From a microscopic inertial active matter model to the Schrödinger equation

Abstract

Active field theories, such as the paradigmatic model known as ‘active model B+’, are simple yet very powerful tools for describing phenomena such as motility-induced phase separation. No comparable theory has been derived yet for the underdamped case. In this work, we introduce active model I+, an extension of active model B+ to particles with inertia. The governing equations of active model I+ are systematically derived from the microscopic Langevin equations. We show that, for underdamped active particles, thermodynamic and mechanical definitions of the velocity field no longer coincide and that the density-dependent swimming speed plays the role of an effective viscosity. Moreover, active model I+ contains an analog of the Schrödinger equation in Madelung form as a limiting case, allowing one to find analoga of the quantum-mechanical tunnel effect and of fuzzy dark matter in active fluids. We investigate the active tunnel effect analytically and via numerical continuation.

Free and enclosed inertial active gas

“…Fig.2Comparison between theoretical results (black-dashed lines) for the mean-square displacement given by eqn(13), and Langevin dynamics simulations (solid lines) of three IABPs gases, equipped with Pe = U 2 t R /D T = 1 Â 10 4 , S R = {2.47, 3.7, 4.99} and S T = {6.59, 9.88, 13.18}. The theoretical non-inertial case (S R = 0) is also illustrated as a pink-dashed line.…”

“…This coefficient grows as rotational inertia increases, hence the MSD of an inertial ABP also grows with inertia, as intuitively expected. To validate eqn (13), Langevin dynamics simulations employing a Verlettype method were carried out. For the simulations, we define t ˜= t/ t R , v ˜= v/U, x ˜= x/Ut R , and used a total time T = 360, N = 2000 particles, a time step Dt ˜= 1 Â 10 À2 , and a Peclet number, Pe = U 2 t R /D T = 1 Â 10 4 , which is the ratio of advection due to activity and diffusion.…”

“…thus reducing eqn (13) to the well-known long-time MSD expression for overdamped ABPs, namely hxÁxi = (U 2 /D O )t + 6D T t.…”

“…Hecht et al 12 introduced the idea of an inertial active refrigerator which uses inertia to reduce by two orders of magnitude its kinetic energy. Finally Vrugt et al 13 proposed a connection between the Schrodinger equation and active matter with inertia.…”

Thermodynamics of an Empty Box

“…As will be shown, applying elementary operations to a thermodynamic model for the box results in several equations that formally resemble known equations used in special and general relativity, as well as quantum mechanics. Analogies between general relativity and condensed matter systems [23] are known to be useful for investigating gravitational systems, and finding analogies between systems from classical statistical mechanics and quantum mechanics is a rapidly growing field of research [24]. Moreover, the successes of entropic gravity [11] indicate that the mathematical relation between thermodynamics and general relativity might not be merely an analogy, but reflect a deeper connection.…”