90
W.-M. Zheng
potential, its height should be ∼ N
(d −1)/d by an estimation from the surface area
of magnetic domain. Provided the height is scaled as some positive power of N , it
approaches infinity at the thermodynamic limit. Besides the thermodynamic limit,
an equilibrium state involves the dynamic limit that time approaches infinity. An
infinitely high potential barrier would result in nonequivalence of the ordering of the
two limits:
lim
t→∞
lim
V →∞
,
lim
V →∞
lim
t→∞
.
In the latter ordering, as long as the temperature is not extremely low, the system has
a chance to visit both sides of the barrier, but this is not the case when the former
ordering is taken. As for a real system, both its volume and the observation time are
finite. Which ordering should be taken depends on the specific undergoing process.
In a word, the phase transition accompanies the breaking of ergodicity.
In numerical simulations for gas or liquid, a uniform initial conformation is usually
taken. If the parameters of the system correspond to the region of the gas-liquid
phase transition, in general no gas-liquid coexistence can be seen, in analog to the
case of the Ising model. If conformations of coexistence phase are wanted, say in
simulation of interface behavior, quite different initial conditions must be chosen.
For example, we can let all particles be locate in one side of the container, and then
relax the system. The most remarkable difference between gas and liquid phases is
their densities, especially at a temperature far from the critical point. When a proper
cutoff radius is chosen, a criterion to associate a particle with a gas or liquid phase
may be the number of particles inside the cutoff sphere centered at the reference
particle. As the demonstration of the breaking of ergodicity, the two opposite kind of
initial conditions mentioned above behave quite differently in evolution. A particle
attributed to the gaseous state will remain in the gas phase for a long period, and
so will a particle attributed to the liquid state remain in the liquid phase. When the
size of the simulated system is not very large, the phenomenon of the ergodicity
breaking is not so conspicuous. On increasing the size of the system, the lifetime of
a particle in its particular gas or liquid phase will be greatly extended. It is worth
further investigating whether and how a sharp change would occur.
2.5 Thermodynamic Limit and Supporting Set of
Distribution
To prove conclusively the correspondence between statistical mechanics and thermodynamics, the existence of the thermodynamic limit is an essential prerequisite,
and this existence, demonstrated a posteriori, depends on the nature of the Hamiltonian of the system [8–11]. Taking the grand canonical ensemble as an example,
we need to prove the existence of the limit lim V →∞ V
−1 log . Lee and Yang conducted a proof of the existence of this limit for a rather general Hamiltonian. S.-G.
Ma suggested possible extensions of the proof and also pointed out the limitation of
W.-M. Zheng
potential, its height should be ∼ N
(d −1)/d by an estimation from the surface area
of magnetic domain. Provided the height is scaled as some positive power of N , it
approaches infinity at the thermodynamic limit. Besides the thermodynamic limit,
an equilibrium state involves the dynamic limit that time approaches infinity. An
infinitely high potential barrier would result in nonequivalence of the ordering of the
two limits:
lim
t→∞
lim
V →∞
,
lim
V →∞
lim
t→∞
.
In the latter ordering, as long as the temperature is not extremely low, the system has
a chance to visit both sides of the barrier, but this is not the case when the former
ordering is taken. As for a real system, both its volume and the observation time are
finite. Which ordering should be taken depends on the specific undergoing process.
In a word, the phase transition accompanies the breaking of ergodicity.
In numerical simulations for gas or liquid, a uniform initial conformation is usually
taken. If the parameters of the system correspond to the region of the gas-liquid
phase transition, in general no gas-liquid coexistence can be seen, in analog to the
case of the Ising model. If conformations of coexistence phase are wanted, say in
simulation of interface behavior, quite different initial conditions must be chosen.
For example, we can let all particles be locate in one side of the container, and then
relax the system. The most remarkable difference between gas and liquid phases is
their densities, especially at a temperature far from the critical point. When a proper
cutoff radius is chosen, a criterion to associate a particle with a gas or liquid phase
may be the number of particles inside the cutoff sphere centered at the reference
particle. As the demonstration of the breaking of ergodicity, the two opposite kind of
initial conditions mentioned above behave quite differently in evolution. A particle
attributed to the gaseous state will remain in the gas phase for a long period, and
so will a particle attributed to the liquid state remain in the liquid phase. When the
size of the simulated system is not very large, the phenomenon of the ergodicity
breaking is not so conspicuous. On increasing the size of the system, the lifetime of
a particle in its particular gas or liquid phase will be greatly extended. It is worth
further investigating whether and how a sharp change would occur.
2.5 Thermodynamic Limit and Supporting Set of
Distribution
To prove conclusively the correspondence between statistical mechanics and thermodynamics, the existence of the thermodynamic limit is an essential prerequisite,
and this existence, demonstrated a posteriori, depends on the nature of the Hamiltonian of the system [8–11]. Taking the grand canonical ensemble as an example,
we need to prove the existence of the limit lim V →∞ V
−1 log . Lee and Yang conducted a proof of the existence of this limit for a rather general Hamiltonian. S.-G.
Ma suggested possible extensions of the proof and also pointed out the limitation of