-\left

\cdot % NEW REWORKED \left

$ } % NEW REWORKED % % NEW REWORKED between the function $\eta$ at time $t$ and the derivative of the function % NEW REWORKED $\eta$ with respect to stress, $\sigma$, at time $t-\tau$ by $\Pi(x,t,\tau)$ as, % NEW REWORKED % % NEW REWORKED \begin{eqnarray} % NEW REWORKED \nonumber % NEW REWORKED \lefteqn{\Pi (x,t,\tau) % NEW REWORKED = % NEW REWORKED Cov_0\left[\vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D(x_t), q(x_t), r(x_t), % NEW REWORKED \epsilon(x_t,t)); \right.} \\ % NEW REWORKED \label{eqno_S10b} % NEW REWORKED && \left. \displaystyle \frac{\partial \eta (\sigma(x_{t-\tau}, t-\tau), D(x_{t-\tau}), q(x_{t-\tau}), % NEW REWORKED r(x_{t-\tau}), \epsilon(x_{t-\tau}, t-\tau))}{\partial \sigma} \right] % NEW REWORKED \end{eqnarray} % NEW REWORKED % % NEW REWORKED and the time-ordered covariance between function $\eta$ at time $t$ and function % NEW REWORKED $\eta$ at time $t-\tau$ by $\Omega(x,t,\tau)$ as, % NEW REWORKED % % NEW REWORKED \begin{eqnarray} % NEW REWORKED \nonumber % NEW REWORKED \lefteqn{\Omega (x, t,\tau) = Cov_0\left[\vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D(x_t), q(x_t), r(x_t), % NEW REWORKED \epsilon(x_t,t)); \right.} \\ % NEW REWORKED \label{eqno_S10c} % NEW REWORKED && \left. \eta (\sigma(x_{t-\tau}, t-\tau), D (x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}), \epsilon % NEW REWORKED (x_{t-\tau}, t-\tau)) \vphantom{\int_{0}^{t} d\tau} \right] % NEW REWORKED \end{eqnarray} % NEW REWORKED % % NEW REWORKED one can write the ensemble average form of Eq.~(\ref{eqno_S7}) as % NEW REWORKED % % NEW REWORKED \begin{eqnarray} % NEW REWORKED \nonumber % NEW REWORKED \lefteqn{\frac{\partial \left<\rho (\sigma(x_t,t), t)\right>}{\partial t} =} \\ % NEW REWORKED \nonumber % NEW REWORKED & - & \displaystyle \frac{\partial}{\partial \sigma}\left[ \left\{ \Xi (x,t) % NEW REWORKED - \int_{0}^{t} d\tau \Pi(x,t,\tau) \right\} \left<\rho (\sigma(x_t,t), t) \right>\right] \\ % NEW REWORKED \label{eqno_S10} % NEW REWORKED &+& \displaystyle \frac{\partial}{\partial \sigma } \left[\int_{0}^{t} d\tau % NEW REWORKED \Omega(x,t,\tau) \vphantom{\int_{0}^{t}} % NEW REWORKED \displaystyle \frac{\partial \left<\rho (\sigma (x_t,t), t)\right>}{\partial \sigma} \right ] % NEW REWORKED \end{eqnarray} % NEW REWORKED % % NEW REWORKED %\begin{eqnarray} % NEW REWORKED %\nonumber % NEW REWORKED %\lefteqn{\frac{\partial \left<\rho (\sigma(x_t,t), t)\right>}{\partial t}=} \\ % NEW REWORKED %\nonumber % NEW REWORKED %&-&\displaystyle \frac{\partial}{\partial \sigma}\left\{ \left[\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D(x_t), % NEW REWORKED %q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right. \right. \\ % NEW REWORKED %\nonumber % NEW REWORKED %&-& \left. \left. \int_{0}^{t} d\tau Cov_0\left[\vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D(x_t), q(x_t), r(x_t), % NEW REWORKED %\epsilon(x_t,t)); \right. \right. \right. \\ % NEW REWORKED %\nonumber % NEW REWORKED %& & \left. \left. \left. \displaystyle \frac{\partial \eta (\sigma(x_{t-\tau}, t-\tau), D(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}) \epsilon(x_{t-\tau}, t-\tau)}{\partial \sigma}\right]\right]\left<\rho (\sigma(x_t,t), t)\right>\right\} \\ % NEW REWORKED %\nonumber % NEW REWORKED %&+& \displaystyle \frac{\partial}{\partial \sigma }\left\{ \left[\int_{0}^{t} d\tau Cov_0\left[\eta(\sigma (x_t,t), D(x_t), q(x_t), % NEW REWORKED %r (x_t), \epsilon(x_t,t)); \right. \right. \right. \\ % NEW REWORKED %\nonumber % NEW REWORKED %& & \left. \left. \left. \eta (\sigma(x_{t-\tau}, t-\tau), D (x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}), \epsilon % NEW REWORKED %(x_{t-\tau}, % NEW REWORKED %t-\tau)) \right] \vphantom{\int_{0}^{t}} \right] \displaystyle \frac{\partial \left<\rho (\sigma (x_t,t), t)\right>}{\partial \sigma} % NEW REWORKED %\right \} \\ % NEW REWORKED %\label{eqno_S10} % NEW REWORKED %\end{eqnarray} % NEW REWORKED % % NEW REWORKED to exact second order (to the order of the covariance time of $ \eta $). By % NEW REWORKED combining Eqs.~(\ref{eqno_S9}) and ~(\ref{eqno_S10}) and rearranging the terms % NEW REWORKED yields an Eulerian--Lagrangian form of Fokker-Planck equation \footnote{also % NEW REWORKED known as Forward--Kolmogorov Equation or Fokker--Planck--Kolmogorov (FPK) % NEW REWORKED equation \citep{book:Risken}, \citep{book:Gardiner}, % NEW REWORKED \citep{Schueller:1997}} (FPE) (for details, refer to Kavvas % NEW REWORKED \citep{Kavvas:2003}), which, for writing convenience, after denoting the % NEW REWORKED covariance between the derivative of the function $\eta$ with respect to stress, % NEW REWORKED $\sigma$ at time $t$ and the function $\eta$ at time $t-\tau$ by % NEW REWORKED $\Phi(x,t,\tau)$, as, % NEW REWORKED % % NEW REWORKED \begin{eqnarray} % NEW REWORKED \nonumber % NEW REWORKED \lefteqn{\Phi (x, t,\tau) = % NEW REWORKED Cov_0 \left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D(x_t), % NEW REWORKED q(x_t), r(x_t) \epsilon(x_t,t))}{\partial \sigma} ; \right.} \\ % NEW REWORKED \label{eqno_S12a} % NEW REWORKED && \left. \eta(\sigma(x_{t-\tau}, t-\tau), D(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}), % NEW REWORKED \epsilon(x_{t-\tau}, t-\tau)) \vphantom{\int_{0}^{t} d\tau} \right], % NEW REWORKED \end{eqnarray} % NEW REWORKED % NEW REWORKED \noindent can be written as, % NEW REWORKED % NEW REWORKED \begin{eqnarray} % NEW REWORKED \nonumber % NEW REWORKED \lefteqn{\frac{\partial P(\sigma(x_t,t), t)}{\partial t} =} \\ % NEW REWORKED \nonumber % NEW REWORKED & - & \displaystyle \frac{\partial}{\partial \sigma}\left[ \left\{ \Xi (x,t) % NEW REWORKED + \int_{0}^{t} d\tau \Phi(x,t,\tau) \right\} P(\sigma(x_t,t), t) \right] \\ % NEW REWORKED \label{eqno_S12} % NEW REWORKED &+& \displaystyle \frac{\partial^2}{\partial \sigma^2 } \left[\int_{0}^{t} d\tau % NEW REWORKED \Omega(x,t,\tau) \vphantom{\int_{0}^{t}} % NEW REWORKED P (\sigma (x_t,t), t) \right ] % NEW REWORKED \end{eqnarray} % NEW REWORKED % % NEW REWORKED % % NEW REWORKED %\begin{eqnarray} % NEW REWORKED %\nonumber % NEW REWORKED %\lefteqn{\displaystyle \frac{\partial P(\sigma(x_t,t), t)}{\partial t}=} \\ % NEW REWORKED %\nonumber % NEW REWORKED %&-& \displaystyle \frac{\partial}{\partial \sigma} \left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D(x_t), % NEW REWORKED %q(x_t), r(x_t) \epsilon(x_t,t)) \right> \right. \right. \\ % NEW REWORKED %\nonumber % NEW REWORKED %&+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D(x_t), q(x_t), r(x_t) % NEW REWORKED %\epsilon(x_t,t))}{\partial \sigma}; \right. \right. \right. \\ % NEW REWORKED %\nonumber % NEW REWORKED %& & \left. \left. \left. \eta(\sigma(x_{t-\tau}, t-\tau), D(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}), % NEW REWORKED %\epsilon(x_{t-\tau}, t-\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \} P(\sigma(x_t,t),t) \right] \\ % NEW REWORKED %\nonumber % NEW REWORKED %&+& \displaystyle \frac{\partial^2}{\partial \sigma^2} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[ \vphantom{\int_{0}^{t}} % NEW REWORKED %\eta(\sigma(x_t,t), D(x_t,t), q(x_t,t), r (x_t,t), \epsilon(x_t,t)); \right. \right. \right. \\ % NEW REWORKED %\nonumber % NEW REWORKED %& & \left. \left. \left. \eta_1 (\sigma(x_{t-\tau}, t-\tau), D(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}), % NEW REWORKED %\epsilon(x_{t-\tau}, t-\tau)) \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma (x_t,t),t) \right] \\ % NEW REWORKED %\label{eqno_S12} % NEW REWORKED %\end{eqnarray} % NEW REWORKED % % NEW REWORKED % % NEW REWORKED to exact second order. The terms $\Xi(x,t)$, % NEW REWORKED $\Phi(x,t,\tau)$, and $\Omega(x,t,\tau)$ in % NEW REWORKED Eq.~(\ref{eqno_S12}) are given by Eqs.~(\ref{eqno_S10a}), % NEW REWORKED (\ref{eqno_S12a}), and (\ref{eqno_S10c}) respectively. % NEW REWORKED Eq.~(\ref{eqno_S12}) is the most general relation for % NEW REWORKED probabilistic behavior of inelastic (non--linear) 1-D % NEW REWORKED constitutive rate equation. % NEW REWORKED % % NEW REWORKED % % NEW REWORKED % % NEW REWORKED % % % % % % % % % % % % %ORIGINAL %ORIGINAL % % %ORIGINAL % % %ORIGINAL % % %ORIGINAL % % %ORIGINAL This ensemble average was recently derived as %ORIGINAL (for detailed derivation, refer to \citet{Kavvas:1996}) %ORIGINAL % %ORIGINAL \begin{eqnarray} %ORIGINAL \nonumber %ORIGINAL \lefteqn{\frac{\partial \left<\rho (\sigma(x_t,t), t)\right>}{\partial t}=} \\ %ORIGINAL \nonumber %ORIGINAL &-&\displaystyle \frac{\partial}{\partial \sigma}\left\{ \left[\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D(x_t), %ORIGINAL q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right. \right. \\ %ORIGINAL \nonumber %ORIGINAL &-& \left. \left. \int_{0}^{t} d\tau Cov_0\left[\vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D(x_t), q(x_t), r(x_t), %ORIGINAL \epsilon(x_t,t)); \right. \right. \right. \\ %ORIGINAL \nonumber %ORIGINAL & & \left. \left. \left. \displaystyle \frac{\partial \eta (\sigma(x_{t-\tau}, t-\tau), D(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}) \epsilon(x_{t-\tau}, t-\tau)}{\partial \sigma}\right]\right]\left<\rho (\sigma(x_t,t), t)\right>\right\} \\ %ORIGINAL \nonumber %ORIGINAL &+& \displaystyle \frac{\partial}{\partial \sigma }\left\{ \left[\int_{0}^{t} d\tau Cov_0\left[\eta(\sigma (x_t,t), D(x_t), q(x_t), %ORIGINAL r (x_t), \epsilon(x_t,t)); \right. \right. \right. \\ %ORIGINAL \nonumber %ORIGINAL & & \left. \left. \left. \eta (\sigma(x_{t-\tau}, t-\tau), D (x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}), \epsilon %ORIGINAL (x_{t-\tau}, %ORIGINAL t-\tau)) \right] \vphantom{\int_{0}^{t}} \right] \displaystyle \frac{\partial \left<\rho (\sigma (x_t,t), t)\right>}{\partial \sigma} %ORIGINAL \right \} \\ %ORIGINAL \label{eqno_S10} %ORIGINAL \end{eqnarray} %ORIGINAL % %ORIGINAL to exact second order (to the order of the covariance time of $ \eta $). %ORIGINAL % %ORIGINAL In Eq.