2 On the Foundational Principles of Statistical Mechanics
85
worth noticing. It is not difficult to distinguish conformational states, thermodynamical states and states of statistical mechanics, but to fully understand the concepts
behind them in analysing problems is often overlooked.
2.2 Canonical Ensemble
A state of statistical mechanics of a system means a distribution over the phase space.
In general, the time scale of the dynamical evolution of molecular orbits is much
shorter than that of the evolution of distributions. A system in statistical mechanics
is always in an environment and of a large number of degrees of freedom. This makes
exactly solving the dynamics of molecular orbits impossible and also unnecessary.
On the same basis, a simple thermodynamical description exists for macroscopic
equilibrium states. As a dynamical system, a microscopic system with a large number
of degrees of freedom does not have any independent integrals of motions except for
the Hamiltonian. In the ideal case, only the Hamiltonian, and only under a statistical
average, is a conservative observable of the system. As a result of the wildly different
time scales of the orbits and distributions, time plays a less important role in the
equilibrium statistical mechanics. A principle of statistical mechanics implied by
statistical laws states that a so-called canonical distribution depending only on the
Hamiltonian exists for a system in thermodynamical equilibrium.
In 1902 Gibbs introduced the concept of ensemble, which is a collection of copies
of a physical system under certain statistical distribution. Gibbs’ theory of ensembles is axiomatic. The ensemble is not a physical thing. Gibbs did not attempt to
derive distributions; the canonical ensemble is introduced by postulation. He merely
designed a logical apparatus that produces relations which can be interpreted as analogues to the thermodynamical ones. The canonical distribution may be explained
with the principle of maximum entropy, which claims that the probability distribution
best representing the current state of knowledge, in the context of precisely stated
testable information, is the one with largest entropy [4].
Suppose that the external parameters determining the Hamiltonian of a system,
such as the number of particles and the volume, are given. In addition, the only
testable information is the fixed value of
U =
d r
N d p
N E(r
N
, p
N
)P(r
N
, p
N
) ≡ ≡E(r
N
, p
N
),
(2.2)
where E is the energy of the system. According to the principle of maximum entropy,
the distribution of the system should be
P(r
N
, p
N
) = Z(β, N , V )
−1 exp[−βE(r
N
, p
N
)],
(2.3)
where Z is a normalization factor or the partition function, and is defined as
85
worth noticing. It is not difficult to distinguish conformational states, thermodynamical states and states of statistical mechanics, but to fully understand the concepts
behind them in analysing problems is often overlooked.
2.2 Canonical Ensemble
A state of statistical mechanics of a system means a distribution over the phase space.
In general, the time scale of the dynamical evolution of molecular orbits is much
shorter than that of the evolution of distributions. A system in statistical mechanics
is always in an environment and of a large number of degrees of freedom. This makes
exactly solving the dynamics of molecular orbits impossible and also unnecessary.
On the same basis, a simple thermodynamical description exists for macroscopic
equilibrium states. As a dynamical system, a microscopic system with a large number
of degrees of freedom does not have any independent integrals of motions except for
the Hamiltonian. In the ideal case, only the Hamiltonian, and only under a statistical
average, is a conservative observable of the system. As a result of the wildly different
time scales of the orbits and distributions, time plays a less important role in the
equilibrium statistical mechanics. A principle of statistical mechanics implied by
statistical laws states that a so-called canonical distribution depending only on the
Hamiltonian exists for a system in thermodynamical equilibrium.
In 1902 Gibbs introduced the concept of ensemble, which is a collection of copies
of a physical system under certain statistical distribution. Gibbs’ theory of ensembles is axiomatic. The ensemble is not a physical thing. Gibbs did not attempt to
derive distributions; the canonical ensemble is introduced by postulation. He merely
designed a logical apparatus that produces relations which can be interpreted as analogues to the thermodynamical ones. The canonical distribution may be explained
with the principle of maximum entropy, which claims that the probability distribution
best representing the current state of knowledge, in the context of precisely stated
testable information, is the one with largest entropy [4].
Suppose that the external parameters determining the Hamiltonian of a system,
such as the number of particles and the volume, are given. In addition, the only
testable information is the fixed value of
U =
d r
N d p
N E(r
N
, p
N
)P(r
N
, p
N
) ≡ ≡E(r
N
, p
N
),
(2.2)
where E is the energy of the system. According to the principle of maximum entropy,
the distribution of the system should be
P(r
N
, p
N
) = Z(β, N , V )
−1 exp[−βE(r
N
, p
N
)],
(2.3)
where Z is a normalization factor or the partition function, and is defined as