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Patent Issued for Method, Apparatus and Computer Program for Multiple Time Stepping Simulation of a Thermodynamic System Using Shadow Hamiltonians

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".OMEGA..OMEGA..times..times..times..times..function..beta..times..times..- DELTA..times..times..DELTA..times..times..OMEGA..times..times..times..time- s..DELTA..times..times..DELTA..times..times..DELTA..times..times..function- ..times..XI..times..times..XI..DELTA..times..times..function..times..XI..t- imes..times..XI. ##EQU00002## in which H.sub..DELTA.t is the shadow Hamiltonian and .DELTA.H.sub..DELTA.t.sup.e is the change in shadow Hamiltonian due to the mixing operation.

"Turning to the conservative dynamics process, each iteration of the process will usually include describing the forces of the atoms of the molecules of the molecular system using the chosen force field, integrating Newton's equation to predict the positions and velocities and recalculation of the forces.

"The multiple time stepping method is carried out so that the calculations for more quickly varying forces (usually those relating to forces over a shorter distance range) are carried out more frequently than calculations for more slowly varying forces.

"The conservative dynamics operation may comprise applying the multiple time stepping method to the current state .OMEGA..sup.+.sub.i=( Y.sub.i.sup.T,t.sub.i).sup.T with Y.sub.i=(X.sub.i.sup.T,1, P.sub.i.sup.T,b.sub.i).sup.T using a time-reversible and volume conserving mapping {circumflex over (.PSI.)}.sub..tau., with the proposed state being defined by: {circumflex over (.OMEGA.)}.sub.i=( .sup.T,t.sub.i+L.DELTA.t).sup.T, with .sub.i={circumflex over (.PSI.)}.sub..tau.( Y.sub.i) with .tau.=.DELTA.tL and given integer L.gtoreq.1; and the subsequent acceptance/rejection operation may comprise obtaining the resulting state .OMEGA..sub.i+1, through a Metropolis accept/reject criterion:

".OMEGA..OMEGA..times..times..times..times..function..function..beta..time- s..times..DELTA..times..times..DELTA..times..times..OMEGA..times..times..t- imes..times..DELTA..times..times..DELTA..times..times..DELTA..times..times- ..function..DELTA..times..times..function. ##EQU00003## in which H.sub..DELTA.t is the shadow Hamiltonian and .DELTA.H.sub..DELTA.t is the change in shadow Hamiltonian due to the conservative dynamics operation; and in which .OMEGA..sub.i.sup.-=( Y.sup.-.sup.T.sub.i,t.sub.i).sup.T, Y.sup.-.sub.i=(X.sub.i.sup.T,1,- P.sub.i.sup.T,b.sub.i).sup.T, .OMEGA..sub.i.sup..+-. denotes either .OMEGA..sub.i.sup.- or .OMEGA..sub.i.sup.+ and .OMEGA..sub.i.sup.- indicates applying a momentum flip to the state .OMEGA..sub.i and .OMEGA..sub.i.sup.+= .OMEGA..sub.i and indicates that no momentum flip is applied.

"Any suitable approximation truncation for the Hamiltonian can be used as the shadow Hamiltonian. Preferably, the shadow Hamiltonian used can be expressed as: H.sub..DELTA.t.sup.[q]=H.sub..DELTA.t.sup.[q],MTS-H.sub..DELTA.t.sup.[q],- MTS,fast+H.sub..delta.t.sup.[q],SV,fast, in which H.sub..DELTA.t.sup.[q],MTS denotes the q th order shadow Hamiltonian for an MTS method, H.sub..DELTA.t.sup.[q],MTS,fast denotes the q th order shadow Hamiltonian for the same MTS method with the more slowly varying forces F.sup.slow set equal to zero and H.sub..delta.t.sup.[q],SV,fast denotes the q th order shadow Hamiltonian for a Stormer-Verlet method applied to the more quickly varying forces in the system, with step-size .delta.t.

"In order to use the method on a computer, initial conditions or parameters need to be input manually or automatically. Thus the method may further comprise a step of initially accepting input of simulation conditions, wherein the simulation conditions include at least one of volume, mass, temperature, pressure, number of particles, and total energy; and/or further comprising a step of initially accepting input of simulation parameters, wherein the simulation parameters include at least one of a number of repetitions of the momentum refreshment process and a number of repetitions of the conservative dynamics process, the larger and smaller time step in conservative dynamics, the number of iterations of the entire method, the current state for the first step in the method, the force field parameters, a time-reversible and volume conserving mapping .PSI..sub..tau., and a constant angle .phi., where 0<.phi..ltoreq..pi./2.

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