~(\ref{eqno_S10}), $Cov_0[\cdot]$ is the time ordered covariance function %ORIGINAL defined by %ORIGINAL % %ORIGINAL \begin{equation} %ORIGINAL \label{eqno_S11} %ORIGINAL Cov_0\left[\eta(x,t_1), \eta(x,t_2)\right] %ORIGINAL = %ORIGINAL \left<\eta(x,t_1) \eta(x,t_2)\right> %ORIGINAL - %ORIGINAL \left<\eta(x,t_1)\right> \cdot \left<\eta(x,t_2)\right> %ORIGINAL \end{equation} %ORIGINAL %ORIGINAL By combining Eqs.~(\ref{eqno_S10}) and ~(\ref{eqno_S9}) and rearranging the terms %ORIGINAL yields the following Fokker-Planck equation (FPE, also known as %ORIGINAL Forward--Kolmogorov Equation or Fokker--Planck--Kolmogorov FPK Equation) %ORIGINAL (\citet{book:Risken}, \citet{book:Gardiner}, \citet{Schueller:1997report}): %ORIGINAL % %ORIGINAL \begin{eqnarray} %ORIGINAL \nonumber %ORIGINAL \lefteqn{\displaystyle \frac{\partial P(\sigma(x_t,t), t)}{\partial t}=} %ORIGINAL \\ \nonumber %ORIGINAL &-& %ORIGINAL \displaystyle \frac{\partial}{\partial \sigma} %ORIGINAL \left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D(x_t), %ORIGINAL q(x_t), r(x_t) \epsilon(x_t,t)) \right> \right. \right. %ORIGINAL \\ \nonumber %ORIGINAL &+& %ORIGINAL \left. \left. \int_{0}^{t} d\tau Cov_0 %ORIGINAL \left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D(x_t), q(x_t), r(x_t) %ORIGINAL \epsilon(x_t,t))}{\partial \sigma}; \right. \right. \right. %ORIGINAL \\ \nonumber %ORIGINAL & & %ORIGINAL \left. \left. \left. \eta(\sigma(x_{t-\tau}, t-\tau), %ORIGINAL D(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}), %ORIGINAL \epsilon(x_{t-\tau}, t-\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \} %ORIGINAL P(\sigma(x_t,t),t) \right] %ORIGINAL \\ \nonumber %ORIGINAL &+& %ORIGINAL \displaystyle \frac{\partial^2}{\partial \sigma^2} %ORIGINAL \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[ \vphantom{\int_{0}^{t}} %ORIGINAL \eta(\sigma(x_t,t), D(x_t,t), q(x_t,t), r (x_t,t), %ORIGINAL \epsilon(x_t,t)); \right. \right. \right. %ORIGINAL \\ \nonumber %ORIGINAL & & %ORIGINAL \left. \left. \left. \eta_1 (\sigma(x_{t-\tau}, t-\tau), %ORIGINAL D(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}), %ORIGINAL \epsilon(x_{t-\tau}, t-\tau)) \vphantom{\int_{0}^{t}} \right] %ORIGINAL \vphantom{\int_{0}^{t}} \right\} P(\sigma (x_t,t),t) \right] %ORIGINAL \\ %ORIGINAL \label{eqno_S12} %ORIGINAL \end{eqnarray} %ORIGINAL % %ORIGINAL to exact second order. %ORIGINAL % %ORIGINAL This is the most general relation for probabilistic %ORIGINAL behavior of inelastic (non--linear, elastic--plastic) %ORIGINAL 1-D stochastic incremental constitutive equation. %ORIGINAL % %ORIGINAL % %ORIGINAL % %ORIGINAL % %ORIGINAL % %ORIGINAL % %ORIGINAL % %ORIGINAL %ORIGINAL %ORIGINAL % % % LATEST REWORKED % LATEST REWORKED % LATEST REWORKED % LATEST REWORKED % % This ensemble average was recently derived as (for detailed derivation, refer to \citet{Kavvas:1996}) % \begin{eqnarray} \nonumber \lefteqn{\frac{\partial \left<\rho (\sigma(x_t,t), t)\right>}{\partial t} =} \\ \nonumber & - & \displaystyle \frac{\partial}{\partial \sigma}\left[ \left\{ \Xi (x,t) - \int_{0}^{t} d\tau \Pi(x,t,\tau) \right\} \left<\rho (\sigma(x_t,t), t) \right>\right] \\ \label{eqno_S10} &+& \displaystyle \frac{\partial}{\partial \sigma } \left[\int_{0}^{t} d\tau \Omega(x,t,\tau) \vphantom{\int_{0}^{t}} \displaystyle \frac{\partial \left<\rho (\sigma (x_t,t), t)\right>}{\partial \sigma} \right ] \end{eqnarray} % to exact second order (to the order of the covariance time of $ \eta $). In Eq.