Hybrid Monte Carlo 299uniform moves of Section 7.3. Typically, in a hybrid Monte Carlo calculation, one aims
for a higher average acceptance probability (40% to 70%) than wasrecommended
in Section 7.3; indeed, largermand smaller ∆tyield better trial moves that are
more likely to be accepted. If force calculations are very expensiveand/or the code is
inefficient, then smallermand larger ∆tare preferable. Finally, if a move is rejected,
the positionsr′are set to their original valuesr. However, if we similarly reset the
momenta to their original values and reapply eqn. (7.4.5), we will end up at the same
point (r′,p′) and reject the move again. Thus, the momenta should be resampled
from a Maxwell-Boltzmann distribution before initiating the next trialmove via eqn.
(7.4.5).
We conclude this section with a short proof that the detailed balancecondition in
eqn. (7.3.23) is satisfied when a time-reversible integrator is used. Since we are only
interested in the configurational distribution, the detailed balancecondition for hybrid
Monte Carlo can be stated as
∫
dNpdNp′T(r′,p′|r,p)A(r′,p′|r,p)f(r,p)
=
∫
dNpdNp′T(r,p|r′,p′)A(r,p|r′,p′)f(r′,p′). (7.4.6)In order to prove this, we first note that
A(r′,p′|r,p)f(r,p) =1
Q(N,V,T)
min[
1 ,e−β[H(r′,p′)−H(r,p)]]
e−βH(r,p)=
1
Q(N,V,T)
min[
e−βH(r,p),e−βH(r′,p′)]. (7.4.7)
Similarly,
A(r,p|r′,p′)f(r′,p′) =1
Q(N,V,T)
min[
1 ,e−β[H(r,p)−H(r′,p′)]]
e−βH(r′,p′)=
1
Q(N,V,T)
min[
e−βH(r′,p′)
,e−βH(r,p)]
. (7.4.8)
Therefore
A(r′,p′|r,p)f(r,p) =A(r,p|r′,p′)f(r′,p′). (7.4.9)
Multiplying both sides of eqn. (7.4.9) byT(r′,p′|r,p) and integrating over momenta
gives
∫
dNpdNp′T(r′,p′|r,p)A(r′,p′|r,p)f(r,p)
=
∫
dNpdNp′T(r′,p′|r,p)A(r,p|r′,p)f(r′,p′). (7.4.10)By using the property of the integrator thatT(r′,p′|r,p) =T(r,−p|r′,−p′), changing
the integration variables on the right side frompandp′to−pand−p′, and noting