Verlet, L. (1967). Computer "experiments" on classical fluids. I. Thermodynamic properties of Lennard-Jones molecules. Phys. Rev. 159, 98--105.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....following equation: ma (160) that is it consists of integrating Newton s equations of motion for all particles in lock step over a series of time steps, the size of the step being chosen small enough to give converged dynamics. Time step integration is often done using the Verlet al..gorithm [68 70]. Lattice theory makes predictions that are hard to observe in experiment, and one of the reasons for doing computer simulations is to relate the two. The simulations I will describe here in detail were carried out in silicon. Silicon is extremely brittle, and highquality macroscopic single ....

L. Verlet, "Computer "experiments" on classical fluids. i. thermodynamical properties of lennard-jones molecules," Physical Review, vol. 159, p. 98, 1967.

....such as temperature does not actually mean that the instantaneous temperature defined T = 1 3kB N m i kv i k 2 (2. 13) is held constant along trajectories (although in fact such a constraint can be imposed and certain quantities can be accurately computed from such an isokinetic model [46]) If molecular dynamics is to be used to sample from a given ensemble, it is essential that the averages taken along trajectories reproduce the appropriate ensemble average. This requires ergodicity: that a given trajectory will eventually visit all states with ae 0. We can only reasonably ....

D.J. Evans, Computer "experiment" for nonlinear thermodynamics of Couette flow, J. Chem. Phys., 78 (1983), pp. 3297--3302

.... in this area, for example extending to hard ellipsoids and to hard sphere systems in soft potentials [16 19] In this preliminary version of the test set, we concern ourselves only with smooth potential models such as the monatomic fluids which were first discussed by Rahman [20] and Verlet [21]. These early computer experiments demonstrated that the equilibrium state of argon could be well described by the two body Lennard Jones potential V (r) 4ffl (1.1) with r being the distance between a pair of particles, ffl giving well depth, and oe being proportional to the ....

L. Verlet, Computer "experiments" on classical fluids. I. Thermodynamical properties of Lennard-Jones Molecules, Phys. Rev., 159 (1967), pp. 98--103

....that the velocities are given only by the particle positions and therefore changes in position directly affect the velocity of the next time step. This increases the stability of the numerical integration. This is not a surprise, because this integration scheme is equal to the Verlet integration [Ver97] which updates the position without computing any velocities by = 2x 2 f . 15) This can be seen by substituting Eq. 12 into Eq. 13. This yields i dt(v ) 16) By further substituting Eq. 14 into Eq. 16 and rewriting we get i x (17) which is equal to Eq. 15. As ....

Loup Verlet. Computer "experiments" on classical fluids. i. thermodynamical properties of lennard-jones molecules. Physical Review, 159(1):98--103, 1997.

....a list for each atom of all other atoms from which it is excluded, and additional improvements are described below. In most molecular dynamics programs, not every pair of atoms is tested against the cuto# distance at every timestep, since this operation scales as N 2 . Often, a pair list [48] a list of atom pairs within or near the cuto#, is generated periodically. Alternately, link cells [17] may be used, grouping atoms into cells by location and calculating interactions between atoms in pairs of nearby cells. Link cells may also be used to e#ciently generate a pair list. The ....

L. Verlet. Computer `experiments' on classical fluids: I. Thermodynamical properties of Lennard-Jones molecules. Physical Review, 159:98--103, 1967.

.... p; 1) d dt p = GammarV (q) Gamma 1 ffl rU ffl (q=ffl) 2) and the solutions exhibit rapid motion on time scales of order ffl due to the force term F strong (q) Gamma 1 ffl rU ffl (q=ffl) If the equations of motions are directly discretized by an explicit method such as Verlet [17], then the step size Deltat has to satisfy Deltat ffl which leads to extremely long simulation times. In various previous publications [8] 10] 15] it has been assumed that U ffl is a convex function with the minima condition rU ffl (q=ffl) 0 defining a m dimensional smooth sub manifold M ....

