210
C. Tang (唐晨宇) and Y. Wang (王延颋)
β(E 1 , V 1 , N 1 ) = β(E 2 , V 2 , N 2 )
(5.3.5)
The maximum of ln Ω(N , E, V ) also indicates that the system has found its equilibrium state, which coincides with the thermodynamic entropy S that also reaches
the maximum when the system is at thermodynamic equilibrium. Thus, we can have
the Boltzmann’s entropic equation:
S = k B ln Ω
(5.3.6)
where k B is the Boltzmann constant. From Eq. (3.5), we can also see that β 1 = β 2
when the two subsystems are at equilibrium, where it is easy to find the definition of
the temperature T :
1
T
=
∂S
∂E
N ,V
(5.3.7)
and then
β =
1
k B T
(5.3.8)
Another important concept that should be mentioned here is the potential energy
surface, which indicates the set of potential energies for all configurations that a
particular system may reach. These concepts will help us better understand not only
computer simulations, but also the properties of many-body systems that we may be
interested in.
5.3.2 Common Ensembles
We have introduced the concept of ensemble in the last segment, and to acquire
knowledge about certain ensembles are of much significance in dealing with manybody systems and thus implementing computer simulations. We will briefly introduce several important ensembles, but to acquire further and detailed information,
readers should check textbooks of statistical physics in order to develop a fundamental
understanding of them.
A thermodynamic system with a constant particle number N , a constant energy
E, and a constant volume V is defined as in the microcanonical ensemble. A system
with a constant particle numbers N , a constant volume V , and a constant temperature T is defined as in the canonical ensemble. A system with a constant chemical
potential μ, a constant volume V , and a constant temperature T is defined to be in
the grandcanonical ensemble, where the chemical potential follows:
C. Tang (唐晨宇) and Y. Wang (王延颋)
β(E 1 , V 1 , N 1 ) = β(E 2 , V 2 , N 2 )
(5.3.5)
The maximum of ln Ω(N , E, V ) also indicates that the system has found its equilibrium state, which coincides with the thermodynamic entropy S that also reaches
the maximum when the system is at thermodynamic equilibrium. Thus, we can have
the Boltzmann’s entropic equation:
S = k B ln Ω
(5.3.6)
where k B is the Boltzmann constant. From Eq. (3.5), we can also see that β 1 = β 2
when the two subsystems are at equilibrium, where it is easy to find the definition of
the temperature T :
1
T
=
∂S
∂E
N ,V
(5.3.7)
and then
β =
1
k B T
(5.3.8)
Another important concept that should be mentioned here is the potential energy
surface, which indicates the set of potential energies for all configurations that a
particular system may reach. These concepts will help us better understand not only
computer simulations, but also the properties of many-body systems that we may be
interested in.
5.3.2 Common Ensembles
We have introduced the concept of ensemble in the last segment, and to acquire
knowledge about certain ensembles are of much significance in dealing with manybody systems and thus implementing computer simulations. We will briefly introduce several important ensembles, but to acquire further and detailed information,
readers should check textbooks of statistical physics in order to develop a fundamental
understanding of them.
A thermodynamic system with a constant particle number N , a constant energy
E, and a constant volume V is defined as in the microcanonical ensemble. A system
with a constant particle numbers N , a constant volume V , and a constant temperature T is defined as in the canonical ensemble. A system with a constant chemical
potential μ, a constant volume V , and a constant temperature T is defined to be in
the grandcanonical ensemble, where the chemical potential follows: