van Kampen, N.G. and Lodder, J.J., Constraints, Am. J. Phys., 52, 419--424, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....degrees of motion and then to solve the reduced equations numerically. This allows one to use larger timesteps and the computation of the long term dynamics of macromolecules could become feasible. Several methods for the removal of the bonded interactions have been suggested [15] 13] 5] [18], 14] In this paper we derive the reduced equations of motion by calculating the free energy in terms of appropriately chosen reaction coordinates. We also give a stochastic embedding of the reduced dynamics by using a generalized Langevin approach [11] 1] 2 The Equations of Motion Let us ....

.... constrained formulation, defined in local coordinates by q 1 = p 1 = 0, to make sure that, in the limit jjK Gamma1 jj 0, the unconstrained system (4) and the corresponding constrained system possess the same reduced density function ae (2) ens (q 2 ; p 2 ) Similar results can be found in [18] and [13] The free energy in the variable (q 2 ; p 2 ) is thus (approximately) given by H(q 2 ; p 2 ) p T 2 B(q 2 )M Gamma1 B(q 2 ) T p 2 2 U(q 2 ) U F (q 2 ) 16) or, in terms of the cartesian coordinates (q; p) 2 IR 2n , by the Hamiltonian H(q; p) p T M Gamma1 p 2 ....

van Kampen, N.G. and Lodder, J.J., Constraints, Am. J. Phys., 52, 419--424, 1984.

....with the minima condition rU ffl (q=ffl) 0 defining a m dimensional smooth sub manifold M of the coordinate space q 2 IR n . Then, under some additional technical assumptions, the dynamics can be reduced to a constrained Hamiltonian systems on the tangent space TM of M in the limit ffl 0 [16], 6] 8] This eliminates the highly oscillatory components in the solutions and the constrained equations of motion can be discretized by the SHAKE modification [10] of the Verlet scheme using a step size Deltat AE ffl. In this paper we are concerned with the situation that U ffl possesses ....

....paper we are concerned with the situation that U ffl possesses many local minima that are separated by an average barrier height E b . In this case 2 we will again observe rapid motions on a time scale s ffl and, on top of this, transitions between nearby local minima over time scales of length [16] e exp( E b kB T ) s : Here we are interested in the case ffi : exp( E b kB T ) AE 1: 3) Thus a diffusion process over length scales of order O(1) will require a timescale of order O(ffi) which again, simulated as such, would lead to extremely long simulations. For example, in the ....

van Kampen, N.G. and Lodder, J.J., Constraints, Am. J. Phys. 159, 98--103, 1984.

.... show how to reformulate (1) as a singularly perturbed problem (5) In Section 3, we will then derive a constrained Hamiltonian system that approximates the smoothed dynamics of (1) The approximation of (1) by a constrained Hamiltonian systems has been considered before (see, for example, 16] [22]) In a naive approach, one would introduce the new variable : 1 ffl 2 Kg(q) and rewrite (1) as d dt q = M Gamma1 p d dt p = GammarV (q) Gamma G(q) T ffl 2 K Gamma1 = g(q) 9) In the limit ffl 0, we obtain the constrained system d dt q = M Gamma1 p d dt p = GammarV ....

.... [7] One can show that, up to terms of order (ffl 2 ) the slow solutions of (1) are given by the constrained equations (27) which differ from (29) by the Fixman potential (28) and thus by a term of order O(ffi) ii) A similar result to Theorem 1 has been published before, e.g. by van Kampen [22] and Pear Weiner [13] in the context of statistical mechanics. In [16] Rubin Unger considered in detail the case p 1 (0) 0 which leads to the formulation (27) and the case p 1 (0) 6= 0 for a single constraint; i.e. m = 1. Smoothed Dynamics 14 The constrained equations (29) yield ....

van Kampen, N.G. and Lodder, J.J., Constraints, Am. J. Phys., 52, 419--424, 1984.

....corresponding (smooth) solutions (Q(t) P (t) satisfy hqi p ffl (t) Gamma Q(t) O(ffi ffl 2 ) and hpi p ffl (t) Gamma P (t) O(ffi ffl 2 ) over bounded intervals of time. The approximation of (1) by a constrained Hamiltonian systems has been considered before (see, for example, 21] [27]) In a naive approach, one would introduce the new variable : 1 ffl 2 Kg(q) and rewrite (1) as d dt q = M Gamma1 p d dt p = GammarV (q) Gamma G(q) T ffl 2 K Gamma1 = g(q) 12) In the limit ffl 0, we obtain the constrained system d dt Q = M Gamma1 P d dt P = GammarV ....

....statistical mechanics. He showed that (30) has to be included into the constrained formulation (13) to make sure that, in the limit ffl 0, the unconstrained system (1) and the constrained system (13) possess the same reduced density function ae ens (q 2 ; p 2 ) Similar results can be found in [27] and [17] Smoothed Dynamics 17 6 Constraint Formulations The coordinates (q 1 ; p 1 ; q 2 ; p 2 ) were only introduced for theoretical purposes. This leaves us with the task of reformulating (31) in terms of the Cartesian coordinates (q; p) In fact, this turns out to be straightforward and we ....

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van Kampen, N.G. and Lodder, J.J., Constraints, Am. J. Phys., 52, 419--424, 1984.

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