Nos# ee S. Unified formulation of the constant temperature molecular-dynamics methods. J Chem Phys 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... in the design of efficient algorithms for problems such as fast summation methods for computing non bonded atomic interactions [1,2] long time numerical integration of equations of motion[3 5] non Newtonian dynamical formulations for simulation in a variety of statistical mechanical ensembles [6 8], and optimization. These projects have often involved mathematicians and or computer scientists who, though adept at algorithm and software development, may possess limited or no physical and chemical knowledge. An important step in the development of a computational method is testing on model ....

....discussion of the choice of mass. The artificial scaling of the kinetic term impedes sampling, particularly the recovery of time correlated averages; various reformulations of the Nos e dynamical formulation have been proposed which correct the timescale and facilitate the use of time reversible [6] or lately symplectic [8] discretization. One of the problems with using low dimensional model problems is the lack of ergodicity that such systems tend to exhibit. Any of the model problems given here could be treated using Langevin dynamics or a Nos e thermostat. Moreover, it is possible to ....

S. Nos' e, A unified formulation of the constant temperature molecular dynamics methods, J. Chem. Phys., 81 (1984), pp. 511--519

.... In this case we would fit the Rubin Ungar potential to the Fixman potential at the gauche conformations ( Sigma2=3) 7 Application to constant temperature molecular dynamics An interesting application of the ideas developed in this paper is related to constant temperature molecular dynamics [12], 10] In the Hoover formulation, the equations of motion are d dt q = M Gamma1 p ; 30) d dt p = Gammar q U(q) Gamma flp ; 31) d dt fl = h p T M Gamma1 p Gamma XkBT i =D ; 32) where fl is now a time dependent friction constant, D a parameter, and X the number of degrees of ....

.... the kinetic energy, then one can use the constraint formulation d dt Q = M Gamma1 P ; d dt P = Gammar QV (Q) Gamma rQ q G(Q)M Gamma1 G(Q) T Gamma flP Gamma G(Q) T ; 0 = g(Q) d dt = Gammafl ; 0 = EN P T M Gamma1 P Gamma XkBT which is obtained by formally setting D 0 [12]. Furthermore, taking into account that, for the macrocanonical distribution function, we have hEN i = kB T , this simplifies to d dt Q = M Gamma1 P ; d dt P = Gammar QV (Q) Gamma kBT 2 rQ ln[G(Q)M Gamma1 G(Q) T ] Gamma flP Gamma G(Q) T ; 0 = g(Q) 0 = P T M Gamma1 P ....

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S. Nos'e, A unified formulation of the constant temperature molecular dynamics methods, J. Chem Phys. 81(1994), 511--517.

....was the same as in the pure water case. The water phase extended over about 40 A and the metal phase over about 25 A. The forces were integrated by the fifth order predictor corrector scheme of Gear. The time step was 0.2 fs. The temperature was held constant at 298 K by the thermostat of Nos e [19]. Image charge interactions [20] were calculated between the ion image and the water molecules. The other possible image charge interactions were neglected. All interaction potentials were modified using the shifted force method with a cutoff distance of half of the shortest sidelength of the ....

Nos'e S. A unified formulation of the constant temperature molecular dynamics methods. // J. Chem. Phys., 1984, vol. 81, p. 511-519.

....(now) matrix valued function G(q)M Gamma1 G(q) T . See [20, 5, 18] for first results 6 on the special case fl = 0. 4 Application to constant temperature molecular dynamics An interesting application of the ideas developed in this paper is related to constant temperature molecular dynamics [14, 10]. In the Hoover formulation, the equations of motion are d dt q = M Gamma1 p ; 41) d dt p = Gammar q U(q) Gamma flp ; 42) d dt fl = Theta p T M Gamma1 p Gamma XkBT =D ; 43) where fl is now a time dependent friction constant, D a parameter, and X the total number of degrees ....

.... then one can use the formulation d dt Q = M Gamma1 P ; 49) d dt P = Gammar QV (Q) Gamma J rQ q G(Q)M Gamma1 G(Q) T Gamma flP Gamma G(Q) T ; 50) 0 = g(Q) 51) d dt J = Gammafl J ; 52) 0 = EN P T M Gamma1 P Gamma XkBT (53) which is obtained by formally setting D 0 [14]. Thus the parameter fl takes now the role of a Langrange multiplier that forces the time evolution of (Q; P ) to satisfy the constraint (53) Furthermore, taking into account that, for the macro canonical distribution function, we have hEN i = kBT , this simplifies to d dt Q = M Gamma1 P ; d ....

S. Nos'e, A unified formulation of the constant temperature molecular dynamics methods, J. Chem. Phys. 81(1994), 511--517.

