Theoretical framework bridging classical and quantum mechanics for the dynamics of cryogenic liquid helium-4 using smoothed-particle hydrodynamics

Abstract

Our recent study suggested that a fully classical mechanical approximation of the two-fluid model of superfluid helium-4 based on smoothed-particle hydrodynamics (SPH) is equivalent to solving a many-body quantum mechanical equation under specific conditions. This study further verifies the existence of this equivalence. First, we derived the SPH form of the motion equation for the superfluid component of the two-fluid model, i.e., the motion equation driven by the chemical potential gradient obtained using the Gibbs–Duhem equation. We then derived the SPH form of the motion equation for condensates based on the Gross–Pitaevskii theory, i.e., the motion equation driven by the chemical potential gradient obtained from the Schrödinger equation of interacting bosons. Following this, we compared the two discretized equations. Consequently, we discovered that a condition maintaining zero internal energy for each fluid particle ensures the equivalence of the equations when the quantum pressure is negligible. Moreover, their equivalence holds even when the quantum pressure is non-negligible if the quantum pressure gradient force equals the mutual friction force. A zero internal energy indicates the thermodynamic ground state, which includes an elementary excitation state. Therefore, the condition can be sufficiently satisfied when the velocities of fluid particles do not exceed the Landau critical velocity, which is not a stringent condition for simulations with a characteristic velocity of a few [Formula: see text] in a laboratory system. Based on the above, we performed a simulation of rotating liquid helium-4 and succeeded in generating a vortex lattice with quantized circulation, known as a quantum lattice.

Three-dimensional analysis of vortex-lattice formation in rotating Bose–Einstein condensates using smoothed-particle hydrodynamics

“…In addition, our recent study showed that the SPH simulation of the angular momentum-conserving two-fluid model incorporating vortex dynamics reproduces vortex lattices [13]. Our another study provided a theory that the equation of motion for inviscid fluids in the two-fluid model becomes equal to the quantum hydrodynamic equation derived from the GP equation under specific conditions in SPH form [6]. Therefore, describing the GP equation in SPH form may help to find some connection with these studies.…”

“…Because the GP equation is a nonlinear Schro dinger equation of interacting bosons, its Hamiltonian is described as a many-body interaction system [4][5][6]. Therefore, compared to other Euler methods such as finite element (FE) [7][8][9] and finite difference (FD) [10][11][12] methods, the mechanical picture of the SPH scheme is closer to real BEC physics; we can describe the GP equations as a many-body interaction system in SPH.…”