User:IantoC/sandbox

=HEOM=

Equation
=position space= The time dependent Hierarchy Equations of Motion in position space are


 * $$\frac{\partial \rho_{j_1,j_2,\cdots,j_K}^{(n)}(\mathbf{q},\mathbf{q}';t)}{\partial t}= -\left[ \frac{i}{\hbar}L(\mathbf{q},\mathbf{q}')

+ \Xi(\mathbf{q},\mathbf{q}') + n\gamma + \sum_{k=1}^Kj_k\nu_k \right] \rho_{j_1,j_2,\cdots,j_K}^{(n)}(\mathbf{q},\mathbf{q}';t) $$ $$- n\gamma \Theta_0(\mathbf{q},\mathbf{q}')\rho_{j_1,j_2,\cdots,j_K}^{(n-1)}(\mathbf{q},\mathbf{q}';t) - \sum_{k=1}^{K}j_k\nu_k\Theta_k(\mathbf{q},\mathbf{q}')\rho_{j_1,j_2,\cdots,j_k-1,\cdots,j_K}^{(n)}(\mathbf{q},\mathbf{q}';t)$$ $$- \Phi(\mathbf{q},\mathbf{q}') \Bigg[ \rho_{j_1,j_2,\cdots,j_K}^{(n+1)}(\mathbf{q},\mathbf{q}';t+{\delta t}) + \sum_{k=1}^{K}\rho_{j_1,j_2,\cdots,j_k+1,\cdots,j_K}^{(n)}(\mathbf{q},\mathbf{q}';t) \Bigg],$$

Where the operators $$L$$ and $$\Theta_0$$ are defined as follows,


 * $$ L(\mathbf{q},\mathbf{q}') = - \frac{\hbar^2}{2m}\left( \frac{\partial^2}{\partial \mathbf{q}^2} - \frac{\partial^2}{\partial \mathbf{q}'^2} \right) + U(\mathbf{q};t) - U(\mathbf{q}';t)$$

and


 * $$\Theta_0(\mathbf{q},\mathbf{q}') = \frac{i\hbar\zeta}{2} \Bigg[ \left( \frac{dV(\mathbf{q})}{d\mathbf{q}}\frac{\partial}{\partial\mathbf{q}} - \frac{dV(\mathbf{q}')}{d\mathbf{q}'}\frac{\partial}{\partial\mathbf{q}'} \right)

+ \frac{1}{2} \left( \frac{d^2V(\mathbf{q})}{d\mathbf{q}^2} - \frac{d^2V(\mathbf{q}')}{d\mathbf{q}'^2} \right) + \frac{m\zeta}{\hbar}\cot\left(\frac{\beta\hbar\gamma}{2}\right)V^{\times}(\mathbf{q},\mathbf{q}') \Bigg]$$

=energy space=
 * $$\frac{\partial }{\hat \rho _0} = - {\hat H_A^ \times}{\hat \rho _0} - i\hat \Phi {\hat \rho _1}$$


 * $$\frac{\partial }{\hat \rho _1} = - \left( {{\hat H_A^ \times } + \gamma } \right){\hat \rho _1} - i\hat \Phi \,{\hat \rho _2} - i\hat \Theta \,{\hat \rho _0}$$




 * $$\frac{\partial }{\hat\rho_n} = - \left(\hat H_A^ \times + n \gamma \right) \hat\rho_{n} - i \hat\Phi \hat\rho_{n + 1} - in \hat \Theta \hat\rho_{n - 1}$$




 * $$\frac{\partial }{\hat \rho _N} = - \left( {{{\hat L}_{QM}} + N\gamma } \right){\hat \rho _N} + \frac{1}{\gamma }\;\hat \Phi \,\hat \Theta \,{\hat \rho _N} + N\hat \Theta {\hat \rho _{N - 1}}$$


 * $$\hat \Phi = \frac{\eta }{\hat V^ \times }$$


 * $$\hat \Theta = \left( {i{{\dot \hat V}^\bigcirc } - \frac{1}{{\hat V}^ \times }} \right)$$