Monday, January 15, 2018

quantum mechanics - Schrodinger equation in term of Fokker-Planck equation


From Wikipedia on the Fokker-Planck equation:



$$\tag{1}\frac{\partial }{\partial t}f\left( x^{\prime },t\right) ~=~\int_{-\infty}^\infty dx\left( \left[ D_{1}\left( x,t\right) \frac{\partial }{\partial x}+D_2 \left( x,t\right) \frac{\partial^2}{\partial x^2}\right] \delta\left( x^{\prime}-x\right) \right) f\left( x,t\right).\qquad $$


Integrate over a time interval $\varepsilon$,


$$f\left( x^\prime ,t+\varepsilon \right) $$ $$~=~\int_{-\infty }^\infty \, dx\left(\left( 1+\varepsilon \left[ D_{1}\left(x,t\right) \frac{\partial }{\partial x}+D_{2}\left( x,t\right) \frac{\partial^{2}}{\partial x^{2}}\right]\right) \delta \left( x^\prime - x\right) \right) f\left( x,t\right)$$ $$\tag{2}+O\left( \varepsilon ^{2}\right).\qquad $$



OK, but Fokker-Planck equation for one dimension is usually


$$\tag{0} \frac{\partial}{\partial t}f(x,t) = -\frac{\partial}{\partial x}\left[\mu(x,t)f(x,t)\right] + \frac{\partial^2}{\partial x^2}\left[ D(x,t)f(x,t)\right].$$



I was not able to understand how one gets from the original equation (0) to the above (1) and how does the first equation (1) lead to the second equation (2). Can anyone explain this?




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