M.R. Pear and J.H. Weiner, Brownian dynamics study of a polymer chain of linked rigid bodies, J. Chem. Phys. 71(1979), 212--224.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for (t) will be derived in Sec. 5. We will see that (t) can be considered as constant over time intervals of order 1=fl. If this time interval is short compared to the dynamics due to the force term Gammar QV (Q) then the correcting potential V c can be approximated by the Fixman potential [6] [13] as used in statistical mechanics [1] i.e. V c (Q) kB T 2 ln[G(Q)M Gamma1 G(Q) T ] 14) These results seem particularly interesting in the context of molecular dynamics for the following reasons: ffl Often it is more realistic to solve the Langevin equations (8) 9) instead of the ....

....of time (Fig. 1) Note that the maximum energy in the Lennard Jones potential is less than kB T which is a realistic value for Lennard Jones interactions. Example 2. Let us now consider an artificial four bead three bond structure. To make the effect of the correcting potentials more pronounced [13], we set the ratio of the masses of the outer and inner beads equal to ten. For equal masses, like in butane, this effect would be about four times smaller. The bond stretching and bond angle bending motions are modeled by strong harmonic potentials with a force constant corresponding to 1=ffl ....

M.R. Pear and J.H. Weiner, Brownian dynamics study of a polymer chain of linked rigid bodies, J. Chem. Phys. 71(1979), 212--224.

....the fastest 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 ....

.... 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 U(q) U ....

Pear, M.R. and Weiner, J.H., Brownian dynamics study of a polymer chain of linked rigid bodies, J. Chem. Phys., 71, 212--224, 1979.

....bond vibrations appear on a timescale of about 1 femtosecond and are the fastest degrees of freedom of the molecule. Careful investigations have shown that the bond vibrations are an essential part of the nonlinear dynamics of the molecule, i.e. they cannot simply be eliminated or modelled [3][8]. Thus, if we are interested in the accuracy of the numerical solution of (1.2) we have to resolve this timescale, i.e. we have to choose stepsizes 1 fs in the time discretization. And even if accuracy is less important we have to use 1 fs in order to ensure numerical stability for the ....

M.R. Pear and J.H. Weiner. Brownian dynamics study of a polymer chain of linked rigid bodies. J. Chem. Phys., 71:212--224, 1979.

.... 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 satisfying results only for ....

....of (34) and the discretization of (32) 31) respectively, by less expensive methods can be found in [2] Note that one could also discretize (32) by a proper modification of the energy momentum methods proposed in [20] Example 2. In this example we consider a four bead three bond structure [13] where the structure is restricted to move in a finite volume by the potential V r (q) X i K r r i oe 6 Here r i denotes the distance of each of the four beads to the origin, oe = 2, and K r = 0:1. We set the mass of all four beads equal to m = 1 and choose r 0 = 1 as the equilibrium ....

Pear, M.R. and Weiner, J.H., Brownian dynamics study of a polymer chain of linked rigid bodies, J. Chem. Phys., 71, 212--224, 1979.

....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 obtain ....

.... up to terms of order (ffl 2 ) the slow solutions of (1) are given by the constrained equations (13) which differ from (34) by the Fixman potential (30) and thus by a term of order O(ffi) ii) A similar result to Corollary 1 has been published before, e.g. by van Kampen [27] and Pear Weiner [17] in the context of statistical mechanics. In [21] Rubin Unger considered in detail the case p 1 (0) 0 which leads to the formulation (13) and the case p 1 (0) 6= 0 for a single constraint; i.e. m = 1. An important aspect of Hamiltonian systems is the presence of symmetries which imply the ....

Pear, M.R. and Weiner, J.H., Brownian dynamics study of a polymer chain of linked rigid bodies, J. Chem. Phys., 71, 212--224, 1979.

....differential equation (16) As we will discuss in x2.3, the expectation value of the energy in the highly oscillatory degree of freedom satisfies hEi kB T provided fl is large enough. This suggests that the correcting force term F c can be approximated by a potential force term due to Fixman [7, 15, 1], i.e. F c (Q) Gammar QV c (Q) V c (Q) kBT ln q G(Q)M Gamma1 G(Q) T : 18) This result seem particularly interesting in the context of molecular dynamics and extends the analysis of Helfand [9] on the Fixman potential and its application to molecular dynamics. Note that the equations ....

M.R. Pear and J.H. Weiner, Brownian dynamics study of a polymer chain of linked rigid bodies, J. Chem. Phys. 71(1979), 212--224.

....can replace the constraint function g by g(Q) g(Q) g q (Q)M Gamma1 g q (Q) T ] Gamma1 g q (Q) M Gamma1 r q V hard (Q) where V hard would include perhaps only the Lennard Jones and the torsion potentials. Example In this example we consider a four bead three bond structure [28] where the structure is restricted to move in a finite volume by the potential V r (q) X i K r r i oe 6 : Here r i denotes the distance of each of the four beads to the origin, oe = 2, and K r = 0:1. We set the mass of all four beads equal to m = 1 and choose r 0 = 1 as the ....

Pear, M.R. and Weiner, J.H., Brownian dynamics study of a polymer chain of linked rigid bodies, J. Chem. Phys. 71, 212--224, 1979.

....can replace the constraint function g by g(Q) g(Q) g q (Q)M Gamma1 g q (Q) T ] Gamma1 g q (Q) M Gamma1 r q V hard (Q) where V hard would include perhaps only the Lennard Jones and the torsion potentials. Example In this example we consider a four bead three bond structure [28] where the structure is restricted to move in a finite volume by the potential V r (q) X i K r r i oe 6 : Here r i denotes the distance of each of the four beads to the origin, oe = 2, and K r = 0:1. We set the mass of all four beads equal to m = 1 and choose r 0 = 1 as the ....

Pear, M.R. and Weiner, J.H., Brownian dynamics study of a polymer chain of linked rigid bodies, J. Chem. Phys. 71, 212--224, 1979.

....in x5. In particular, we will see that J(t) can be considered as constant over time intervals of order 1=fl. If this time interval is short compared to the dynamics in the slow variable (strong thermal coupling) then the correcting potential V c can be approximated by a potential due to Fixman [7, 14, 1], i.e. V c (Q) kBT ln q G(Q)M Gamma1 G(Q) T : 17) These results seem particularly interesting in the context of molecular dynamics and extends the analysis of Helfand [9] on the Fixman potential and its application to molecular dynamics. Note that the equations (13) 15) can be solved ....

M.R. Pear and J.H. Weiner, Brownian dynamics study of a polymer chain of linked rigid bodies, J. Chem. Phys. 71(1979), 212--224.

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: Pear, M.R. and Weiner, J.H., Brownian dynamics study of a polymer chain of linked rigid bodies, J. Chem. Phys., 71, 212--224, (1979).

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