1549380323-Statistical Mechanics Theory and Molecular Simulation

184 Canonical ensemble H ̃N= ( HN(r,s,p,ps)−H (0) N ) s = (N ∑ i=1 p^2 i 2 mis^2 +U(r 1 ,...,rN) + p^2 s 2 Q +gkTlns−H(0)N ) s, ...

Non-Hamiltonian statistical mechanics 185 4.8.3 The Nos ́e–Hoover equations In 1985, W. G. Hoover introduced a reformulation of ...

186 Canonical ensemble most situations, when the surroundings are integrated out in this way, the microscopic equations of motio ...

Non-Hamiltonian statistical mechanics 187 or that there exists a function whose derivative yields the compressibility. Substitut ...

188 Canonical ensemble the volume element, when √ ghas no explicit time dependence, so that the average can be performed at any ...

Non-Hamiltonian statistical mechanics 189 4.9.3 Equilibrium solutions In equilibrium, bothf(xt,t) and √ g(xt,t) have no explicit ...

190 Canonical ensemble 4.9.4 Analysis of the Nos ́e–Hoover equations We now turn to the analysis of eqns. (4.8.19). Our goal is ...

Non-Hamiltonian statistical mechanics 191 quantity. Unfortunately, this is not the typical situation. In the absence of external ...

192 Canonical ensemble -3 -2 -1 0 1 2 3 x -3 -2 -1 0 1 2 3 p -3 -2 -1 0 1 2 3 x -3 -2 -1 0 1 2 3 p -4 -2 0 2 4 p 0 0.1 0.2 0.3 0 ...

Nos ́e–Hoover chains 193 equations of motion can be expressed as r ̇i= pi mi p ̇i=Fi− pη 1 Q 1 pi η ̇j= pηj Qj j= 1,...,M p ̇η 1 ...

194 Canonical ensemble Here, we have introduced the variableηc= ∑M j=2ηjas a convenience since this par- ticular combination of ...

Nos ́e–Hoover chains 195 -4 -2 0 2 4 x -4 -2 0 2 4 p -4 -2 0 2 4 x -4 -2 0 2 4 p -4 -2 0 2 4 p 0 0.1 0.2 0.3 0.4 0.5 f( p) -4 -2 ...

196 Canonical ensemble p ̇η 1 ,i= [ p^2 i mi −dkT ] − pη 2 ,i Q 2 pη 1 ,i p ̇ηj,i= [ p^2 ηj− 1 ,i Qj− 1 −kT ] − pηj+1,i Qj+1 pηj ...

Integrating Nos ́e–Hoover chains 197 no longer exist in the system, a fact which leads to a simplification of the proof that the ...

198 Canonical ensemble iL 2 = ∑N i=1 Fi· ∂ ∂pi iLNHC=− ∑N i=1 pη 1 Q 1 pi· ∂ ∂pi + ∑M j=1 pηj Qj ∂ ∂ηj + M∑− 1 j=1 ( Gj−pηj pηj+ ...

Integrating Nos ́e–Hoover chains 199 algorithm could improve the accuracy of RESPA without adding significantly to the computati ...

200 Canonical ensemble In the present discussion, we will letS(∆t/2) be a primitive factorization of the operator exp(iLNHC∆t/2) ...

Integrating Nos ́e–Hoover chains 201 Consider the action of the operator exp(cx∂/∂x) onx. We can work this out using a Taylor se ...

202 Canonical ensemble part of the propagator acts on the outside but with the small time step. We de- note the schemes in eqn. ...

Isokinetic ensemble 203 not canonical is of little consequence. Nevertheless, since the momentum- and position- dependent parts ...