Chapter 8
Synchronization of Coupled
Oscillators—Phase Transitions
and Entropy Production
Steven Yuvan and Martin Bier
Abstract Over the last half century, a good understanding of liquid-gas phase transitions and magnetization phase transitions has been developed. After an order parameter, r , is defined, it can be derived how r = 0 for T > T c and how r ∝ (T c − T )
γ at
lowest order for T < T c . Here T is the temperature and T c represents a critical temperature. The value of γ appears to not depend on physical details of the system, but very
much on dimensionality. No phase transitions exist for one-dimensional systems. For
systems of four or more dimensions, each unit is interacting with sufficiently many
neighbors to warrant a mean-field approach. The mean-field approximation leads to
γ = 1/2. In this article we formulate a realistic, nonequilibrium system of coupled
oscillators. Each oscillator moves forward through a cyclic 1D array of n states and
the rate at which an oscillator proceeds from state i to state i + 1 depends on the
populations in states i + 1 and i − 1. We study how the phase transitions occur from
a homogeneous distribution over the states to a clustered distribution. A clustered
distribution means that oscillators have synchronized. We define an order parameter
and we find that the critical exponent takes on the mean-field value of 1/2 for any
number of states n. However, as n increases, the phase transition occurs for ever
smaller values of T c . We present rigorous mathematics and simple approximations
to develop an understanding of the phase transitions in this system. We explain why
and how the critical exponent value of 1/2 is expected to be robust and we discuss a
wet-lab experimental setup to substantiate our findings.
S. Yuvan · M. Bier (B)
Department of Physics, East Carolina University, Greenville, NC 27858, USA
e-mail: bierm@ecu.edu
© Springer Nature Switzerland AG 2021
A. Stefanovska and P. V. E. McClintock (eds.), Physics of Biological
Oscillators, Understanding Complex Systems,
https://doi.org/10.1007/978-3-030-59805-1_8
131
Synchronization of Coupled
Oscillators—Phase Transitions
and Entropy Production
Steven Yuvan and Martin Bier
Abstract Over the last half century, a good understanding of liquid-gas phase transitions and magnetization phase transitions has been developed. After an order parameter, r , is defined, it can be derived how r = 0 for T > T c and how r ∝ (T c − T )
γ at
lowest order for T < T c . Here T is the temperature and T c represents a critical temperature. The value of γ appears to not depend on physical details of the system, but very
much on dimensionality. No phase transitions exist for one-dimensional systems. For
systems of four or more dimensions, each unit is interacting with sufficiently many
neighbors to warrant a mean-field approach. The mean-field approximation leads to
γ = 1/2. In this article we formulate a realistic, nonequilibrium system of coupled
oscillators. Each oscillator moves forward through a cyclic 1D array of n states and
the rate at which an oscillator proceeds from state i to state i + 1 depends on the
populations in states i + 1 and i − 1. We study how the phase transitions occur from
a homogeneous distribution over the states to a clustered distribution. A clustered
distribution means that oscillators have synchronized. We define an order parameter
and we find that the critical exponent takes on the mean-field value of 1/2 for any
number of states n. However, as n increases, the phase transition occurs for ever
smaller values of T c . We present rigorous mathematics and simple approximations
to develop an understanding of the phase transitions in this system. We explain why
and how the critical exponent value of 1/2 is expected to be robust and we discuss a
wet-lab experimental setup to substantiate our findings.
S. Yuvan · M. Bier (B)
Department of Physics, East Carolina University, Greenville, NC 27858, USA
e-mail: bierm@ecu.edu
© Springer Nature Switzerland AG 2021
A. Stefanovska and P. V. E. McClintock (eds.), Physics of Biological
Oscillators, Understanding Complex Systems,
https://doi.org/10.1007/978-3-030-59805-1_8
131