5 Basics of Molecular Modeling and Molecular Simulation
209
5.3.1 Basic Concepts
Most of the computer simulations that the following segments are to talk about are
based on that the classical statistical mechanics can be used in describing the detailed
motions of atoms and molecules. This basis to a large extent simplifies most of the
calculations, and is surprisingly appropriate when dealing with most cases in practice.
Thus, the following section of this chapter is going to briefly talk about some of the
basics of classical statistical physics.
Firstly, to understand the basics of thermodynamic and classical equilibrium statistical physics, it is vital to notice two basic hypotheses, equiprobability and ergodicity.
Equiprobability states that, in a thermal system, there is an equal probability for the
system to visit each microstate, whereas ergodicity states that all thermal states will
be visited when the revolution time approaches infinity, namely, the ensemble average
equals to the time average.
For further understanding, there are several concepts that need to be noted. For
an N particles system with a total energy E that is confined in a volume V , we
denote term Ω(N , E, V ) to indicate the number of all the microstates that it may
visit, which also indicates its ensemble, namely, the set of all possible states under
a given thermal condition. Now let us consider a system that is combined with two
weakly interacting subsystems, which indicates that the total energy of the system
follows the rule E = E 1 + E 2 . Thus, for a given E 1 , there is Ω(E) = Ω(E 1 ) × Ω(E 2 ),
and that
ln Ω(E, E − E 1 ) = ln Ω 1 (E 1 ) + ln Ω 2 (E − E 1 )
(5.3.1)
Assuming that the two subsystems can transfer energy, then the most likely
configuration of the distribution of the energy can be given with the equiprobability
hypothesis, which is that the most likely value of E 1 maximizes ln Ω(E, E − E 1 ).
Therefore,
∂ ln Ω(E, E − E 1 )
∂E 1
N ,V ,E
= 0
(5.3.2)
or
∂ ln Ω 1 (E 1 )
∂E 1
N 1 ,V 1
=
∂ ln Ω 2 (E 2 )
∂E 2
N 2 ,V 2
(5.3.3)
We can define
β(E, V , N ) =
∂ ln Ω(E, V , N )
∂E
N ,V ,E
(5.3.4)
So Eq. (3.3) becomes
209
5.3.1 Basic Concepts
Most of the computer simulations that the following segments are to talk about are
based on that the classical statistical mechanics can be used in describing the detailed
motions of atoms and molecules. This basis to a large extent simplifies most of the
calculations, and is surprisingly appropriate when dealing with most cases in practice.
Thus, the following section of this chapter is going to briefly talk about some of the
basics of classical statistical physics.
Firstly, to understand the basics of thermodynamic and classical equilibrium statistical physics, it is vital to notice two basic hypotheses, equiprobability and ergodicity.
Equiprobability states that, in a thermal system, there is an equal probability for the
system to visit each microstate, whereas ergodicity states that all thermal states will
be visited when the revolution time approaches infinity, namely, the ensemble average
equals to the time average.
For further understanding, there are several concepts that need to be noted. For
an N particles system with a total energy E that is confined in a volume V , we
denote term Ω(N , E, V ) to indicate the number of all the microstates that it may
visit, which also indicates its ensemble, namely, the set of all possible states under
a given thermal condition. Now let us consider a system that is combined with two
weakly interacting subsystems, which indicates that the total energy of the system
follows the rule E = E 1 + E 2 . Thus, for a given E 1 , there is Ω(E) = Ω(E 1 ) × Ω(E 2 ),
and that
ln Ω(E, E − E 1 ) = ln Ω 1 (E 1 ) + ln Ω 2 (E − E 1 )
(5.3.1)
Assuming that the two subsystems can transfer energy, then the most likely
configuration of the distribution of the energy can be given with the equiprobability
hypothesis, which is that the most likely value of E 1 maximizes ln Ω(E, E − E 1 ).
Therefore,
∂ ln Ω(E, E − E 1 )
∂E 1
N ,V ,E
= 0
(5.3.2)
or
∂ ln Ω 1 (E 1 )
∂E 1
N 1 ,V 1
=
∂ ln Ω 2 (E 2 )
∂E 2
N 2 ,V 2
(5.3.3)
We can define
β(E, V , N ) =
∂ ln Ω(E, V , N )
∂E
N ,V ,E
(5.3.4)
So Eq. (3.3) becomes