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S. Yuvan and M. Bier
and J (i+1)→(i+2) = (1/n)(1 + ε) exp [2αε/n]. Upon going from state (i + m) to
state (i + m + 1), there is a probability decrease from (1 + ε)/n to (1 − ε)/n.
This leads to J (i+m)→(i+m+1) = (1/n)(1 + ε) exp [−2αε/n] and J (i+m+1)→(i+m+2) =
(1/n)(1 − ε) exp [−2αε/n]. The remaining fluxes along the horizontal axis in Fig.
8.3b are (1 + ε)/n and (1 − ε)/n in the elevated and lowered part, respectively. They
average to 1/n as there are just as many elevated as lowered probabilities. It is readily verified that the four fluxes adjacent to the upward and downward jump average
to (1/n) cosh [2αε/n]. For any nonzero real value of x, we have cosh x > 1. This
means that having a “bump” leads to a higher throughput for the cycle in Fig. 8.3a
and to a larger production of entropy.
The phase transition from a homogeneous distribution to one with “bumps” constitutes a symmetry breaking and an establishment of an order. However, this lower
entropy structure leads to a larger throughput and a larger entropy production for
the system as a whole. We can thus view the bumps as a self-organized dissipative
structure as described by Prigogine in the 1970s [20].
The increase of four fluxes from an average of 1/n to an average of (1/n)
cosh [2αε/n] for every bump can help us understand why the phase transition is
pushed out to α → ∞ for n → ∞. We have |ε| < 1. So for n → ∞, the argument
2αε/n vanishes (leading to (1/n) cosh [2αε/n] → 1/n) and the enhanced flux disappears, unless α changes proportionally with n.
The subject matter of this subsection can be the starting point for ample mathematical analysis. There is, for instance, a large body of work on how the flux through
an entire cycle as in Fig. 8.3a is affected if just a few transitions are speeded up
[14, 16]. As Eq. (8.7) is a rough approximation already, it would be somewhat
excessive to substantially elaborate in this direction.
8.3.3 Why Mean-Field Works for a 1D System
It may at first seem surprising that a 1D system as in Fig. 8.3a gives rise to a value
of the critical exponent that is characteristic of the mean-field approximation. After
all, the 1D Ising model features no phase transition at all if the number of involved
spins, N , is finite. Only if the 1D Ising array has infinitely many spins is there a phase
transition, but it occurs at the T → 0 limit if N → ∞.
However, upon closer consideration our mean-field result makes sense. With the
system in Fig. 8.3a we face a large number of oscillators, N tot , that is distributed
over n states. For the mass action approach of Eq. (8.10) to apply, we need N tot n.
Let N i be the number of oscillators in state i. We then have p i = N i /N tot . The rate
of change of N i depends on the numbers N i−2 , N i−1 , N i , and N i+1 . In this way
every individual oscillator is interacting with infinitely many other oscillators. The
legitimacy of the mean-field approach can thus be understood.
The 1D Ising model can be evaluated analytically and the pertinent derivation
is shown in many authoritative textbooks [10, 22, 24]. A rigorous treatment shows
that at finite temperature, the magnetization (which is taken as an order parameter in
S. Yuvan and M. Bier
and J (i+1)→(i+2) = (1/n)(1 + ε) exp [2αε/n]. Upon going from state (i + m) to
state (i + m + 1), there is a probability decrease from (1 + ε)/n to (1 − ε)/n.
This leads to J (i+m)→(i+m+1) = (1/n)(1 + ε) exp [−2αε/n] and J (i+m+1)→(i+m+2) =
(1/n)(1 − ε) exp [−2αε/n]. The remaining fluxes along the horizontal axis in Fig.
8.3b are (1 + ε)/n and (1 − ε)/n in the elevated and lowered part, respectively. They
average to 1/n as there are just as many elevated as lowered probabilities. It is readily verified that the four fluxes adjacent to the upward and downward jump average
to (1/n) cosh [2αε/n]. For any nonzero real value of x, we have cosh x > 1. This
means that having a “bump” leads to a higher throughput for the cycle in Fig. 8.3a
and to a larger production of entropy.
The phase transition from a homogeneous distribution to one with “bumps” constitutes a symmetry breaking and an establishment of an order. However, this lower
entropy structure leads to a larger throughput and a larger entropy production for
the system as a whole. We can thus view the bumps as a self-organized dissipative
structure as described by Prigogine in the 1970s [20].
The increase of four fluxes from an average of 1/n to an average of (1/n)
cosh [2αε/n] for every bump can help us understand why the phase transition is
pushed out to α → ∞ for n → ∞. We have |ε| < 1. So for n → ∞, the argument
2αε/n vanishes (leading to (1/n) cosh [2αε/n] → 1/n) and the enhanced flux disappears, unless α changes proportionally with n.
The subject matter of this subsection can be the starting point for ample mathematical analysis. There is, for instance, a large body of work on how the flux through
an entire cycle as in Fig. 8.3a is affected if just a few transitions are speeded up
[14, 16]. As Eq. (8.7) is a rough approximation already, it would be somewhat
excessive to substantially elaborate in this direction.
8.3.3 Why Mean-Field Works for a 1D System
It may at first seem surprising that a 1D system as in Fig. 8.3a gives rise to a value
of the critical exponent that is characteristic of the mean-field approximation. After
all, the 1D Ising model features no phase transition at all if the number of involved
spins, N , is finite. Only if the 1D Ising array has infinitely many spins is there a phase
transition, but it occurs at the T → 0 limit if N → ∞.
However, upon closer consideration our mean-field result makes sense. With the
system in Fig. 8.3a we face a large number of oscillators, N tot , that is distributed
over n states. For the mass action approach of Eq. (8.10) to apply, we need N tot n.
Let N i be the number of oscillators in state i. We then have p i = N i /N tot . The rate
of change of N i depends on the numbers N i−2 , N i−1 , N i , and N i+1 . In this way
every individual oscillator is interacting with infinitely many other oscillators. The
legitimacy of the mean-field approach can thus be understood.
The 1D Ising model can be evaluated analytically and the pertinent derivation
is shown in many authoritative textbooks [10, 22, 24]. A rigorous treatment shows
that at finite temperature, the magnetization (which is taken as an order parameter in