Long-Time Convergence of an Adaptive Biasing Force Method: The Bi-Channel Case |
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Authors: | T Lelièvre K Minoukadeh |
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Institution: | 1.CERMICS, école des Ponts Paris Tech,Marne-la-Vallée cedex 2,France;2.MICMAC Project-Team, INRIA Rocquencourt,Le Chesnay,France |
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Abstract: | We present convergence results for an adaptive algorithm to compute free energies, namely the adaptive biasing force (ABF)
method (Darve and Pohorille in J Chem Phys 115(20):9169–9183, 2001; Hénin and Chipot in J Chem Phys 121:2904, 2004). The free energy is the effective potential associated to a so-called reaction coordinate ξ(q), where q = (q
1, … , q
3N
) is the position vector of an N-particle system. Computing free energy differences remains an important challenge in molecular dynamics due to the presence
of metastable regions in the potential energy surface. The ABF method uses an on-the-fly estimate of the free energy to bias
dynamics and overcome metastability. Using entropy arguments and logarithmic Sobolev inequalities, previous results have shown
that the rate of convergence of the ABF method is limited by the metastable features of the canonical measures conditioned
to being at fixed values of ξ (Lelièvre et al. in Nonlinearity 21(6):1155–1181, 2008). In this paper, we present an improvement on the existing results in the presence of such metastabilities, which is a generic
case encountered in practice. More precisely, we study the so-called bi-channel case, where two channels along the reaction
coordinate direction exist between an initial and final state, the channels being separated from each other by a region of
very low probability. With hypotheses made on ‘channel-dependent’ conditional measures, we show on a bi-channel model, which
we introduce, that the convergence of the ABF method is, in fact, not limited by metastabilities in directions orthogonal
to ξ under two crucial assumptions: (i) exchange between the two channels is possible for some values of ξ and (ii) the free energy is a good bias in each channel. This theoretical result supports recent numerical experiments (Minoukadeh et al. in J Chem Theory Comput 6:1008–1017, 2010), where the efficiency of the ABF approach is demonstrated for such a multiple-channel situation. |
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