138
S. Yuvan and M. Bier
There is no explicit Brownian noise in the system of Fig. 8.3a. However, a constant rate out of a state implies that an individual oscillator has an exponentially
distributed waiting time in that state. Effectively, this gives the system a stochasticity
and a temperature: for α = 0 the n-state system will, over time, forget any initial
distribution over the n states and evolve towards a homogeneous distribution over
the states, i.e. p i = 1/n where i = 1, 2, ... n. The parameter α denotes the coupling
strength. For α = 0 there is no coupling. As in the case of the gas-liquid transition
and the magnetization, the randomization is opposed by a coupling that drives the
system to an ordered state.
There is a fundamental difference between the liquid-gas transition (Fig. 8.1)
and the magnetization (Fig. 8.2) on the one hand and our setup (Fig. 8.3a) on the
other hand. With the first two systems we are looking at a thermal equilibrium.
The system in Fig. 8.3a, however, goes through a chemical cycle. As there are no
clockwise transitions, there is no detailed balance and there is continuous production
of entropy.
8.2 Numerical Simulation and Mathematical Analysis
8.2.1 A Heuristic Approach
A simple heuristic model for the behavior close to the phase transition leads to some
concise formulae that well approximate the actual behavior. The simple model can,
furthermore, help build intuition for the mechanisms behind the phase transition. A
less general form of the model was also presented in Ref. [5] in the context of n = 4,
i.e., oscillators going through a 4-state cycle.
As was mentioned before, when α is small the probability distribution over the
n states will over time approach p i = 1/n for all initial conditions. For sufficiently
large values of α, clusters as in Fig. 8.3b will persist. For the clustering and the
homogenization to be in balance, a “bump of length m” as in Fig. 8.3b needs to have
as much influx (J
in
i = p i k i ) as outflux (J
out
i+m = p i+m k i+m ). We are interested in the
region near the phase transition, so we consider values of the probability variation,
ε (cf. Fig. 8.3b), that are small relative to 1. For simplicity, we allow two values for
p i : (1 + ε)/n and (1 − ε)/n. Each value is taken on by half of the probabilities in
the distribution. In the vicinity of the phase transition, equating the fluxes into and
out of the “bump” leads to:
1 − ε
n
exp [2αε/n] ≈
1 + ε
n
exp [−2αε/n] .
(8.6)
This formula is only valid for a bump that consists of two or more neighboring states
with a population (1 + ε)/n. Furthermore, to the left of this bump there need to be at
least two neighboring states with a population (1 − ε)/n. Expressing α/n in terms of
S. Yuvan and M. Bier
There is no explicit Brownian noise in the system of Fig. 8.3a. However, a constant rate out of a state implies that an individual oscillator has an exponentially
distributed waiting time in that state. Effectively, this gives the system a stochasticity
and a temperature: for α = 0 the n-state system will, over time, forget any initial
distribution over the n states and evolve towards a homogeneous distribution over
the states, i.e. p i = 1/n where i = 1, 2, ... n. The parameter α denotes the coupling
strength. For α = 0 there is no coupling. As in the case of the gas-liquid transition
and the magnetization, the randomization is opposed by a coupling that drives the
system to an ordered state.
There is a fundamental difference between the liquid-gas transition (Fig. 8.1)
and the magnetization (Fig. 8.2) on the one hand and our setup (Fig. 8.3a) on the
other hand. With the first two systems we are looking at a thermal equilibrium.
The system in Fig. 8.3a, however, goes through a chemical cycle. As there are no
clockwise transitions, there is no detailed balance and there is continuous production
of entropy.
8.2 Numerical Simulation and Mathematical Analysis
8.2.1 A Heuristic Approach
A simple heuristic model for the behavior close to the phase transition leads to some
concise formulae that well approximate the actual behavior. The simple model can,
furthermore, help build intuition for the mechanisms behind the phase transition. A
less general form of the model was also presented in Ref. [5] in the context of n = 4,
i.e., oscillators going through a 4-state cycle.
As was mentioned before, when α is small the probability distribution over the
n states will over time approach p i = 1/n for all initial conditions. For sufficiently
large values of α, clusters as in Fig. 8.3b will persist. For the clustering and the
homogenization to be in balance, a “bump of length m” as in Fig. 8.3b needs to have
as much influx (J
in
i = p i k i ) as outflux (J
out
i+m = p i+m k i+m ). We are interested in the
region near the phase transition, so we consider values of the probability variation,
ε (cf. Fig. 8.3b), that are small relative to 1. For simplicity, we allow two values for
p i : (1 + ε)/n and (1 − ε)/n. Each value is taken on by half of the probabilities in
the distribution. In the vicinity of the phase transition, equating the fluxes into and
out of the “bump” leads to:
1 − ε
n
exp [2αε/n] ≈
1 + ε
n
exp [−2αε/n] .
(8.6)
This formula is only valid for a bump that consists of two or more neighboring states
with a population (1 + ε)/n. Furthermore, to the left of this bump there need to be at
least two neighboring states with a population (1 − ε)/n. Expressing α/n in terms of