Istituto Sistemi Complessi, Consiglio Nazionale delle Ricerche, UOS Sapienza, 00185 Rome, Italy.
Dipartimento di Fisica, Università Sapienza, 00185 Rome, Italy.
Phys Rev E. 2019 Dec;100(6-1):062130. doi: 10.1103/PhysRevE.100.062130.
Motivated by the collective behavior of biological swarms, we study the critical dynamics of field theories with coupling between order parameter and conjugate momentum in the presence of dissipation. Under a fixed-network approximation, we perform a dynamical renormalization group calculation at one loop in the near-critical disordered region, and we show that the violation of momentum conservation generates a crossover between an unstable fixed point, characterized by a dynamic critical exponent z=d/2, and a stable fixed point with z=2. Interestingly, the two fixed points have different upper critical dimensions. The interplay between these two fixed points gives rise to a crossover in the critical dynamics of the system, characterized by a crossover exponent κ=4/d. The crossover is regulated by a conservation length scale R_{0}, given by the ratio between the transport coefficient and the effective friction, which is larger as the dissipation is smaller: Beyond R_{0}, the stable fixed point dominates, while at shorter distances dynamics is ruled by the unstable fixed point and critical exponent, a behavior which is all the more relevant in finite-size systems with weak dissipation. We run numerical simulations in three dimensions and find a crossover between the exponents z=3/2 and z=2 in the critical slowdown of the system, confirming the renormalization group results. From the biophysical point of view, our calculation indicates that in finite-size biological groups mode coupling terms in the equation of motion can significantly change the dynamical critical exponents even in the presence of dissipation, a step toward reconciling theory with experiments in natural swarms. Moreover, our result provides the scale within which fully conservative Bose-Einstein condensation is a good approximation in systems with weak symmetry-breaking terms violating number conservation, as quantum magnets or photon gases.
受生物群体的集体行为启发，我们研究了在存在耗散的情况下，具有序参量和共轭动量之间耦合的场论的临界动力学。在固定网络近似下，我们在近临界无序区域进行了一个循环的动态重整化群计算，结果表明，动量守恒的违反会导致不稳定固定点和稳定固定点之间的交叉，其中不稳定固定点的特征是动态临界指数 z=d/2，而稳定固定点的特征是 z=2。有趣的是，这两个固定点具有不同的上临界维度。这两个固定点之间的相互作用导致了系统临界动力学的交叉，其特征是交叉指数 κ=4/d。交叉由一个守恒长度尺度 R_0 来调节，R_0 由输运系数和有效摩擦之间的比值给出，随着耗散的减小，R_0 会增大：在 R_0 之外，稳定固定点占主导地位，而在较短的距离内，动力学由不稳定固定点和临界指数决定，这种行为在具有弱耗散的有限尺寸系统中更为相关。我们在三维空间中进行了数值模拟，并在系统的临界减速中发现了指数 z=3/2 和 z=2 之间的交叉，这证实了重整化群的结果。从生物物理的角度来看，我们的计算表明，在有限尺寸的生物群体中，即使存在耗散，运动方程中的模式耦合项也可以显著改变动力学临界指数，这是朝着在自然群体中协调理论与实验迈出的一步。此外，我们的结果提供了一个尺度，在这个尺度内，对于具有弱对称性破坏项的系统，完全保守的玻色-爱因斯坦凝聚在违反数守恒的情况下是一个很好的近似，例如量子磁体或光子气体。
