The recent inflow of empirical data about the collective behaviour of strongly correlated biological systems has brought field theory and the renormalization group into the biophysical arena. Experiments on bird flocks and insect swarms show that social forces act on the particles’ velocity through the generator of its rotations, namely the spin, indicating that mode-coupling field theories are necessary to reproduce the correct dynamical behaviour. Unfortunately, a theory for three coupled fields—density, velocity and spin—has a prohibitive degree of intricacy. A simplifying path consists in getting rid of density fluctuations by studying incompressible systems. This requires imposing a solenoidal constraint on the primary field, an unsolved problem even for equilibrium mode-coupling theories. Here, we perform an equilibrium dynamic renormalization group analysis of a mode-coupling field theory subject to a solenoidal constraint; using the classification of Halperin and Hohenberg, we can dub this case as a solenoidal Model G. We demonstrate that the constraint produces a new vertex that mixes static and dynamical coupling constants, and that this vertex is essential to grant the closure of the renormalization group structure and the consistency of dynamics with statics. Interestingly, although the solenoidal constraint leads to a modification of the static universality class, we find that it does not change the dynamical universality class, a result that seems to represent an exception to the general rule that dynamical universality classes are narrower than static ones. Our results constitute a solid stepping stone in the admittedly large chasm towards developing an off-equilibrium mode-coupling theory of biological groups.

最近关于强相关生物系统的集体行为的经验数据的流入将场论和重整化组带入了生物物理领域。对鸟群和昆虫群的实验表明，社会力通过粒子的旋转发生器（即自旋）作用于粒子的速度，这表明模式耦合场理论对于再现正确的动力学行为是必要的。不幸的是，关于三个耦合场——密度、速度和自旋——的理论过于复杂。一条简化的路径在于通过研究不可压缩系统来消除密度波动。这需要对初级场施加螺线管约束，即使对于平衡模式耦合理论也是一个未解决的问题。这里，我们对受螺线管约束的模式耦合场论进行平衡动态重整化群分析；使用 Halperin 和 Hohenberg 的分类，我们可以将这种情况称为螺线管模型 G。我们证明约束产生了一个混合静态和动态耦合常数的新顶点，并且该顶点对于授予重整化组的闭包是必不可少的结构和动力学与静力学的一致性。有趣的是，虽然螺线管约束导致了静态普遍性类的修改，但我们发现它并没有改变动态普遍性类，这一结果似乎代表了动态普遍性类比静态类更窄的一般规则的一个例外。