96
W.-M. Zheng
time evolution of distribution and macroscopic thermodynamical variables, whose
time scale is much greater than the microscopic dynamical time scale of molecular
orbits (in 6N -dimensional phase space). No time is involved in Gibbs’ theory of
ensemble, which specifies the equilibrium distribution function, assigns it with the
minimal free energy, and makes no indication about approaching equilibrium. The
Liouville equation is essentially equivalent to the dynamical equation of molecular
orbits, and is not a proper starting point for statistical mechanics. The principle about
the contrast between the time scales of orbits and distributions in Gibbs’ theory
is valid also for general nonequilibrium distributions. Continuing in the spirit of
Gibbs’ thinking, we have to give up the attempt to derive the time evolution equation
of distributions from the dynamical equation of molecular orbits like the Liouville
equation. A new principle is required to describe the evolution of distributions, in the
same manner as in quantum mechanics where we do not attempt to derive Schrödinger
equation from the Hamiltonian equation.
Necessary conditions for a legitimate time evolution equation of distributions
include satisfying the conservation of probability, with the equilibrium distribution
being its solution. It is most natural to take the master equation as a candidate. Let
us consider the master equation for the time evolution of distributions specified by
the transition rates T (z → z
):
P t+1 (z) =
d z
P t (z
)T (z
→ z),
(2.18)
where z is an abbreviated notation for conformation, while the transition probability
T (z → z
) is completely determined by the Hamiltonian of the system and satisfies
the following detailed balance condition:
P eq (z
)T (z
→ z) = P eq (z)T (z → z
),
T (z → z
)
T (z → z)
=
P eq (z
)
P eq (z)
=
e
−βH (z
)
e −βH (z) .
(2.19)
The above distribution dynamics guarantees that a system will approach equilibrium
P eq (z). In summary, we propose a principle of statistical mechanics as follows:
The evolution dynamics for the distribution of a system in an environment or
heatbath is described by the master equation (2.18) with transition rates satisfying
the detailed balance condition (2.19).
Note that the detailed balance condition here is associated with the original 6N -
dimensional conformations; the condition need not be valid for a reduced distribution
in a lower dimensional space.
The transition probability matrix T satisfies
d z T k (z
→ z) = 1, exhibiting a
right eigenvector 1 (whose components are all equal to a positive number). If T
is irreducible, accoding to the Perron-Frobenius theorem then its spectrum radius
equals 1. The left eigenvector dual to 1 is a non-negative vector, corresponding to
the equilibrium distribution. The continuous state extension of the finite Markov
chain is the theory of compact transition operators. Furthermore, discrete time may
be extended to continuous time.
W.-M. Zheng
time evolution of distribution and macroscopic thermodynamical variables, whose
time scale is much greater than the microscopic dynamical time scale of molecular
orbits (in 6N -dimensional phase space). No time is involved in Gibbs’ theory of
ensemble, which specifies the equilibrium distribution function, assigns it with the
minimal free energy, and makes no indication about approaching equilibrium. The
Liouville equation is essentially equivalent to the dynamical equation of molecular
orbits, and is not a proper starting point for statistical mechanics. The principle about
the contrast between the time scales of orbits and distributions in Gibbs’ theory
is valid also for general nonequilibrium distributions. Continuing in the spirit of
Gibbs’ thinking, we have to give up the attempt to derive the time evolution equation
of distributions from the dynamical equation of molecular orbits like the Liouville
equation. A new principle is required to describe the evolution of distributions, in the
same manner as in quantum mechanics where we do not attempt to derive Schrödinger
equation from the Hamiltonian equation.
Necessary conditions for a legitimate time evolution equation of distributions
include satisfying the conservation of probability, with the equilibrium distribution
being its solution. It is most natural to take the master equation as a candidate. Let
us consider the master equation for the time evolution of distributions specified by
the transition rates T (z → z
):
P t+1 (z) =
d z
P t (z
)T (z
→ z),
(2.18)
where z is an abbreviated notation for conformation, while the transition probability
T (z → z
) is completely determined by the Hamiltonian of the system and satisfies
the following detailed balance condition:
P eq (z
)T (z
→ z) = P eq (z)T (z → z
),
T (z → z
)
T (z → z)
=
P eq (z
)
P eq (z)
=
e
−βH (z
)
e −βH (z) .
(2.19)
The above distribution dynamics guarantees that a system will approach equilibrium
P eq (z). In summary, we propose a principle of statistical mechanics as follows:
The evolution dynamics for the distribution of a system in an environment or
heatbath is described by the master equation (2.18) with transition rates satisfying
the detailed balance condition (2.19).
Note that the detailed balance condition here is associated with the original 6N -
dimensional conformations; the condition need not be valid for a reduced distribution
in a lower dimensional space.
The transition probability matrix T satisfies
d z T k (z
→ z) = 1, exhibiting a
right eigenvector 1 (whose components are all equal to a positive number). If T
is irreducible, accoding to the Perron-Frobenius theorem then its spectrum radius
equals 1. The left eigenvector dual to 1 is a non-negative vector, corresponding to
the equilibrium distribution. The continuous state extension of the finite Markov
chain is the theory of compact transition operators. Furthermore, discrete time may
be extended to continuous time.