~(\ref{eqno_S10}), $\Xi (x,t)$ is the ensemble average of the function $\eta$ at time $t$ and is given as, % \begin{equation} \nonumber %\label{eqno_S10a} \Xi (x,t) = \left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D(x_t), q(x_t), r(x_t), \epsilon(x_t,t)) \right>; \end{equation} % and $\Pi(x,t,\tau)$ is the covariance between the function $\eta$ at time $t$ and the derivative of the function $\eta$ with respect to stress, $\sigma$, at time $t-\tau$ and is given as, % \begin{eqnarray} \nonumber \lefteqn{\Pi (x,t,\tau) = Cov_0\left[\vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D(x_t), q(x_t), r(x_t), \epsilon(x_t,t)); \right.} \\ \nonumber %\label{eqno_S10b} && \left. \displaystyle \frac{\partial \eta (\sigma(x_{t-\tau}, t-\tau), D(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}), \epsilon(x_{t-\tau}, t-\tau))}{\partial \sigma} \right]; \end{eqnarray} % while $\Omega(x,t,\tau)$ is the covariance between function $\eta$ at time $t$ and function $\eta$ at time $t-\tau$ and is given as, % \begin{eqnarray} \nonumber \lefteqn{\Omega (x, t,\tau) = Cov_0\left[\vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D(x_t), q(x_t), r(x_t), \epsilon(x_t,t)); \right.} \\ \nonumber %\label{eqno_S10c} && \left. \eta (\sigma(x_{t-\tau}, t-\tau), D (x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}), \epsilon (x_{t-\tau}, t-\tau)) \vphantom{\int_{0}^{t} d\tau} \right] \end{eqnarray} % % % %\begin{eqnarray} %\nonumber %\lefteqn{\frac{\partial \left<\rho (\sigma(x_t,t), t)\right>}{\partial t}=} \\ %\nonumber %&-&\displaystyle \frac{\partial}{\partial \sigma}\left\{ \left[\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D(x_t), %q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right. \right. \\ %\nonumber %&-& \left. \left. \int_{0}^{t} d\tau Cov_0\left[\vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D(x_t), q(x_t), r(x_t), %\epsilon(x_t,t)); \right. \right. \right. \\ %\nonumber %& & \left. \left. \left. \displaystyle \frac{\partial \eta (\sigma(x_{t-\tau}, t-\tau), D(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}) \epsilon(x_{t-\tau}, t-\tau)}{\partial \sigma}\right]\right]\left<\rho (\sigma(x_t,t), t)\right>\right\} \\ %\nonumber %&+& \displaystyle \frac{\partial}{\partial \sigma }\left\{ \left[\int_{0}^{t} d\tau Cov_0\left[\eta(\sigma (x_t,t), D(x_t), q(x_t), %r (x_t), \epsilon(x_t,t)); \right. \right. \right. \\ %\nonumber %& & \left. \left. \left. \eta (\sigma(x_{t-\tau}, t-\tau), D (x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}), \epsilon %(x_{t-\tau}, %t-\tau)) \right] \vphantom{\int_{0}^{t}} \right] \displaystyle \frac{\partial \left<\rho (\sigma (x_t,t), t)\right>}{\partial \sigma} %\right \} \\ %\label{eqno_S10} %\end{eqnarray} % %to exact second order (to the order of the covariance time of $ \eta $). %% It is important to note that the covariances appearing in the above equations are time ordered covariances, which for two dummy variables $p(x,t_1)$ and $p(x,t_2)$ can be written as, % \begin{equation} \label{eqno_S11} Cov_0\left[p(x,t_1); p(x,t_2)\right] = \left

