300 Monte Carlo
that d(−p) d(−p′) = dpdp′, we obtain eqn. (7.4.6), which emphasizes the importance
of using a reversible, measure-preserving integration algorithm such as velocity Verlet.
As a final comment, we note that the trial moves in eqn. (7.4.5) neednot be purely
Hamiltonian in nature. We could have used thermostatted equationsof motion, for
example, as described in Chapter 4 and still generated a proper canonical sampling
using the acceptance criterion in eqn. (7.4.1) or a modified acceptance criterion based
on a conserved extended energy (see eqn. (4.10.3), for example).
7.5 Replica exchange Monte Carlo
One of the most challenging computational problems met by researchers in statistical
mechanics is the development of methods capable of sampling a canonical distribution
when the potential energyU(r 1 ,...,rN) is characterized by a large number of local
minima separated by high barriers. Such potential energy functions describe many
physical systems including proteins, glasses, polymer membranes,and polymer blends,
to name just a few. An illustration of such a surface is shown in Fig. 7.2. Potential
energy surfaces that resemble Fig. 7.2 (but in 3Ndimensions) are referred to asrough
energy landscapes. The ongoing development of general and robust techniques capable
of adequately sampling statistically relevant configurations on sucha surface continues
to impact computational biology and materials science in important ways as newer and
more sophisticated methods become available.
Fig. 7.2A two-dimensional rough potential energy surface.