Adaptive Biasing Force Method: Difference between revisions
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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 by free diffusions alongside these collective variables. 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: | 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 by free diffusions alongside these collective variables. 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: | ||
<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} | <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> | ||
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=== | ===Instanteous Collective Forces=== | ||
PMFLib implements two | PMFLib implements two algorithms to calculate Equation 3: simplified and original ABF algorithms. The original ABF algorithm is theoretically more accurate than the simplified one but it works only for MD without constraints. Therefore, the simplified algorithm is the default one as it is numerically more robust at all MD setups. | ||
====Simplified Algorithm==== | |||
====Simplified ABF Algorithm==== | |||
The simplified ABF algorithm (fmode=1) uses the result of the product rule for derivatives applied on Equation 3: | |||
<center><math>\frac {d} {dt} \left( \bold Z^{-1} \frac{d \boldsymbol \xi} {dt} \right) \big|_{(t)} = + \left( \bold Z^{-1}(t) \nabla \boldsymbol \xi (t) \right) \cdot \bold a(t) + \frac{d}{dt} \left( \bold Z^{-1}(t) \nabla \boldsymbol \xi (t)) \right) \cdot \bold v(t) = \bold F_{ICF,pot} + \bold F_{ICF,kin}</math> ... (11)</center> | |||
The accelerations <math>\bold a(t)</math> are calculated from the velocities to incorporate the effect of SHAKE constraints in [[the leap-frog integration algorithm]]: | |||
<center>\bold a(t) = \frac{\bold v(t+dt/2) - \bold v(t-dt/2)}{dt}<math> | |||
The accel | |||
====Original ABF Algorithm==== | ====Original ABF Algorithm==== | ||
The original ABF algorithm (fmode=2) implements the algorithm presented by Darve ''at al''.<ref name="darve2008"/><ref name="darve2007">Darve, E. Thermodynamic Integration Using Constrained and Unconstrained Dynamics. In Free Energy Calculations Theory and Applications in Chemistry and Biology; Springer Series in CHEMICAL PHYSICS; Springer: Berlin, 2007; Vol. 86, pp 119–170.</ref>. This algorithm should not be used when SHAKE is enabled because the accelerations entering into the calculations are not corrected for constraints. The algorithm uses a history of values collected in four consecutive time steps. Thus, the first result is available from the fourth time step. | |||
<center><math>\frac {d} {dt} \left( \bold Z^{-1} \frac{d \boldsymbol \xi} {dt} \right) \big|_{(t)} = \frac{1}{2} \left( \frac{\bold p^+_{\xi}(t+dt) - \bold p^+_{\xi}(t) } {dt} + \frac{\bold p^-_{\xi}(t+dt) - \bold p^-_{\xi}(t) } {dt} \right)</math> ... (11)</center> | |||
where | |||
<center><math>\bold p^+_{\xi}(t) = \bold Z^{-1}(t) \nabla \boldsymbol \xi(t) \cdot \left[ \bold v(t+dt/2) - \frac{dt}{6} \bold a(t+dt) \right]</math> ... (12)</center> | |||
<center><math>\ | <center><math>\bold p^-_{\xi}(t) = \bold Z^{-1}(t) \nabla \boldsymbol \xi(t) \cdot \left[ \bold v(t-dt/2) + \frac{dt}{6} \bold a(t-dt) \right]</math> ... (13)</center> | ||
==References== | ==References== | ||
<references /> | <references /> | ||