-\left

\cdot \left

\end{equation}
%
%In Eq.~(\ref{eqno_S10}), $Cov_0[\cdot]$ is the time ordered covariance function
%defined by
%%
%\begin{equation}
%\label{eqno_S11}
%Cov_0\left[\eta(x,t_1), \eta(x,t_2)\right]
%=
%\left<\eta(x,t_1) \eta(x,t_2)\right>
%-
%\left<\eta(x,t_1)\right> \cdot \left<\eta(x,t_2)\right>
%\end{equation}
By combining Eqs.~(\ref{eqno_S9}) and ~(\ref{eqno_S10}) and rearranging the terms
yields the following
Eulerian--Lagrangian form of second-order accurate Fokker-Planck equation \footnote{also
known as Forward--Kolmogorov Equation or Fokker--Planck--Kolmogorov (FPK)
equation \citep{book:Risken}, \citep{book:Gardiner},
\citep{Schueller:1997}} (FPE) (for details, refer to Kavvas
\citep{Kavvas:2003}):
%
\begin{eqnarray}
\nonumber
\lefteqn{\frac{\partial P(\sigma(x_t,t), t)}{\partial t} =} \\
\nonumber
& - & \displaystyle \frac{\partial}{\partial \sigma}\left[ \left\{ \Xi (x,t)
+ \int_{0}^{t} d\tau \Phi(x,t,\tau) \right\} P(\sigma(x_t,t), t) \right] \\
\label{eqno_S12}
&+& \displaystyle \frac{\partial^2}{\partial \sigma^2 } \left[\int_{0}^{t} d\tau
\Omega(x,t,\tau) \vphantom{\int_{0}^{t}}
P (\sigma (x_t,t), t) \right ]
\end{eqnarray}
%
where, $\Phi(x,t,\tau)$ is the time-ordered covariance between the derivative of the function
$\eta$ with respect to stress, $\sigma$ at time $t$ and the function $\eta$
at time $t-\tau$ and is given as,
%
\begin{eqnarray}
\nonumber
\lefteqn{\Phi (x, t,\tau) =
Cov_0 \left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D(x_t),
q(x_t), r(x_t) \epsilon(x_t,t))}{\partial \sigma} ; \right.} \\
\nonumber
%\label{eqno_S12a}
&& \left. \eta(\sigma(x_{t-\tau}, t-\tau), D(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
\epsilon(x_{t-\tau}, t-\tau)) \vphantom{\int_{0}^{t} d\tau} \right]
\end{eqnarray}
%\begin{eqnarray}
%\nonumber
%\lefteqn{\displaystyle \frac{\partial P(\sigma(x_t,t), t)}{\partial t}=}
%\\ \nonumber
%&-&
%\displaystyle \frac{\partial}{\partial \sigma}
%\left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D(x_t),
%q(x_t), r(x_t) \epsilon(x_t,t)) \right> \right. \right.
%\\ \nonumber
%&+&
%\left. \left. \int_{0}^{t} d\tau Cov_0
%\left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D(x_t), q(x_t), r(x_t)
%\epsilon(x_t,t))}{\partial \sigma}; \right. \right. \right.
%\\ \nonumber
%& &
%\left. \left. \left. \eta(\sigma(x_{t-\tau}, t-\tau),
%D(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
%\epsilon(x_{t-\tau}, t-\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \}
%P(\sigma(x_t,t),t) \right]
%\\ \nonumber
%&+&
%\displaystyle \frac{\partial^2}{\partial \sigma^2}
%\left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[ \vphantom{\int_{0}^{t}}
%\eta(\sigma(x_t,t), D(x_t,t), q(x_t,t), r (x_t,t),
%\epsilon(x_t,t)); \right. \right. \right.
%\\ \nonumber
%& &
%\left. \left. \left. \eta_1 (\sigma(x_{t-\tau}, t-\tau),
%D(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
%\epsilon(x_{t-\tau}, t-\tau)) \vphantom{\int_{0}^{t}} \right]
%\vphantom{\int_{0}^{t}} \right\} P(\sigma (x_t,t),t) \right]
%\\
%\label{eqno_S12}
%\end{eqnarray}
%
%to exact second order.
%
%Eq.~(\ref{eqno_S12}) is the most general relation for probabilistic
%behavior of inelastic (non--linear, elastic--plastic)
%1-D stochastic incremental constitutive equation.