Verlet, L., Computer `experiments' on classical fluids. I. thermodynamical properties of Lennard-Jones molecules, Phys. Rev. 159, 98--103, 1967.

....that are intrinsic to Hamiltonian systems, the loss of which is extremely detrimental to long time integrations. We are most interested in efficient symplectic integration methods for solving the N body problem in molecular dynamics. The preferred method has been the leapfrog Stormer Verlet method [17] which is known to be symplectic, time reversible, and second order accurate. The leapfrog method has the advantages of being explicit and requiring only a single force evaluation per time step. The computation of the force is the most expensive part of the integration. Considering the ....

....kinetic energy. The system of first order ODEs associated with (1) is q = M Gamma1 p; p = f(q) 2) where the force is f(q) GammarV (q) For the usual existence and uniqueness of solution, H is assumed to be sufficiently smooth with respect to q and p. The leapfrog Stormer Verlet method [17] integrates the system from time nh to (n 1)h. Starting the step with values q n , p n , and f n = f(q n ) obtained from the previous step, we compute p n 1 2 = p n h 2 f n ; 3) q n 1 = q n hM Gamma1 p n 1 2 ; 4) f n 1 = f(q n 1 ) 5) p n 1 = p n ....

L. Verlet, Computer `experiments' on classical fluids I. Thermodynamical properties of Lennard--Jones molecules, Phys. Rev. 159, 98--103, 1967. 12

....to the number of nodes. The program is input output compatible with CHARMm(Brooks et al. 1983) and X PLOR(Br unger 1988; Br unger 1990) EGO employs the CHARMm force field and uses the same parameter files and protein structure files as X PLOR. The program uses a modified Verlet al..gorithm (Verlet 1967) with a distance class algorithm (Grubm uller et al. 1991) for the non bonded interactions, i.e. it does not truncate the Coulomb and van der Waals forces. A parallel version of the Shake algorithm (Ryckaert et al. 1977; Raine 1990) constrains the bond length for hydrogen atoms and allows an ....

Verlet, L. (1967) Computer "experiments" on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules Phys. Rev. 159:98--103.

....MC step. If we draw the momenta from a Gaussian distribution exp[ Gammafi T (p) according to (1) and propose new coordinates x 0 and new momenta p 0 by integrating the system through phase space with a reversible and volume preserving discretized flow Psi (e.g. the Verlet discretization [20]) the new coordinates x 0 are accepted with a probability of P acc = min Gamma 1; exp[ Gammafi (H(x 0 ; p 0 ) Gamma H(x; p) Delta = min 1; x 0 ) exp[ Gammafi T (p 0 ) x) exp[ Gammafi T (p) If we want to use the HMC scheme to sample a generalized ensemble ....

....angles, which is necessary for a rough reconstruction of the molecule s configuration. The torsion angles of the ribose can be approximated by the pseudorotation angle P and the phase [1] 7 Results and Discussion The n butane molecule For simulations of n butane, we used the Verlet scheme [20] with n = 40 iterations and a time step of = 15 fs, which results in a trajectory length of 600 fs for each update step. Performing a simulation over 10 5 steps at a temperature of T = 100 K with HMC (not shown here) we observed a trapping of the Markov chain in the gauche conformation, where ....

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L. Verlet. Computer "experiments" on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Phys. Rev., 159:98--103, 1967. 15

....structures and the even more basic and accurate modeling of chemical reactions by multigrid Monte Carlo methods for real time Feynman path integrals. This first report summarizes our long term objectives (Sec. 2) and our initial work on multiscale energy minimization methods (Sec. 3) References [1] [13] were used by us in getting acquainted with existing approaches in molecular simulations. We have also been delighted to discover several articles, such as [11] 13] where some rudimentary forms of multiscaling have already been attempted, motivated by physical insights, with considerable ....

