as

$$\begin{aligned} H(t) = \sum _{k=1}^K a_k(t) H_k, \end{aligned}$$

(48)

where $K$ is finite and $K \le L$.

The Wei–Norman method finds an expression of the time-evolution operator $U(t)$ with the following form:

$$\begin{aligned} U(t) = \exp \left( g_1(t) H_1 \right) \exp \left( g_2(t) H_2 \right) \cdots \exp \left( g_L(t) H_L \right) , \end{aligned}$$

(49)

where $g_l(0) = 0$ for all $l \in \{1,2,\dots , L\}$. The time derivative of Eq. (49) gives

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t} U(t) = \sum _{l=0}^L \dot{g}_l(t) \left( \prod _{j=1}^{l-1} \exp (g_j(t) H_j) \right) H_i \left( \prod _{j=i}^{L} \exp (g_j(t) H_j) \right) . \end{aligned}$$

(50)

Performing a post-multiplication by the inverse operator $U^{-1}$, and employing the Baker–Campbell–Hausdorff formula,

$$\begin{aligned} \mathrm {e}^{H_i} H_j \mathrm {e}^{- H_i} = \mathrm {e}^{(\mathrm {ad} H_i)} H_j, \end{aligned}$$

(51)

the following expression is obtained:

$$\begin{aligned} \left( \frac{\mathrm {d}}{\mathrm {d}t} U(t) \right) U^{-1}(t) = \sum _{l=0}^L \dot{g}_l(t) \left( \prod _{j=1}^{l-1} \exp \left( g_j(t) (\mathrm {ad} H_j) \right) \right) H_l. \end{aligned}$$

(52)

On the other hand, from Eqs. (47) and (48), we have

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t} U(t) = \sum _{l=0}^L a_l(t) H_l U(t), \end{aligned}$$

(53)

where $a_l(t) \equiv 0$ for $l > K$. Hence, the following expression is obtained:

$$\begin{aligned} \left( \frac{\mathrm {d}}{\mathrm {d}t} U(t) \right) U^{-1}(t) = \sum _{l=0}^L a_l(t) H_l. \end{aligned}$$

(54)

Comparing Eqs. (52) with (54), we finally obtain

$$\begin{aligned} \sum _{l = 0}^L a_l(t) H_l = \sum _{l=0}^L \dot{g}_l(t) \left( \prod _{j=1}^{l-1} \exp \left( g_j(t) (\mathrm {ad} H_j) \right) \right) H_l. \end{aligned}$$

(55)

That is, we have a linear relation between $a_l(t)$ and $\dot{g}_l(t)$. Hence, comparing the coefficients of each $H_l$ in the left and right hand sides, the coupled ordinary differential equations for $\{\dot{g}_l(t)\}$ are derived.