%
%
%
%
%
%
%
%
% This ensemble average was recently derived as
% (for detailed derivation, refer to \citet{Kavvas:1996})
% %
% \begin{eqnarray}
% \nonumber
% \lefteqn{\frac{\partial \left<\rho (\sigma(x_t,t), t)\right>}{\partial t}=} \\
% \nonumber
% &-&\displaystyle \frac{\partial}{\partial \sigma}\left\{ \left[\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D(x_t),
% q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right. \right. \\
% \nonumber
% &-& \left. \left. \int_{0}^{t} d\tau Cov_0\left[\vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D(x_t), q(x_t), r(x_t),
% \epsilon(x_t,t)); \right. \right. \right. \\
% \nonumber
% & & \left. \left. \left. \displaystyle \frac{\partial \eta (\sigma(x_{t-\tau}, t-\tau), D(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}) \epsilon(x_{t-\tau}, t-\tau)}{\partial \sigma}\right]\right]\left<\rho (\sigma(x_t,t), t)\right>\right\} \\
% \nonumber
% &+& \displaystyle \frac{\partial}{\partial \sigma }\left\{ \left[\int_{0}^{t} d\tau Cov_0\left[\eta(\sigma (x_t,t), D(x_t), q(x_t),
% r (x_t), \epsilon(x_t,t)); \right. \right. \right. \\
% \nonumber
% & & \left. \left. \left. \eta (\sigma(x_{t-\tau}, t-\tau), D (x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}), \epsilon
% (x_{t-\tau},
% t-\tau)) \right] \vphantom{\int_{0}^{t}} \right] \displaystyle \frac{\partial \left<\rho (\sigma (x_t,t), t)\right>}{\partial \sigma}
% \right \} \\
% \label{eqno_S10}
% \end{eqnarray}
% %
% to exact second order (to the order of the covariance time of $ \eta $).
% %
% In Eq.~(\ref{eqno_S10}), $Cov_0[\cdot]$ is the time ordered covariance function
% defined by
% %
% \begin{equation}
% \label{eqno_S11}
% Cov_0\left[\eta(x,t_1), \eta(x,t_2)\right]
% =
% \left<\eta(x,t_1) \eta(x,t_2)\right>
% -
% \left<\eta(x,t_1)\right> \cdot \left<\eta(x,t_2)\right>
% \end{equation}
%
% By combining Eqs.~(\ref{eqno_S10}) and ~(\ref{eqno_S9}) and rearranging the terms
% yields the following Fokker-Planck equation (FPE, also known as
% Forward--Kolmogorov Equation or Fokker--Planck--Kolmogorov FPK Equation)
% (\citet{book:Risken}, \citet{book:Gardiner}, \citet{Schueller:1997report}):
% %
% \begin{eqnarray}
% \nonumber
% \lefteqn{\displaystyle \frac{\partial P(\sigma(x_t,t), t)}{\partial t}=}
% \\ \nonumber
% &-&
% \displaystyle \frac{\partial}{\partial \sigma}
% \left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D(x_t),
% q(x_t), r(x_t) \epsilon(x_t,t)) \right> \right. \right.
% \\ \nonumber
% &+&
% \left. \left. \int_{0}^{t} d\tau Cov_0
% \left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D(x_t), q(x_t), r(x_t)
% \epsilon(x_t,t))}{\partial \sigma}; \right. \right. \right.
% \\ \nonumber
% & &
% \left. \left. \left. \eta(\sigma(x_{t-\tau}, t-\tau),
% D(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
% \epsilon(x_{t-\tau}, t-\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \}
% P(\sigma(x_t,t),t) \right]
% \\ \nonumber
% &+&
% \displaystyle \frac{\partial^2}{\partial \sigma^2}
% \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[ \vphantom{\int_{0}^{t}}
% \eta(\sigma(x_t,t), D(x_t,t), q(x_t,t), r (x_t,t),
% \epsilon(x_t,t)); \right. \right. \right.
% \\ \nonumber
% & &
% \left. \left. \left. \eta_1 (\sigma(x_{t-\tau}, t-\tau),
% D(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
% \epsilon(x_{t-\tau}, t-\tau)) \vphantom{\int_{0}^{t}} \right]
% \vphantom{\int_{0}^{t}} \right\} P(\sigma (x_t,t),t) \right]
% \\
% \label{eqno_S12}
% \end{eqnarray}
% %
% to exact second order.
