Universality class for a nonequilibrium state of matter: A <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn><mml:mo>−</mml:mo><mml:mi>ε</mml:mi></mml:mrow></mml:math> expansion study of Malthusian flocks
Leiming Chen, Chiu Fan Lee, John Toner
Abstract
We show that ``Malthusian flocks''---i.e., coherently moving collections of self-propelled entities (such as living creatures) which are being ``born'' and ``dying'' during their motion---belong to a new universality class in spatial dimensions $d>2$. We calculate the universal exponents and scaling laws of this new universality class to $O(\ensuremath{\epsilon})$ in a $d=4\ensuremath{-}\ensuremath{\epsilon}$ expansion and find these are different from the ``canonical'' exponents previously conjectured to hold for ``immortal'' flocks (i.e., those without birth and death) and shown to hold for incompressible flocks with spatial dimensions in the range of $2<d\ensuremath{\le}4$. We also obtain a universal amplitude ratio relating the damping of transverse and longitudinal velocity and density fluctuations in these systems. Furthermore, we find a universal separatrix in real space ($\mathbf{r}$) between two regions in which the equal-time density correlation $\ensuremath{\langle}\ensuremath{\delta}\ensuremath{\rho}(\mathbf{r},t)\ensuremath{\delta}\ensuremath{\rho}(0,t)\ensuremath{\rangle}$ has opposite signs. Our expansion should be quite accurate in $d=3$, allowing precise quantitative comparisons between our theory, simulations, and experiments.