On the using cumulant expansion method and van Kampen’s lemma for stochastic differential equations with forcing

Abstract

Second-order exact ensemble averaged equation for linear stochastic differential equations with multiplicative randomness and random forcing is obtained by using the cumulant expansion ensemble averaging method and by taking the time dependent sure part of the multiplicative operator into account. It is shown that the satisfaction of the commutativity and the reversibility requirements proposed earlier for linear stochastic differential equations without forcing are necessary for the linear stochastic differential equations with forcing when the cumulant expansion ensemble averaging method is used. It is shown that the applicability of the operator equality, which is used for the separation of operators in the literature, is also subjected to the satisfaction of the commutativity and the reversibility requirements. The van Kampen’s lemma, which is proposed for the analysis of nonlinear stochastic differential equations, is modified in order to make the probability density function obtained through the lemma depend on the forcing terms too. The second-order exact ensemble averaged equation for linear stochastic differential equations with multiplicative randomness and random forcing is also obtained by using the modified van Kampen’s lemma in order to validate the correctness of the modified lemma. Second-order exact ensemble averaged equation for one dimensional convection diffusion equation with reaction and source is obtained by using the cumulant expansion ensemble averaging method. It is shown that the van Kampen’s lemma can yield the cumulant expansion ensemble averaging result for linear stochastic differential equations when the lemma is applied to the interaction representation of the governing differential equation. It is found that the ensemble averaged equations given for one the dimensional convection diffusion equation with reaction and source in the literature obtained by applying the lemma to the original differential equation are restricted with small sure part of multiplicative operator. Second-order exact differential equations for the evolution of the probability density function for the one dimensional convection diffusion equation with reaction and source and one dimensional nonlinear overland flow equation with source are obtained by using the modified van Kampen’s lemma. The equation for the evolution of the probability density function for one dimensional nonlinear overland flow equation with source given in the literature is found to be not second-order exact. It is found that the differential equations for the evolution of the probability density functions for various hydrological processes given in the literature are not second-order exact. The significance of the new terms found due to the second-order exact ensemble averaging performed on the one dimensional convection diffusion equation with reaction and source and during the application of the van Kampen’s lemma to the one dimensional nonlinear overland flow equation with source is investigated.

Appendices

Appendix 1

Using the following identity given by van Kampen (1981) for any random matrix B and any random vector b as

$$ \left\langle e^{B} b \right\rangle = \left\langle e^{B} \right\rangle \{ \left\langle b \right\rangle + \left\langle \left\langle Bb \right\rangle \right\rangle + \frac{1}{2} \left\langle \left\langle B^{2} b \right\rangle \right\rangle + \cdots \} $$

(64)

Eq. 9 can be written in terms of cumulants after partial ordering as

$$ \left\langle H \right\rangle = \\ \left\lceil {\exp \left[\begin{gathered} \alpha \int\limits_{0}^{t} {dt_{1} \left\langle{V(t_{1} )} \right\rangle } + \alpha^{2} \int\limits_{0}^{t}{dt_{1} \int\limits_{0}^{{t_{1} }} {dt_{2} } \ll V(t_{1} )V(t_{2})} \gg + \ldots \hfill \\ \ldots + \alpha^{m} \int\limits_{0}^{t}{dt_{1} \int\limits_{0}^{{t_{1} }} {dt_{2} } \ldots\int\limits_{0}^{{t_{m - 1} }} {dt_{m} } \ll V(t_{1} )V(t_{2} )}\ldots V(t_{m} ) \gg \hfill \\ \end{gathered} \right]}\right\rceil a \\ + \int\limits_{0}^{t} {dt^{\prime }\left\lceil\begin{gathered} \exp \left[\begin{gathered} \alpha \int\limits_{{t^{\prime } }}^{t} {dt_{1}\left\langle {V(t_{1} )} \right\rangle } + \alpha^{2}\int\limits_{{t^{\prime } }}^{t} {dt_{1} \int\limits_{{t^{\prime }}}^{{t_{1} }} {dt_{2} } \ll V(t_{1} )V(t_{2} )} \gg + \ldots\hfill \\ \ldots + \alpha^{m} \int\limits_{{t^{\prime}}}^{t}{dt_{1} \int\limits_{{t^{\prime}}}^{{t_{1} }} {dt_{2} } \ldots\int\limits_{{t^{\prime}}}^{{t_{m - 1} }} {dt_{m} } \ll V(t_{1})V(t_{2} )} \ldots V(t_{m} ) \gg \hfill \\ \end{gathered} \right]\hfill \\ \left\{ {\left\langle {g(t^{\prime } )} \right\rangle +\alpha \int\limits_{{t^{\prime } }}^{t} {dt^{\prime \prime } \ll V(t^{\prime \prime } )g(t^{\prime } ) \gg + \ldots } } \right\}\hfill \\ \end{gathered} \right\rceil } $$

