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Nosé - Hoover thermostatgiving the equations of motionin the new time t´:m s x f 2 m ssx , i 1,...,N2i i i i iN21Qs misx i ( m 1) kTsi1The entire system is conservative andits ensemble is microcanonical.127Nosé - Hoover thermostatsDefining one can also express the equationssin realfrictiontime t: fixi xdesiredimeasuredmkineticiNkineticenergy1 1 21Q mx i i ( m1) kTenergy 2 2 2i1The ln s that is need to calculate thetotal energy is obtained integrating:dlnsdt 128
Nosé - Hoover thermostatCoupling to heat baths has thermal inertia Q .measuredkineticenergy xifimN1 1 21 i i ( 1)2 2 i12ixQ mx m kTifrictiondesiredkineticenergyApplet129Nosé - Hoover thermostatQ must be chosen empirically.If Q is too large equilibration is too slow andfor Q → one recovers microcanonical MD.If Q is too small the temperature exhibits spuriousoscillations. Therefore one must check that thewidth of the temperature distribution follows:2T TNdwhere d is the dimensionand N the particle number.130
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