L. Verlet, Computer `experiments' on classical fluids, I. Thermodynamical properties of Lennard-Jones molecules, Phys. Rev. 98 (1967). -- 12 --

.... for the parallel solution of long range interaction problems were given in [1, 2] and applied for the parallel calculation of the potential energy of a protein [7] The idea was further devised and implemented in the parallel molecular dynamics program, based on fundamental sequential technique [3], described in details in the next sections. Our target system was a mesh of 20MHz INMOS transputers T800 connected with a PC host. Instead of the usual generic programming language Occam, the parallel computational library PCL 2 in connection with 3L s Parallel C was used [4] PCL gives to the ....

....integration is explained and the principle of mirror forces is introduced. In Section 4 the estimation of the time complexity for the proposed parallel algorithm is devised. In Conclusions we evaluate the results and mention some further research directions. 2 Physical model The implemented model [3, 2] is based on a cube of side L and density ae = N=L 3 . The lengths are expressed in units of oe (oe = 3:405A ffi for argon) and the energies in units of ffl (ffl = 119:8 ffi K for argon) 2 The source code of PCL is available from authors. The equation of motion m d 2 r i dt 2 = ....

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L. Verlet, Computer "Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules, Physical Review Vol. 159 1 (1967) 98-103.

....due to a result by Ge and Marsden [4] those integrators cannot, in general, also preserve energy exactly. Thus, a second option is to consider energy conserving schemes [11] 6] 15] If one wants to resolve all the details in the motion of (1) explicit symplectic integrators, such as Verlet [16] q n 1 = q n DeltatM Gamma1 p n 1=2 p n 1=2 = pn Gamma Deltat 2 rV (q n ) pn 1 = p n 1=2 Gamma Deltat 2 rV (q n 1 ) 3) seem preferable. They show excellent long time behavior, approximately preserve energy over long periods of time, and are computational inexpensive. In contrast to ....

....the weak contributions by an explicit symplectic method like (3) In Section 4, we show how this can be achieved in practice. Finally, in Section 5, we demonstrate the good stability behavior of our method by means of a simple simulation of particles that interact through Lennard Jones potentials [16]. A completely different approach to the integration of a system like (5) is to average over the fast degrees of motion and to derive an approximation to the slowly varying components of (5) in terms of a constrained Hamiltonian system [13] 2 An energy conserving method The modified implicit ....

L. Verlet, Computer `experiments' on classical fluids. I. thermodynamical properties of Lennard-Jones molecules, Phys. Rev. 159, 98--103, 1967.

....which is convenient when dealing with heterogeneous mixtures. 2.2 Integration Algorithms The dynamical equations 2.3 and 2. 4 are integrated using this author s modi cation[43] of the Beeman algorithm[3] For atomic systems the accuracy is of the same order as the commonly used Verlet al..gorithm[51]. The accuracy of common integration algorithms was discussed by Rodger[45] who showed that the Beeman algorithm is the most accurate of the supposedly Verlet equivalent algorithms. The Beeman algorithm is the only one accurate to O( t 4 ) in the co ordinates and O( t 3 ) in the velocities ....

L. Verlet, Computer `experiments' on classical uids. I. thermodynamical properties of lennard-jones molecules, Physical Review 165 (1967), 201-214.

.... vector field x 0 = fx; Hg If Psi h is a scheme of order , then H = H O(h ) Yoshida [31] first noticed that for unconstrained systems (1) higher order methods can be constructed by a proper composition of second order symmetric methods such as the implicit midpoint [14] or Verlet [30] method. The idea is the following one: Let Psi h denote the time h flow defined by a symmetric second order method. Then the composed mapping Psi c 1 h Delta Psi c 2 h Delta Psi c 1 h (13) with 2c 1 c 2 = 1 and 2c 3 1 c 3 2 = 0 is of fourth order. More generally, if Psi h is ....