(65)

The symbol “$\langle \langle \quad \rangle \rangle$” in Eqs. 64 and 65 stands for the cumulant averaging. When Eq. 65 is differentiated with respect to time and a partial truncation is applied based on the assumption that the Kubo number, ατ _c , and α smaller than unity the resultant equation is

$$ \frac{{\partial \langle H \rangle }}{{\partial t}} = \left[{\alpha \langle V(t) \rangle + \alpha ^{2} \int\limits_{0}^{t}{d\tau \langle \langle V(t)V(t - \tau ) \rangle \rangle } }\right]\left\lceil {\exp \left[ {\alpha \int\limits_{0}^{t} {dt_{1} \langle V(t_{1} ) \rangle + \alpha ^{2}\int\limits_{0}^{t} {dt_{1} \int\limits_{0}^{{t_{1} }} {dt_{2} }\langle \langle V(t_{1} )V(t_{2} ) \rangle \rangle + \cdots + \alpha ^{m} \int\limits_{0}^{t} {dt_{1} \int\limits_{0}^{{t_{1} }}{dt_{2} \ldots \int\limits_{0}^{{t_{{m - 1}} }} {dt_{m} \langle \langle V(t_{1} )V(t_{2} ) \ldots V(t_{m} ) \rangle \rangle } } } } } } \right]} \right\rceil a + \langle g(t) \rangle + \left[{\alpha \langle V(t) \rangle + \alpha ^{2} \int\limits_{0}^{t} {d\tau \langle \langle V(t)V(t - \tau ) \rangle \rangle } }\right]\int\limits_{0}^{t} {dt^{\prime } \left\lceil {\exp \left[{\alpha \int\limits_{{t^{\prime}}}^{t} {dt_{1} \langle V(t_{1} )\rangle } + \alpha ^{2} \int\limits_{{t^{\prime}}}^{t} {dt_{1}\int\limits_{{t^{\prime}}}^{{t_{1} }} {dt_{2} } \langle \langle V(t_{1} )V(t_{2} ) \rangle \rangle + \cdots + \alpha ^{m} \int\limits_{{t^{\prime}}}^{t} {dt_{1} \int\limits_{{t^{\prime}}}^{{t_{1} }} {dt_{2} } \ldots \int\limits_{{t^{\prime}}}^{{t_{{m - 1}} }} {dt_{m} } \langle \langle V(t_{1} )V(t_{2} ) \ldots V(t_{m} ) \rangle \rangle } } } \right]\left\{ { \langle g(t^{\prime } ) \rangle + \alpha \int\limits_{{t^{\prime } }}^{t} {dt^{{\prime \prime }} \langle \langle V(t^{{\prime \prime }} )g(t^{\prime } ) \rangle \rangle } } \right\}} \right\rceil } + \int\limits_{0}^{t} {dt^{\prime } \left\lceil {\exp \left[ {\alpha \int\limits_{{t^{\prime}}}^{t} {dt_{1} \langle V(t_{1} ) \rangle } + \alpha ^{2} \int\limits_{{t^{\prime}}}^{t} {dt_{1} \int\limits_{{t^{\prime}}}^{{t_{1} }} {dt_{2} } \langle \langle V(t_{1} )V(t_{2} )} \rangle \rangle + \cdots + \alpha ^{m} \int\limits_{{t^{\prime}}}^{t} {dt_{1} \int\limits_{{t^{\prime}}}^{{t_{1} }} {dt_{2} \ldots \int\limits_{{t^{\prime}}}^{{t_{{m - 1}} }} {dt_{m} \langle \langle V(t_{1} )V(t_{2} ) \ldots V(t_{m} ) \rangle \rangle } } } } \right] \alpha \langle \langle V(t)g(t^{\prime } ) \rangle \rangle } \right\rceil } $$

(66)

