2 On the Foundational Principles of Statistical Mechanics
89
the crucial reason for this non-equivalence. The energy of a thermodynamical system fluctuates around its internal energy. When simulating Hamiltonian dynamics
in molecular dynamics, the energy is conservative, but only when the total energy
of the system is kept near its internal energy is the simulation meaningful. However,
the internal energy of a system is usually unknown in advance, and uncontrollable.
In order to meet a preset temperature, energy is often adjusted by scaling the average
kinetic energy or by means of a simulated thermostat.
2.4 Phase Transition and Breaking of Ergodicity
The Ising model at temperature T = 0 exhibits a state with the lowest energy, where
all spins are parallel. Due to the ferromagnetic interaction J > 0, the same orientation
of nearby spins favours energy, but disfavours entropy. However, at a low temperature
the contribution of entropy to free energy is suppressed, which makes it possible for
the spin orientation to be consistent across a macroscopic distance. That is, a long
range order or coherent spin orientation occurs, and M ≡ ≡
i σ i does not vanish
even in the absence of a field. This phenomenon is known as spontaneous magnetization. The highest temperature allowing this to appear is the critical temperature of
the Ising model.
The Ising model in the absence of a field is symmetric with respect to the spin
orientation being up and down. As a consequence of this symmetry, the exact calculation of M always results in zero since for every conformation with a total spin
m =
i σ i being positive there must be a symmetric conformation of negative m,
and they cancel each other [7]. Then how does the spontaneous magnetization occur?
One mechanism is to introduce an ‘auxiliary field’, which approaches zero after an
initial nonsymmetrical distribution is formed. The breaken symmetry remains after
the auxiliary field is removed. However, a more natural mechanism is the breaking
of ergodicity, which is discussed below.
By summing over the Boltzmann factors of the conformations whose total spin
m =
i σ i is set at a specified value μ, we may define a ‘state-sum’ y(μ) and a
‘sub-free-energy’ g(μ) as follows:
y(μ) =
m=μ
e
βE({σ i })
, g(μ) ≡ −T log y(μ).
Obviously, the partition function equals Y (T , h, N ) =
μ y(μ) =
μ e
−βg(μ) , and
y(μ)/Y is the probability to observe the conformations with total spin μ. From the
above analysis of the spontaneous magnetization, function g(μ) which we regard as
a one-dimensional effective potential should behave as follows: it is a single well
at high temperature, but becomes a double-well when the temperature is decreased
below the critical temperature. An external field can make the potential asymmetric,
such that the depths of the two wells will be unequal. Once a barrier appears in the
89
the crucial reason for this non-equivalence. The energy of a thermodynamical system fluctuates around its internal energy. When simulating Hamiltonian dynamics
in molecular dynamics, the energy is conservative, but only when the total energy
of the system is kept near its internal energy is the simulation meaningful. However,
the internal energy of a system is usually unknown in advance, and uncontrollable.
In order to meet a preset temperature, energy is often adjusted by scaling the average
kinetic energy or by means of a simulated thermostat.
2.4 Phase Transition and Breaking of Ergodicity
The Ising model at temperature T = 0 exhibits a state with the lowest energy, where
all spins are parallel. Due to the ferromagnetic interaction J > 0, the same orientation
of nearby spins favours energy, but disfavours entropy. However, at a low temperature
the contribution of entropy to free energy is suppressed, which makes it possible for
the spin orientation to be consistent across a macroscopic distance. That is, a long
range order or coherent spin orientation occurs, and M ≡ ≡
i σ i does not vanish
even in the absence of a field. This phenomenon is known as spontaneous magnetization. The highest temperature allowing this to appear is the critical temperature of
the Ising model.
The Ising model in the absence of a field is symmetric with respect to the spin
orientation being up and down. As a consequence of this symmetry, the exact calculation of M always results in zero since for every conformation with a total spin
m =
i σ i being positive there must be a symmetric conformation of negative m,
and they cancel each other [7]. Then how does the spontaneous magnetization occur?
One mechanism is to introduce an ‘auxiliary field’, which approaches zero after an
initial nonsymmetrical distribution is formed. The breaken symmetry remains after
the auxiliary field is removed. However, a more natural mechanism is the breaking
of ergodicity, which is discussed below.
By summing over the Boltzmann factors of the conformations whose total spin
m =
i σ i is set at a specified value μ, we may define a ‘state-sum’ y(μ) and a
‘sub-free-energy’ g(μ) as follows:
y(μ) =
m=μ
e
βE({σ i })
, g(μ) ≡ −T log y(μ).
Obviously, the partition function equals Y (T , h, N ) =
μ y(μ) =
μ e
−βg(μ) , and
y(μ)/Y is the probability to observe the conformations with total spin μ. From the
above analysis of the spontaneous magnetization, function g(μ) which we regard as
a one-dimensional effective potential should behave as follows: it is a single well
at high temperature, but becomes a double-well when the temperature is decreased
below the critical temperature. An external field can make the potential asymmetric,
such that the depths of the two wells will be unequal. Once a barrier appears in the