84
W.-M. Zheng
Gibbs’ ensemble concept, and the partition function to which the concept of
ensemble naturally leads, can be regarded as an axiomatic representation of statistical mechanics. From the very beginning, he avoided the problem of the origin of
statistical distributions. In this article several problems concerning the fundamental
principles of statistical mechanics will be briefly discussed.
2.1 Statistical Laws
Consider a system of volume V containing N particles. Assuming it follows Hamiltonian dynamics, its state is described by the positions and momenta of the particles:
(r 1 , r 2 , . . . , r N ; p 1 , p 2 , . . . , p N ) ≡ (r
N
, p
N
), which is also called a microscopic conformation, or simply conformation. A conformation corresponds to a point in the
6N -dimensional phase space spanned by r
N and p
N . Suppose the Hamiltonian of the
system is H(r
N
, p
N
) = K(p
N
) + U (r
N
), then the equations of motion are
˙
r j =
∂H
∂p j
,
˙
p j = −
∂H
∂r j
.
(2.1)
The change of conformations in time depicts a trajectory in the phase space, or a
molecular orbit.
It is easy to write down the equations of motion, but it is impossible to solve or
integrate the equations for any given initial conditions due to the extremely large
number of degrees of freedom. However, this large number of degrees of freedom
leads to totally new laws. As an object of the thermodynamics, a macroscopic system is always in some environment. Intrinsic (due to the unpredictability of chaotic
dynamics) and extrinsic (from environmental noises) complications ‘mix’ up the
molecular orbits. The picture of molecular orbit is destroyed, and an exact solution to the dynamics is no more necessary. New laws, which are statistical ones,
appear in macroscopic systems. For example, the number of particles within a large
enough volume element inside a container is rather steady. The laws result in the
phenomenon observed in thermodynamics: a large system behaves quite simply and
regularly, and can be characterized with a few independent variables. The language
of molecular orbit is then replaced by that of probability distribution. A new kind
of states, the state of statistical mechanics, can be defined as a distribution over the
phase space specifying the probability for a system to appear around any point of
the phase space. It should be emphasized that such statistical properties are independent of any details of the underlying microscopic laws. Whether they are classical
or quantum, the framework of the theory of statistical mechanics does not change.
States of thermodynamics are the macroscopic states in thermal equilibrium. They
form the space of thermodynamical states, in which a path corresponds to a process
of thermodynamics. As for quantities in thermodynamics, some can be obtained
by averaging with the help of distributions of statistical mechanics; others are not
averages and should be derived directly and entirely from the distributions, which is
W.-M. Zheng
Gibbs’ ensemble concept, and the partition function to which the concept of
ensemble naturally leads, can be regarded as an axiomatic representation of statistical mechanics. From the very beginning, he avoided the problem of the origin of
statistical distributions. In this article several problems concerning the fundamental
principles of statistical mechanics will be briefly discussed.
2.1 Statistical Laws
Consider a system of volume V containing N particles. Assuming it follows Hamiltonian dynamics, its state is described by the positions and momenta of the particles:
(r 1 , r 2 , . . . , r N ; p 1 , p 2 , . . . , p N ) ≡ (r
N
, p
N
), which is also called a microscopic conformation, or simply conformation. A conformation corresponds to a point in the
6N -dimensional phase space spanned by r
N and p
N . Suppose the Hamiltonian of the
system is H(r
N
, p
N
) = K(p
N
) + U (r
N
), then the equations of motion are
˙
r j =
∂H
∂p j
,
˙
p j = −
∂H
∂r j
.
(2.1)
The change of conformations in time depicts a trajectory in the phase space, or a
molecular orbit.
It is easy to write down the equations of motion, but it is impossible to solve or
integrate the equations for any given initial conditions due to the extremely large
number of degrees of freedom. However, this large number of degrees of freedom
leads to totally new laws. As an object of the thermodynamics, a macroscopic system is always in some environment. Intrinsic (due to the unpredictability of chaotic
dynamics) and extrinsic (from environmental noises) complications ‘mix’ up the
molecular orbits. The picture of molecular orbit is destroyed, and an exact solution to the dynamics is no more necessary. New laws, which are statistical ones,
appear in macroscopic systems. For example, the number of particles within a large
enough volume element inside a container is rather steady. The laws result in the
phenomenon observed in thermodynamics: a large system behaves quite simply and
regularly, and can be characterized with a few independent variables. The language
of molecular orbit is then replaced by that of probability distribution. A new kind
of states, the state of statistical mechanics, can be defined as a distribution over the
phase space specifying the probability for a system to appear around any point of
the phase space. It should be emphasized that such statistical properties are independent of any details of the underlying microscopic laws. Whether they are classical
or quantum, the framework of the theory of statistical mechanics does not change.
States of thermodynamics are the macroscopic states in thermal equilibrium. They
form the space of thermodynamical states, in which a path corresponds to a process
of thermodynamics. As for quantities in thermodynamics, some can be obtained
by averaging with the help of distributions of statistical mechanics; others are not
averages and should be derived directly and entirely from the distributions, which is