Biology Reference
In-Depth Information
the environment. Indeed, the canonical distribution function can be
defined as the distribution maximizing the entropy
16
and is expressed as
-
b
Hrp
(, )
r
NVT
(, )
rp
µ
e
.
The Helmholtz free energy
F
corresponding to the canonical ensemble is
FUTS
=-.
(11)
where
U
represents the average energy of the system, and
S
the
entropy.
A number of factors impede a precise computer simulation of the
NVT statistical mechanical ensemble. First, one can obviously not simu-
late a realistic heat bath with a large number of degrees of freedom.
Second, the nature of the coupling between the system and the bath,
which is deliberately disregarded in standard statistical mechanical theory,
needs to enter explicitly the MD equations of motion. The modified
dynamics that serve this purpose are called thermostats.
A thermostat maintains the average temperature of the system at
T
by absorbing any excess energy that might appear in the simulation.
Spurious energy sources are due to various inaccuracies in the MD algo-
rithm used. These include the use of cut-offs for long-range interac-
tions, the skipping of time steps for determination of atom pairs within
the cut-off, the use of a special algorithm to constrain bond length, and
numerical drift if the equations of motion are integrated with large time
steps. All of these effects tend to heat up the system. If, in addition,
work is performed by an external agent to drive a given process, dissi-
pation will tend to increase the temperature. A thermodynamic picture
of an MD system coupled to a thermostat is shown in Fig. 4. There, the
energy variation
U
of the system expected from Eq. (11) is mediated
by its different couplings to the outside. These couplings are expressed
as different kinds of work: the external work
W
ext
, the work
W
thermo
pro-
vided by the thermostat, or
W
MD
resulting from the side effects of the
algorithms used.
An example of thermostating dynamics is the Nosé-Hoover (NH)
thermostat.
17
∆
In addition to the physical degrees of freedom (
r
,
p
), an
