Integrators time reversibility

This latter modified midpoint method does work well, however, for the long time integration of Hamiltonian systems which are not highly oscillatory. Note that conservation of any other first integral can be enforced in a similar manner. To our knowledge, this method has not been considered in the literature before in the context of Hamiltonian systems, although it is standard among methods for incompressible Navier-Stokes (where its time-reversibility is not an issue, however). 

One of the advantages of the Verlet integrator is that it is time reversible and symplectic[30, 31, 32]. Reversibility means that in the absence of numerical round off error, if the trajectory is run for many time steps, say nAt, and the velocities are then reversed, the trajectory will retrace its path and after nAt more time steps it will land back where it started. An integrator can be viewed as a mapping from one point in phase apace to another. If this mapping is applied to a measurable point set of states at on(> time, it will... 

Since many systems of interest in chemistry have intrinsic multiple time scales it is important to use integrators that deal efficiently with the multiple time scale problem. Since our multiple time step algorithm, the so-called reversible Reference System Propagator Algorithm (r-RESPA) [17, 24, 18, 26] is time reversible and symplectic, they are very useful in combination with HMC for constant temperature simulations of large protein systems. 

Beneath the conservation properties of QCMD its equations of motion possess another important geometric structure by being time reversible. As shown in [10], the application of symmotric integrators to reversible problems yields... 

Note that the same results have not been shown for symmetric (time-reversible) integration methods, although symmetric methods seem to perform quite well in practice. For a discussion of symmetric methods in the context of the QCMD model see [16, 17, 13]. 

In this paper we present a number of time integrators for various problems ranging from classical to quantum molecular dynamics. These integrators share some common features they are new, they are second-order accurate and time-reversible, they improve substantially over standard schemes in well-defined model situations — and none of them has been tested on real applications at the time of this writing. This last feature will hopefully change in the near future [20]. 

The algorithm requires only a single force evaluation per integration cycle (computationally, force evaluations are the most expensive part of the simulation). This formulation, which is based on forward and backward expansions, guarantees time reversibility (a property of the equation of motion). 

Time reversibility. Newton s equation is reversible in time. Eor a numerical simulation to retain this property it should be able to retrace its path back to the initial configuration (when the sign of the time step At is changed to —At). However, because of chaos (which is part of most complex systems), even modest numerical errors make this backtracking possible only for short periods of time. Any two classical trajectories that are initially very close will eventually exponentially diverge from one another. In the same way, any small perturbation, even the tiny error associated with finite precision on the computer, will cause the computer trajectories to diverge from each other and from the exact classical trajectory (for examples, see pp. 76-77 in Ref. 6). Nonetheless, for short periods of time a stable integration should exliibit temporal reversibility. 

Equation (9.23) belongs to a elass of non-linear differential equations known as the matrix Rieeati equations. The eoeffieients of P(t) are found by integration in reverse time starting with the boundary eondition... 

Kalman demonstrated that as integration in reverse time proeeeds, the solutions of F t) eonverge to eonstant values. Should t be infinite, or far removed from to, the matrix Rieeati equations reduee to a set of simultaneous equations... 

The emission line is centered at the mean energy Eq of the transition (Fig. 2.2). One can immediately see that I E) = 1/2 I Eq) for E = Eq E/2, which renders r the full width of the spectral line at half maximum. F is called the natural width of the nuclear excited state. The emission line is normalized so that the integral is one f l(E)dE = 1. The probability distribution for the corresponding absorption process, the absorption line, has the same shape as the emission line for reasons of time-reversal invariance. 

After the momenta are selected from the distribution (8.39), the dynamics is propagated by a standard leapfrog algorithm (any symplectic and time-reversible integrator is suitable). The move is then accepted or rejected according to a criterion based on the detailed balance condition... 

Since the molecular dynamics integrator that we use is time-reversible, the reverse move, x —> x, is generated if we pick, at x, exactly the same momentum with which we arrived there, but with opposite sign. In other words, the reverse move is attempted with probability... 

Barth, E., Leimkuhler, B., Reich, S. Time-reversible variable-stepsize integrator for constrained dynamics. SIAM J. Sci. Comput. 1999, 21,1027-44. 

It is also useful to define the transformed operator L whose operation on a function f is L f = L[Peqf). This operator coincides with the time reversed backward operator, further details on these relationships may be found in Refs. 43,44. L operates in the Hilbert space of phase space functions which have finite second moments with respect to the equilibrium distribution. The scalar product of two functions in this space is defined as (f, g) = (fgi q. It is the phase space integrated product of the two functions, weighted by the equilibrium distribution P The operator L is not Hermitian, its spectrum is in principle complex, contained in the left half of the complex plane. 

In fact, the physical picture of the phenomenon is more subtle. The contribution of any trajectory with 0 is cancelled in the first order in B by the contribution of the time-reversed trajectory, since the values of SB — Sq are opposite for these paths [ Fig. 1(d), (e)]. The cancellation does not occur only at very small / . The integration in Eq. 3 over also give a quadratic in (3 contribution to the MR. This contribution is positive and comes from the second order term in the expansion of enSB —enS° in B. It follows from our results [Eqs. (1),(9)] that the contribution of small angles is dominant resulting in a negative parabolic MR [22], We find that the parabolic MR crosses over to linear at very small / ss 0.05/ o, which explains why the parabolic MR was not seen in numerical simulations [15] and experiment [16]. 

Specialized to thermal equilibrium, the velocity distributions for the molecules are the Maxwell-Boltzmann distribution (a special case of the general Boltzmann distribution law). The expression for the rate constant at temperature T, k(T), can be reduced to an integral over the relative speed of the reactants. Also, as a consequence of the time-reversal symmetry of the Schrodinger equation, the ratio of the rate constants for the forward and the reverse reaction is equal to the equilibrium constant (detailed balance). 

We also have an integral equation corresponding to (6.78). We write it for the time-reversed bra vector using analogous arguments. Note that the resolvent for the bra vector is the complex conjugate of the resolvent analogous to (6.70). 

Similiar problems are known in classical MD simulations, where intramolecular and intermolecular dynamics evolve on different time scales. One possible solution to this problem is the method of multiple time scale propagators which is describede in section 5. Berne and co-workers [21] first used different time steps to integrate the intra- and intermolecular degrees of freedom in order to reduce the computational effort drastically. The method is based on a Trotter-factorization of the classical Liouville-operator for the time evolution of the classical system, resulting in a time reversible propagation scheme. The multiple time scale approach has also been used to speed up Car-Parrinello simulations [20] and ab initio molecular dynamics algorithms [21]. 

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