% %
% %
% %
% %
% %
% %
% %
% %
% This is the most general relation for probabilistic
% behavior of inelastic (non--linear, elastic--plastic)
% 1-D stochastic incremental constitutive equation.
% %
% %
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Eq.~(\ref{eqno_S12}) is the most general relation for probabilistic
behavior of inelastic (non--linear, elastic--plastic)
1-D stochastic incremental constitutive equation.
%
The solution
of this deterministic linear FPE (Eq.~(\ref{eqno_S12})), in terms of the
probability density $P(\sigma,t)$,
under appropriate initial and boundary conditions will yield the PDF of the
state variable $\sigma$ of the original 1-D non-linear stochastic constitutive
rate equation (Eq.~(\ref{eqno_S3})).
%
It is important to note that while the
original equation (Eq.~(\ref{eqno_S3})) is non-linear, the FPE
(Eq.~(\ref{eqno_S12})) is linear in terms of its unknown, the probability density
$P(\sigma,t)$ of the state variable $ \sigma $.
%
This linearity, in turn,
provides significant advantages in the solution of the probabilistic behavior of
the incremental constitutive equation (Eq.~(\ref{eqno_S3})).
One should also note that Eq.~(\ref{eqno_S12}) is a mixed Eulerian-Lagrangian
equation. This stems from the fact that while the real space location $x_t$ at
time $t$ is known, the
location $x_{t-\tau}$ is an unknown. {If one assumes small strain theory,
one can relate the unknown location $ x_{t-\tau} $ from the known location
$ x_t $ by using the strain rate, $\dot \epsilon $ (=$d \epsilon/dt$) as},
%
% \begin{equation}
% \label{eqno_S13}
% {d\epsilon=\dot \epsilon \tau = \frac{x_t-x_{t-\tau}}{x_t}}
% \end{equation}
% %
% {Or, rearranging}
%
\begin{equation}
\label{eqno_S14}
{x_{t-\tau}=(1-\dot \epsilon \tau)x_t}
\end{equation}
%
Once the probability density function $P(\sigma,t)$ is obtained it can be used
to obtain the mean of state variable ($\sigma$) by usual expectation operation
%
\begin{equation}
\label{eqno_S16}
<\sigma(t)>=\int \sigma(t) P(\sigma(t)) d\sigma(t)
\end{equation}
Another possible way to obtain the mean of state variable is to use the
equivalence between FPE and It{\^o}--operator stochastic differential equation
\citep{book:Gardiner}. In this case It{\^o}--operator stochastic differential equation
equivalent to Eq.~(\ref{eqno_S12}) is
%
\begin{eqnarray}
\nonumber
d\sigma (x,t)
&=&
\left\{ \left< \vphantom{\displaystyle
\frac{\partial \eta_1 (\sigma(x_t,t), A(x_t,t),
\epsilon(x_t,t))}{\partial \sigma}}
\eta (\sigma(x_t,t), D(x_t), q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right.
\\ \nonumber
&+&
\left. \int_0^t d \tau Cov_0
\left[\displaystyle \frac{\partial
\eta (\sigma(x_t,t), D(x_t), q(x_t), r(x_t), \epsilon(x_t,t))}
{\partial \sigma}; \right. \right.
\\ \nonumber
& &
\left. \left. \eta (\sigma (x_{t-\tau},t-\tau),
D(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
\epsilon (x_{t-\tau},
t-\tau)) \vphantom{\displaystyle
\frac{\partial \eta_1 (\sigma(x_t,t), A(x_t,t),
\epsilon(x_t,t))}{\partial \sigma}}\right] \right\} dt
\\
&+&
b(\sigma,t)dW(t)
\label{eqno_S17}
\end{eqnarray}
%
where,
%
\begin{eqnarray}
\nonumber
b^2(\sigma,t)
&=&
2 \int_0^t d \tau Cov_0
\left[\vphantom{\int_0^t d \tau}
\eta(\sigma(x_t,t), D(x_t), q(x_t), r(x_t),
\epsilon(x_t,t)); \right.
\\
& &
\left. \eta(\sigma (x_{t-\tau},t-\tau),
D (x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}), \epsilon
(x_{t-\tau},t-\tau))
\vphantom{\int_0^t d \tau} \right]
\nonumber \\
\label{eqno_S18}
\end{eqnarray}
%
and, $dW(t)$ is an increment of Wiener process W with $