Computational Physics

8.5 Molecular dynamics methods for different ensembles 223

whereS 2 is the second order integrator, and he fixedαandβby the requirement that
the resulting expression is equal to the continuum operator to fourth order. Higher
order integrators were found similarly. The general result can be written as

fork=1tondo

x(k)=x(k−^1 )−hak∂T[p(k−^1 )]/∂p (8.69)

p(k)=p(k−^1 )−hbk∂U[x(k)]/∂x

end

and the numbersakandbkcan be found in Yoshida’s paper. For the fourth order
case, they read

a 1 =a 4 = 1 /[ 2 ( 2 − 21 /^3 )]; a 2 =a 3 =( 1 − 21 /^3 )a 1 (8.70a)
b 1 =b 3 = 2 a 2 ; b 2 =− 21 /^3 b 1 ; b 4 =0. (8.70b)
From Yoshida’s derivation it follows that there exists a conserved quantity which
acts as the analog of the energy. The integrator is certainly not the same as the exact
time evolution operator, but it deviates from the latter only by a small amount.
Writing the integratorS(h)as

S(h)=exp(hAD) (8.71)

we have a new operatorADwhich deviates from the continuum operatorAonly by
an amount of orderhn+^1 , as the difference can be written as a sum of higher order
CBH commutators. It will be shown in Problem 8.9 that for an operator of the form
exp(tAD)which is symplectic for allt, there exists a HamiltonianHDwhich is the
analogue of the Hamiltonian in the continuum time evolution. This means that, if
we knowHD(which is usually impossible to find, except for the trivial case of the
harmonic oscillator), we could either use the integrator (8.71) to give us the image
at timeh, or the continuum solution of the dynamical system with HamiltonianHD
fort=h: both mappings would give identical results. The HamiltonianHD(z)is
therefore a conserved quantity of the integrator, and it differs from the energy by an
amount of orderhn+^1. The existence of such a conserved quantity is also discussed
inRefs.[ 17 , 18 , 31 ].

8.5 Molecular dynamics methods for different ensembles

8.5.1 Constant temperature

In experimental situations the total energy is often not a control variable as usually
the temperature of the system is kept constant. We know that in the infinite system
the temperature is proportional to the average kinetic energy per degree of freedom