.... k subject to 0 = g(q k 1 ) The p variable is updated by p k 1 = p k Gamma hrV (q k ) Gamma h 2 (G(q k ) t k G(q k 1 ) t k 1 ) where k 1 is now determined by the constraint 0 = G(q k 1 ) M Gamma1 p k 1 A second order algorithm can be obtained by applying the popular Verlet scheme [30] to (24) We obtain q k 1 = q k hM Gamma1 p k 1=2 p k 1=2 = p k Gamma h 2 (rV (q k ) G(q k ) t k ) 0 = g(q k 1 ) and p k 1 = p k 1=2 Gamma h 2 (rV (q k 1 ) G(q k 1 ) t k 1 ) 0 = G(q k 1 ) M Gamma1 p k 1 This scheme is identical to the one derived by Anderson in [2] It was ....

Verlet, L., Computer `experiments' on classical fluids, Phys. Rev., 159(1967), 98--103.

....sophisticated algorithms, software and hardware for parallel computing, because simulations of realistic molecular models typically require very substantial amounts of computer time and memory. Perhaps the best known computational method for studying atomic motion is molecular dynamics simulation [1, 3, 13, 15, 16, 20, 33]. In this method, the forces acting on the atoms are calculated from a model potential energy function, usually a function of simple analytic form, and the atoms are moved iteratively in accordance with Newton s differential equation of motion. Many factors conspire to make such calculations ....

....for angular bonds range from 3 Theta 10 13 per second downward. The above equation (1) and its counterpart for the angle in a water molecule take the form d 2 u dt 2 = f(u) 2) A popular technique for approximately solving such equations (or systems of equations) is the Verlet al..gorithm [33] U n 1 Gamma 2U n U n Gamma1 = Deltat) 2 f(U n ) 3) In the case that f(u) 2 (u Gamma u 0 ) a standard stability analysis indicates that the stability region of the Verlet al..gorithm is the set of Deltat in the interval [ Gamma2i; 2i] on the imaginary axis. Thus the Verlet ....

L. Verlet, Computer `experiments' on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules, Phys. Rev. 159 (1967), 98-103.

....be in O(N 2 ) for N monomers. Instead, a list is built containing all pairs of monomers within a certain distance of each other. Because monomers are moving in the system, this list needs to be updated. To avoid updating it every time step, we employ a mixed method of the Verlet neighbor list [24] and the cell index method [20] which are discussed in detail in the following subsections. As a result, the time required to compute the non bonding forces fF LJ i g is reduced to O(N ) We also discuss this in detail in the following subsections. The neighbor list is used for computing ....

L. Verlet. Computer `experiments' on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules, pages 98--103. ?, 1968.

....a random walk, determined by Delta MAX . 3 Implementation Strategies The force computation is organized around the procedure for enumerating interactions. Fundamentally, two approaches have been taken: the link cell method (also called chaining mesh [6] and the Verlet Neighbor List method [12]. The link cell method uses a mesh to organize the computation. Each box of the mesh contains a list with the particles found in its corresponding subregion of the domain, and the interacting neighbors lie in neighbor boxes, avoiding a costly O(N 2 ) search for neighbors. After the new positions ....

L. Verlet, Computer "Experiments" on Classical Fluids, Physical Review, 159 (1967), pp. 98-- 103.

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Verlet, L. Computer "Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard--Jones Molecules. Physical Review. 1967, 159, 98--103.

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Verlet, L. (1967). Computer "experiments" on classical fluids. I. Thermodynamic properties of Lennard-Jones molecules. Phys. Rev. 159, 98--105.

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L. Verlet. Computer `experiments' on classical fluids. I. thermodynamical properties of Lennard-Jones molecules. Physical Review, 165:201--214, 1967.

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Verlet, L, "Computer "Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules." Physical Review, Vol. 159, 1967.

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L Verlet. Computer "experiments" on classical fluids. Physics Review, 159(1):98--100, July 1967. A Additional theory This appendix contains various pieces of theory not given in the main text. 15 A.1 Verlet algorithm The Verlet algorithm can be derived simply from the Taylor series for the present coordinate, u:

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L. Verlet. Computer "experiments" on classical fluids. I. Thermodynamic properties of Lennard-Jones molecules. Phys. Rev. 159, 98--103 (1967).

No context found.

Verlet, L. Computer "Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules. Physical Review. 1967, 159, 98--103.

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