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Constraint methodHoover‘s original proposal:The condition forconstant temperatureimplies for theLagrangemultiplier ξ :One must start alreadyat a temperature T.NNd 2 T pi pipidt i1 i1Ni1 p f p pi i i iNi1Ni1 fpipii20119Constraint methodBerendsen et al (1984):Define: 1T T HermanBerendsenwhich reaches the temperature T exponentiallywith the coefficient γ . This method is not timereversible and does not give canonical distribution.120
Constraint methodThis method is similar to velocity rescaling as seenthrough a Taylor expansion of : 1 TT 12tT2tTt T 1 1t T for small Δt and small (T – T ), giving:TT121Constraint methodFinally, another alternative would be todetermine the friction coefficient ξ through: mk TTQwhere m is the number of degrees of freedom(m = 3N -3) andQ the „thermal inertia“,a fit parameter describing the relaxation.But the time scale is artificially dominated by Q.122
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