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W.-M. Zheng
its state is fully described by a few physical and chemical variables or state variables characterizing the macroscopic properties. The equilibrium state of a system is
classified according to a finite number of independent state variables, which span a
space of thermodynamic states. A point in the space corresponds to a thermodynamical state. State variables may be either extensive or intensive. Extensive variables
are proportional to the size of the system, and are additive, while the latter are independent of the size. At least one extensive variable is needed to fix the size of the
system. In fact, entropy does not meet the literal description of the definition of an
extensive variable.
There are three kinds of thermodynamical variables [13]. The first are external
parameters attributed to external environment and are independent of the interior of
the system. The second are averages of dynamical functions of microscopic conformations. The third are typical, belong particularly to thermodynamics, have no
microscopic meaning, can be grasped only in a macroscopic sense, and are associated
totally with the distribution function. A representative of such variables is entropy.
The equation of state describes the dependence among several state variables which
are not completely independent of each other. A thermodynamic process describes a
series of thermodynamical states, and a path in the space of thermodynamical states
corresponds an equilibrium process. (A non-equilibrium or irreversible process has
no path representation.) In dealing with thermodynamic processes, independent and
dependent variables should be carefully specified. For example, in an isobaric process
the pressure is selected as an independent variable, and held fixed. Thermodynamics is a phenomenological theory. For example, the equation of state satisfies some
thermodynamical constrains, but its form depends on the property of the matter that
the system consists of, so it is not derivable from thermodynamics. To obtain thermodynamical relations and compute thermodynamical quantities from molecular
knowledge is the task of statistical mechanics.
The first law of thermodynamics defines the internal energy and divides it into
work and heat, but only the internal energy is a state variable while heat and work
are not and depend on the process. The commonest work is the mechanical work
W mech = −pdV , where p is pressure. When a system performs work on some external
environment its volume increases, and W mech is negative. The general form of work
is W =
i f i dX i , where f i is a generalized force and X i a generalized displacement.
In statistical mechanics the internal energy U is the average of energy:
U (β, N , V ) =
d r
N d p
N E(r
N
, p
N
)P(r
N
, p
N
),
(2.8)
where the probability density is P = e
−βE
/Z for the canonical ensemble, while the
temperature β and number N of particles are parameters and volume V is an independent variable. Therefore, the change in internal energy equals
dU =
d r
N d p
N
[P(r
N
, p
N
)dE(r
N
, p
N
) + E(r
N
, p
N
)dP(r
N
, p
N
)].
W.-M. Zheng
its state is fully described by a few physical and chemical variables or state variables characterizing the macroscopic properties. The equilibrium state of a system is
classified according to a finite number of independent state variables, which span a
space of thermodynamic states. A point in the space corresponds to a thermodynamical state. State variables may be either extensive or intensive. Extensive variables
are proportional to the size of the system, and are additive, while the latter are independent of the size. At least one extensive variable is needed to fix the size of the
system. In fact, entropy does not meet the literal description of the definition of an
extensive variable.
There are three kinds of thermodynamical variables [13]. The first are external
parameters attributed to external environment and are independent of the interior of
the system. The second are averages of dynamical functions of microscopic conformations. The third are typical, belong particularly to thermodynamics, have no
microscopic meaning, can be grasped only in a macroscopic sense, and are associated
totally with the distribution function. A representative of such variables is entropy.
The equation of state describes the dependence among several state variables which
are not completely independent of each other. A thermodynamic process describes a
series of thermodynamical states, and a path in the space of thermodynamical states
corresponds an equilibrium process. (A non-equilibrium or irreversible process has
no path representation.) In dealing with thermodynamic processes, independent and
dependent variables should be carefully specified. For example, in an isobaric process
the pressure is selected as an independent variable, and held fixed. Thermodynamics is a phenomenological theory. For example, the equation of state satisfies some
thermodynamical constrains, but its form depends on the property of the matter that
the system consists of, so it is not derivable from thermodynamics. To obtain thermodynamical relations and compute thermodynamical quantities from molecular
knowledge is the task of statistical mechanics.
The first law of thermodynamics defines the internal energy and divides it into
work and heat, but only the internal energy is a state variable while heat and work
are not and depend on the process. The commonest work is the mechanical work
W mech = −pdV , where p is pressure. When a system performs work on some external
environment its volume increases, and W mech is negative. The general form of work
is W =
i f i dX i , where f i is a generalized force and X i a generalized displacement.
In statistical mechanics the internal energy U is the average of energy:
U (β, N , V ) =
d r
N d p
N E(r
N
, p
N
)P(r
N
, p
N
),
(2.8)
where the probability density is P = e
−βE
/Z for the canonical ensemble, while the
temperature β and number N of particles are parameters and volume V is an independent variable. Therefore, the change in internal energy equals
dU =
d r
N d p
N
[P(r
N
, p
N
)dE(r
N
, p
N
) + E(r
N
, p
N
)dP(r
N
, p
N
)].