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Register a new userOrder parameter allows classification of planar graphs based on balanced fixed points in the Kuramoto model
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Phase balanced states are a highly under-explored class of solutions of the Kuramoto model and other coupled oscillator models on networks. So far, coupled oscillator research focused on phase synchronized solutions. Yet, global constraints on oscillators may forbid synchronized state, rendering phase balanced states as the relevant stable state. If for example oscillators are driving the contractions of a fluid filled volume, conservation of fluid volume constraints oscillators to balanced states as characterized by a vanishing Kuramoto order parameter. It has previously been shown that stable, balanced patterns in the Kuramoto model exist on circulant graphs. However, which non-circulant graphs first of all allow for balanced states and what characterizes the balanced states is unknown. Here, we derive rules of how to build non-circulant, planar graphs allowing for balanced states from the simple cycle graph by adding loops or edges to it. We thereby identify different classes of small planar networks allowing for balanced states. Investigating the balanced states characteristics, we find that the variance in basin stability scales linearly with the size of the graph for these networks. We introduce the balancing ratio as a new order parameter based on the basin stability approach to classify balanced states on networks and evaluate it analytically for a subset of the network classes. Our results offer an analytical description of non-circulant graphs supporting stable, balanced states and may thereby help to understand the topological requirements on oscillator networks under global constraints.
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Several important biological processes are initiated by the binding of a protein to a specific site on the DNA. The strategy adopted by a protein, called transcription factor (TF), for searching its specific binding site on the DNA has been investigated over several decades. In recent times the effects obstacles, like DNA-binding proteins, on the search by TF has begun to receive attention. RNA polymerase (RNAP) motors collectively move along a segment of the DNA during a genomic process called transcription. This RNAP traffic is bound to affect the diffusive scanning of the same segment of the DNA by a TF searching for its binding site. Motivated by this phenomenon, here we develop a kinetic model where a `particle, that represents a TF, searches for a specific site on a one-dimensional lattice. On the same lattice another species of particles, each representing a RNAP, hop from left to right exactly as in a totally asymmetric simple exclusion process (TASEP) which forbids simultaneous occupation of any site by more than one particle, irrespective of their identities. Although the TF is allowed to attach to or detach from any lattice site, the RNAPs can attach only to the first site at the left edge and detach from only the last site on the right edge of the lattice. We formulate the search as a {it first-passage} process; the time taken to reach the target site {it for the first time}, starting from a well defined initial state, is the search time. By approximate analytical calculations and Monte Carlo (MC) computer simulations, we calculate the mean search time. We show that RNAP traffic rectifies the diffusive motion of TF to that of a Brownian ratchet, and the mean time of successful search can be even shorter than that required in the absence of RNAP traffic. Moreover, we show that there is an optimal rate of detachment that corresponds to the shortest mean search time.
A swarm of preys when attacked by a predator is known to rely on their cooperative interactions to escape. Understanding such interactions of collectively moving preys and the emerging patterns of their escape trajectories still remain elusive. In this paper, we investigate how the range of cooperative interactions within a prey group affects the survival chances of the group while chased by a predator. As observed in nature, the interaction range of preys may vary due to their vision, age, or even physical structure. Based on a simple theoretical prey-predator model, here, we show that an optimality criterion for the survival can be established on the interaction range of preys. Very short range or long range interactions are shown to be inefficient for the escape mechanism. Interestingly, for an intermediate range of interaction, survival probability of the prey group is found to be maximum. Our analysis also shows that the nature of the escape trajectories strongly depends on the range of interactions between preys and corroborates with the naturally observed escape patterns. Moreover, we find that the optimal survival regime depends on the prey group size and also on the predator strength.
In this paper we propose and realize (the code is publicly available at https://github.com/Thrawn1985/2D-Partition-Function) an algorithm for exact calculation of partition function for planar graph models with binary spins. The complexity of the algorithm is O(N^2). Test experiments shows good agreement with Onsagers analytical solution for two-dimensional Ising model of infinite size.
Driven-dissipative systems are expected to give rise to non-equilibrium phenomena that are absent in their equilibrium counterparts. However, phase transitions in these systems generically exhibit an effectively classical equilibrium behavior in spite of their non-equilibrium origin. In this paper, we show that multicritical points in such systems lead to a rich and genuinely non-equilibrium behavior. Specifically, we investigate a driven-dissipative model of interacting bosons that possesses two distinct phase transitions: one from a high- to a low-density phase---reminiscent of a liquid-gas transition---and another to an antiferromagnetic phase. Each phase transition is described by the Ising universality class characterized by an (emergent or microscopic) $mathbb{Z}_2$ symmetry. They, however, coalesce at a multicritical point, giving rise to a non-equilibrium model of coupled Ising-like order parameters described by a $mathbb{Z}_2 times mathbb{Z}_2$ symmetry. Using a dynamical renormalization-group approach, we show that a pair of non-equilibrium fixed points (NEFPs) emerge that govern the long-distance critical behavior of the system. We elucidate various exotic features of these NEFPs. In particular, we show that a generic continuous scale invariance at criticality is reduced to a discrete scale invariance. This further results in complex-valued critical exponents and spiraling phase boundaries, and it is also accompanied by a complex Liouvillian gap even close to the phase transition. As direct evidence of the non-equilibrium nature of the NEFPs, we show that the fluctuation-dissipation relation is violated at all scales, leading to an effective temperature that becomes hotter and hotter at longer and longer wavelengths. Finally, we argue that this non-equilibrium behavior can be observed in cavity arrays with cross-Kerr nonlinearities.
We compute the partition function of the $q$-states Potts model on a random planar lattice with $pleq q$ allowed, equally weighted colours on a connected boundary. To this end, we employ its matrix model representation in the planar limit, generalising a result by Voiculescu for the addition of random matrices to a situation beyond free probability theory. We show that the partition functions with $p$ and $q-p$ colours on the boundary are related algebraically. Finally, we investigate the phase diagram of the model when $0leq qleq 4$ and comment on the conformal field theory description of the critical points.
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