Please notice that the error made due to the truncation applied to reach Eq. 66 is smaller than α ² τ _c . By using Eq. 65, Eq. 66 can also be written as

$$\frac{\partial \langle H \rangle }{\partial t} = \left[ {\alpha \langle V(t) \rangle + \alpha^{2} \int\limits_{0}^{t} {d\tau \langle \langle V(t)V(t - \tau ) \rangle \rangle } } \right] \langle H \rangle + \langle g(t) \rangle + \int\limits_{0}^{t} {dt^{\prime} \left\lceil {\exp \left[ {\alpha \int\limits_{{t^{\prime}}}^{t} {dt_{1} \langle V(t_{1} ) \rangle } + \alpha^{2} \int\limits_{{t^{\prime}}}^{t} {dt_{1} \int\limits_{{t^{\prime}}}^{{t_{1} }} {dt_{2} } \langle \langle V(t_{1} )V(t_{2} )} \rangle \rangle + \cdots + \alpha^{m} \int\limits_{{t^{\prime}}}^{t} {dt_{1} \int\limits_{{t^{\prime}}}^{{t_{1} }} {dt_{2} } \ldots \int\limits_{{t^{\prime}}}^{{t_{m - 1} }} {dt_{m} } \langle \langle V(t_{1} )V(t_{2} )} \ldots V(t_{m} ) \rangle \rangle } \right] \alpha \langle \langle V(t)g(t^{\prime} ) \rangle \rangle } \right\rceil } $$

(67)

When the terms with order smaller than α ² τ _c are omitted Eq. 67 becomes Eq. 10

Appendix 2

The $D_{c}^{\dag }$ and $D_{e}^{\dag }$ in Eq. 52 are

$$ D_{c}^{\dag } = \frac{\partial }{{\partial c}}\frac{\partial }{{\partial c}}\left[ {2\left[ {\int\limits_{0}^{t} {ds\,Cov\left[ {w\frac{\partial }{{\partial x}}c,w(x + \varsigma ,\,t - s)\frac{\partial }{{\partial x}}c(x + \varsigma ,\,t - s)} \right]} } \right.} \right. - \int\limits_{0}^{t} {ds\,Cov\left[ {w\frac{\partial }{{\partial x}}c\,,r(x + \varsigma ,\,t - s)c(x + \varsigma ,\,t - s)} \right]} - \int\limits_{0}^{t} {ds\,Cov\left[ {w\frac{\partial }{{\partial x}}c\,,f(x + \varsigma ,\,t - s)} \right]} - \int\limits_{0}^{t} {ds\,Cov\left[ {rc,w(x + \varsigma ,\,t - s)\frac{\partial }{{\partial x}}c(x + \varsigma ,\,t - s)} \right]} + \int\limits_{0}^{t} {ds\,Cov[rc,r(x + \varsigma ,\,t - s)c(x + \varsigma ,\,t - s)]} + \int\limits_{0}^{t} {ds\,Cov[rc,f(x + \varsigma ,\,t - s)]} - \int\limits_{0}^{t} {ds\,Cov\left[ {f(t),w(x + \varsigma ,\,t - s)\frac{\partial }{{\partial x}}c(x + \varsigma ,\,t - s)} \right]} + \int\limits_{0}^{t} {ds\,Cov[f(t),r(x + \varsigma ,\,t - s)c(x + \varsigma ,\,t - s)]} \left. { + \left. {\int\limits_{0}^{t} {ds\,Cov[f(t),f(x + \varsigma ,\,\,t - s)]} } \right] \langle \rho ^{\dag } \rangle } \right] $$

(68)

$$D_{e}^{\dag } = \frac{\partial }{{\partial c}}\frac{\partial }{{\partial e}}\left[2\left[ - \int\limits_{0}^{t} {ds\,Cov\left[ {w\frac{\partial }{{\partial x}}c\,,f(x + \varsigma ,t - s)} \right]} + \int\limits_{0}^{t} {ds\,Cov\left[rc,f(x + \varsigma ,\,t - s)\right]} + \int\limits_{0}^{t} {ds\,Cov\left[f(t),f(x + \varsigma ,\,t - s)\right]} \right] \left\langle \rho ^{\dag } \right\rangle \right] + \frac{1}{2}\frac{\partial }{{\partial e}}\frac{\partial }{{\partial c}}\left[2\left[ - \int\limits_{0}^{t} {ds\,Cov\left[ {f(t),w(x + \varsigma ,\,t - s)\frac{\partial }{{\partial x}}c(x + \varsigma ,\,t - s)} \right]} + \int\limits_{0}^{t} {ds\,Cov\left[f(t),r(x + \varsigma ,\,t - s)c(x + \varsigma ,\,t - s)\right]} + \int\limits_{0}^{t} {ds\,Cov\left[f(t)\,,f(x + \varsigma ,\,t - s)\right]} \right]\, \left\langle \rho ^{\dag } \right\rangle \right] + \frac{1}{2}\frac{\partial }{{\partial e}}\frac{\partial }{{\partial e}}\left[2\int\limits_{0}^{t} {ds\,Cov\left[f(t),f(x + \varsigma ,\,t - s)\right]} \left\langle \rho ^{\dag } \right\rangle \right] $$

