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| ===Adaptive Bias=== | | ===Adaptive Bias=== |
| ABF introduces an adaptive bias, which improves the sampling of rare events. The bias removes barriers or higher free energy regions in the space described by predefined collective variables <math>\boldsymbol \xi</math>. As a result, the system evolves alongside these collective variables by free diffusions. The bias is derived from the free energy, which is projected in the form of biasing force <math>\bold F_{bias}</math> to the Cartesian space and removed from force <math>\bold F_{pot}</math> originated from interatomic interaction potential (V(x)). The application of the bias thus leads to the modified equations of motions: | | ABF introduces an adaptive bias, which improves the sampling of rare events. The bias removes barriers or higher free energy regions in the space described by predefined collective variables <math>\boldsymbol \xi</math>. As a result, the system evolves alongside these collective variables by free diffusions. The bias is derived from the free energy, which is projected in the form of biasing force <math>\bold F_{bias}</math> to the Cartesian space and removed from force <math>\bold F_{pot}</math> originated from interatomic interaction potential <math>V(\bold R)</math>. The application of the bias thus leads to the modified equations of motions: |
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| <center><math>m_{i} \frac { d^2 \bold r_i }{dt^2} = \bold F_{pot,i}(\bold r_i) - \bold F_{bias,i}(\bold r_i) = - \frac{\partial V(\bold R)} {\partial \bold r_i} - \frac{\partial G(\boldsymbol \xi^{*})}{\partial \boldsymbol \xi} \frac{\partial \boldsymbol \xi}{\partial \bold r_i}</math> ... (8)</center> | | <center><math>m_{i} \frac { d^2 \bold r_i }{dt^2} = \bold F_{pot,i}(\bold r_i) - \bold F_{bias,i}(\bold r_i) = - \frac{\partial V(\bold R)} {\partial \bold r_i} - \frac{\partial G(\boldsymbol \xi)}{\partial \boldsymbol \xi} \frac{\partial \boldsymbol \xi}{\partial \bold r_i}</math> ... (8)</center> |
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| where <math>m_i</math> is mass of atom i, <math>\bold r_i</math> is atom position, and <math>t</math> is time. | | where <math>m_i</math> is mass of atom i, <math>\bold r_i</math> is atom position, and <math>t</math> is time. |
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| | | ===Instanteous Collective Forces=== |
| The free energy calculation is achieved by introducing a bias, which removes barriers or higher free energy regions alongside predefined collective variables . As a result, the system evolves alongside these collective variables by free diffusions. The bias is derived from the free energy (G(ξ)), which is projected in the form of biasing force (F_bias) to the Cartesian space and removed from force (F_pot) originated from interatomic interaction potential (V(x)). The application of the bias thus leads to the modified equations of motions:
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| m_i (d^2 x_i)/(dt^2 )=F_(pot,i) (x_i )-F_(bias,i) (x_i )=-(∂V(x)/(∂x_i )-∂G(ξ)/∂ξ ∂ξ/(∂x_i )) (1)
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| where m_i is mass of atom i, x_i is atom position, t is time.
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| The Adaptive Biasing Force (ABF) method calculates the derivative of the free energy (PMF) using the following formula:
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| <center><math>\frac{\partial A}{\partial \boldsymbol \xi} = - {\left \langle { \frac{d}{d t} \left( \bold {Z_{\xi}}^{-1} \frac{d \boldsymbol \xi}{d t} \right) } \right \rangle}_{\xi}</math> .... (1)</center>
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| where ξ is the set of collective variables, t is time, and Z<sub>ξ</sub> is the matrix defined as
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| <center><math>\left [ \bold Z_{\xi} \right ]_{i,j} = \sum_{k=1}^{N}{ \frac{1}{m_k}\frac{\partial \xi_i}{\partial \bold x_k}\frac{\partial \xi_j}{\partial \bold x_k}}</math> .... (2)</center>
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| where m<sub>k</sub> is the mass of atom with cartesian coordinate x<sub>k</sub>.
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| The averages over ξ are collected from unconstrained molecular dynamics. The free energy derivatives are practically calculated over discretized ranges of collective variables (CVs). Each CV<sub>i</sub> is discretized into M<sub>i</sub> bins leading into one-dimensional bins for one CV, two-dimensional bins for two CVs, etc. In each such bin, the vector of PMF is accumulated using the formula (1). For one CV this can be written as
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| <center><math>\frac{\partial A}{\partial \xi} \bigg|_{(\xi_{i}+\xi_{i+1})/2} = - {\left \langle { \frac{d}{d t} \left( {Z_{\xi}}^{-1} \frac{d \xi}{d t} \right) } \right \rangle}_{ ( \xi_{i},\xi_{i+1} \rangle }</math> .... (3)</center>
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| In the ABF simulations, the estimated PMF is used to bias molecular dynamics simulations to improve sampling in the regions exhibiting large free energy barriers.
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| ==References== | | ==References== |
| <references /> | | <references /> |