....similar, to some extent, to that of molecular dynamics, in nonequilibrium statistical mechanics. There, some progress took place when infinite reservoirs, or driving boundary conditions were replaced by artificial constraints imposed on the bulk dynamics of N particle systems, with N 1 (see e.g. [3, 4, 5, 6]) This way direct numerical simulations of the particle models become feasible, and dynamical systems theory leads to theoretical predictions [6] which can be tested in numerical simulations or real experiments. The trade off of this approach is that the relevant dynamical equations do not seem ....

....is derived from the observation that in stationary isotropic turbulence, the mean energy in a narrow wave number shell is nearly constant in time. These equations are similar to those used in nonequilibrium molecular dynamics, for driven particle systems subjected to a gaussian thermostat [3, 4, 5]. Therefore, the properties of such particle systems should be observed to some degree in the constrained Euler system of [2] and, perhaps, in still more general settings. This led Gallavotti [8] to revive and put under a new light Ruelle s principle for hydrodynamics [10] This principle had been ....

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S. Nos`e. A unified formulation of the constant temperature molecular dynamics methods. J. Chem. Phys., 81(1):511--519, 1984.

..... This will be discussed in a forthcoming publication. See [19, 5, 17] for first results 4 on the special case fl = 0. 6 Application to constant temperature molecular dynamics An interesting application of the ideas developed in this paper is related to constant temperature molecular dynamics [13, 10]. In the Hoover formulation, the equations of motion are d dt q = M Gamma1 p ; 34) d dt p = Gammar q U(q) Gamma flp ; 35) d dt fl = Theta p T M Gamma1 p Gamma XkBT =D ; 36) 2 The fact that there is only a single fast degree of motion is, however, crucial. 3 Any term of ....

.... then one can use the formulation d dt Q = M Gamma1 P ; 42) d dt P = Gammar QV (Q) Gamma J rQ q G(Q)M Gamma1 G(Q) T Gamma flP Gamma G(Q) T ; 43) 0 = g(Q) 44) d dt J = Gammafl J ; 45) 0 = EN P T M Gamma1 P Gamma XkBT (46) which is obtained by formally setting D 0 [13]. Thus the parameter fl takes now the role of a Langrange multiplier that forces the time evolution of (Q; P ) to satisfy the constraint (46) Furthermore, taking into account that, for the macrocanonical distribution function, we have hEN i = kBT , this simplifies to d dt Q = M Gamma1 P ; d ....

S. Nos'e, A unified formulation of the constant temperature molecular dynamics methods, J. Chem. Phys. 81(1994), 511--517.

....system, one can double the dimension of the space and extend the ODE by cotangent lift to obtain a Hamiltonian system. This does no good numerically as invariant tori are typically half the dimension of the symplectic manifold, i.e. equal to the dimension of the original system. However, Nos e [100] shows that molecular dynamics formulations with constant temperature may be presented in Hamiltonian form by the addition of two more variables. ffl What is the structure of a dynamical system which allows a Hamiltonian formulation by the addition of a small number of variables 14 Assorted ....

Nos'e, S., A unified formulation of the constant temperature molecular dynamics methods, J. Chem. Phys. 81 (1984), 511--519.

....molecular dynamics is treated in the so called constant temperature ensemble. The equations of motion are given in a time reversible formulation due to Nos e and Hoover. The derivation begins with the canonical Nos e extended Hamiltonian which, for a system of particles, takes the form 3 [14]: H Nos e = 1 2z 2 p T M Gamma1 p V (q) 1 2Q 2 z g fi ln z: The equations of motion are thus d dt q = 1 z 2 M Gamma1 p; d dt p = GammarV (q) dz dt = z =Q; d z dt = 1 z 1 z 2 p T M Gamma1 p Gamma g fi : The Hoover formulation is typically ....

.... A symmetric second order discretization has been suggested in [6] The system (36) 39) is time reversible under the involution S 0 B B 2 6 6 4 q p z 3 7 7 5 1 C C A = 0 B B 2 6 6 4 q Gamma p z Gamma 3 7 7 5 1 C C A : 3 The variable z is equivalent to the variable s in [14, 7]. and the time transformation dt=d = z obviously preserves the symmetry. In analogy to rigid body motion, a fully explicit, symmetric, and second order discretization of the equations (36) 39) can be obtained by an appropriate splitting of the differential equation into integrable subproblems. ....

S. Nos' e, A unified formulation of the constant temperature molecular dynamics methods, J. Chem Phys., 81 (1994), pp. 511--517.

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Nos# ee S. Unified formulation of the constant temperature molecular-dynamics methods. J Chem Phys 1984.

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