(69)

in which $\varsigma = \int\nolimits_{t - s}^{t} {d\eta \langle w(x,\eta ) \rangle }$

Appendix 3

In order to replace $\langle c(x + \varsigma ,t - s) \rangle$ with 〈c(x, t)〉 in Eq. 53 and obtain a differential equation for 〈c(x, t)〉 we can write the equivalence of $\langle c(x + \varsigma ,t - s) \rangle$ by using Taylor series expansion with highest order terms as

$$ \left\langle c(x + \varsigma ,t - s) \right\rangle = \left\langle c(x,t) \right\rangle - s\frac{\partial \left\langle c(x,t) \right\rangle }{\partial t} + \varsigma \frac{\partial \left\langle c(x,t) \right\rangle }{\partial x} $$

(70)

Under the assumption that the correlation length among the random operators/variables are τ _c Eq. 70 can become as much as

$$ \left\langle c(x + \varsigma ,t - s) \right\rangle = \left\langle c(x,t) \right\rangle - \tau_{c} \frac{\partial \left\langle c(x,t) \right\rangle }{\partial t} + \tau_{c} \left\langle w \right\rangle \frac{\partial \left\langle c(x,t) \right\rangle }{\partial x} $$

(71)

By using Eq. 11 with the first order terms only we can write

$$ \left\langle c(x + \varsigma ,t - s) \right\rangle = \left\langle c(x,t) \right\rangle - \tau_{c} \left[ {A_{0} (t) \left\langle c \right\rangle + \alpha \left\langle A_{1} (t) \right\rangle \left\langle c \right\rangle + \left\langle f(t) \right\rangle } \right] + \tau_{c} \left\langle w \right\rangle \frac{\partial \left\langle c(x,t) \right\rangle }{\partial x} $$

(72)

The second and the third term in the bracket in Eq. 72 can be omitted by assuming that the Kubo number is smaller than unity as assumed during the cumulant expansion ensemble averaging. In order to omit the first term in the bracket based on the Kubo number restriction again we should require that the order of A ₀(t) should be order of α or less so Eq. 72 becomes

$$ \left\langle c(x + \varsigma ,t - s) \right\rangle = \left\langle c(x,t) \right\rangle + \tau_{c} \left\langle w \right\rangle \frac{\partial \left\langle c(x,t) \right\rangle }{\partial x} $$

(73)

Since we already assumed that the order of A ₀(t) is α or less we can drop the second term at the right hand side of Eq. 73 so that $\langle c(x + \varsigma ,t - s) \rangle$ can be replaced with 〈c(x, t)〉 and the terms due to the sure part of the multiplicative operator in Eq. 53 should also be omitted since A ₀(t) is assumed to be α or less to obtain

$$ \begin{aligned} \frac{\partial \left\langle c \right\rangle }{\partial t} = & - \left\langle w \right\rangle \frac{\partial }{\partial x} \left\langle c \right\rangle + D\frac{{\partial^{2} }}{{\partial x^{2} }} \left\langle c \right\rangle + \left\langle r \right\rangle \left\langle c \right\rangle + \left\langle f(t) \right\rangle \\ & - \int\limits_{0}^{t} {ds\,Cov[r,w(x + \varsigma ,t - s)]\frac{\partial }{\partial x} \left\langle c(x,t) \right\rangle } \\ & - \int\limits_{0}^{t} {ds\,Cov\left[ {w\frac{\partial }{\partial x},r(x + \varsigma ,t - s)} \right] \left\langle c(x,t) \right\rangle } \\ & + \int\limits_{0}^{t} {ds\,Cov\left[ {w\frac{\partial }{\partial x},w(x + \varsigma ,t - s)} \right]\frac{\partial }{\partial x} \left\langle c(x,t) \right\rangle } \\ & + \int\limits_{0}^{t} {ds\,Cov[r,r(x + \varsigma ,t - s)] \left\langle c(x,t) \right\rangle } \\ & - \int\limits_{0}^{t} {ds\,Cov\left[ {w\frac{\partial }{\partial x},f(x + \varsigma ,t - s)} \right]} \\ & + \int\limits_{0}^{t} {ds\,Cov[r,f(x + \varsigma ,t - s)]} \\ \end{aligned} $$

(74)

in which $\xi = \int\nolimits_{t - s}^{\tau } {d\eta \langle w(x,\eta ) \rangle } \quad \varsigma = \int\limits_{t - s}^{t} {d\eta \langle w(x,\eta ) \rangle }$

We can also replace ς with −ς in Eq. 74 under A ₀(t) ≪ α or spatially stationary randomness assumption to reach Eq. 54.