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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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"Preferably, the conservative dynamics process uses a mollified impulse method for stability improvement, for example using a process in which an averaging operator A is applied to a collective position vector X and derives mollified potential energy U.sub.molly.sup.slow related to the more slowly varying potential energy, to yield more slowly varying mollified forces F.sub.molly.sup.slow, and preferably wherein calculated properties are re-weighted at the end of the entire method, to allow for the use of the mollified terms and shadow Hamiltonians.

"Advantageously, the mollified multiple time stepping method applies the operator A to the collective position vector X and derives the mollified potential energy U.sub.molly.sup.slow related to the more slowly varying potential energy to yield more slowly varying forces mollified forces F.sub.molly.sup.slow, wherein:

".times.dd.times..function..times..times..DELTA..times..times..times..time- s..delta..function..times..times..DELTA..times..times..times..function..fu- nction..times..times..delta..times..times..times..times..delta..function..- times..times..delta..times..times..times..function..function. ##EQU00001## for t.di-elect cons.[0,t'=L.DELTA.t] with the slow forces defined by F.sub.molly.sup.slow(X)=-.gradient.U.sub.molly.sup.slow(X)=A.sub.X(X).sup- .TF.sup.slow(A(X)), where M is a diagonal mass matrix of atomic masses, A.sub.X(X) denotes the Jacobian matrix of partial derivatives, the larger time-step for more slowly varying forces F.sup.slow is .DELTA.t, the smaller time-step for more quickly varying forces F.sup.fast is .delta.t, .delta..sub.x is the Dirac delta function, c.sub.m=d.sub.n=1 except when m=n=0 or m=L, n=pL, respectively, in which case c.sub.m=d.sub.n=1/2, integer p>1, and L>0 is a given integer.

"Either the momentum refreshment or the conservative dynamics process is the first process of the method, and the resulting state of any process in the method preferably provides the current state for the next process in the method.

"Advantageously, the acceptance/rejection operation following the mixing operation returns the resulting state of the mixing operation in the case of acceptance and the state before the mixing operation as the replacement state in the case of rejection; and the acceptance/rejection operation following the conservative dynamics operation returns the resulting state of the conservative dynamics operation in the case of acceptance; and, in the case of rejection, the state before the conservative dynamics operation is either returned as the replacement state or undergoes a momentum flip to provide the replacement state.

"Preferably the momentum refreshment process and/or the conservative dynamics process constitutes a multiple iteration process, in which the entire process is repeated a selected number of times consecutively, to provide a final resulting state, which may be used as a current state for the next process. Usually, the method as a whole is also repeated, for example so that the results of the acceptance/rejection operation in the conservative dynamics process are fed into the momentum refreshment process in a further iteration of the method.

"Any suitable methodology can be used for combining the collective momentum vector with a noise vector in the momentum refreshment process. In one suitable method, the states in the method may be denoted by .OMEGA..sub.i=(Y.sub.i.sup.T,t.sub.i).sup.T, i=0, . . . , I, where I is a given integer, Y.sub.i=(X.sub.i.sup.T,1,P.sub.i.sup.T,b.sub.i).sup.T, X.sub.i a collective vector of atomic positions, P.sub.i is a collective vector of atomic momenta, b.sub.i is a scalar, and t.sub.i is time and wherein the mixing operation comprises: given a current state, mixing its collective atomic momentum vector P.sub.i with an independent and identically distributed normal noise vector .XI..sub.i of dimension 3N, so that {tilde over (P)}.sub.i=cos(.phi.)P.sub.i+sin(.phi.).XI..sub.i, {tilde over (.XI.)}.sub.i=cos(.phi.).XI..sub.i-sin(.phi.)P.sub.i, where i is a given integer, N is the number of particles in the system, 0<.phi..ltoreq..pi./2 is a given angle, .XI..sub.i.about.N[0,.beta.M.sup.-1], N[0,.beta.M.sup.-1] denotes the (3N)-dimensional normal distribution with zero mean and covariance matrix .beta.m.sup.-1, M is the diagonal mass matrix of the molecular system, and .beta.=1/k.sub.BT is the inverse temperature, the proposed state vector being denoted by {tilde over (.OMEGA.)}.sub.i=({tilde over (Y)}.sub.i.sup.T,t.sub.i).sup.T, {tilde over (Y)}.sub.i=(X.sub.i.sup.T,1,{tilde over (P)}.sub.i.sup.T,b.sub.i).sup.T; and preferably wherein the subsequent acceptance/rejection operation comprises obtaining the resulting state .OMEGA..sub.i; through a Metropolis accept/reject criterion:

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