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

TL;DR

This paper establishes a theoretical link between classical smoothed-particle hydrodynamics and quantum mechanics for superfluid helium-4, enabling accurate simulations of quantum phenomena like vortex lattices under certain conditions.

Contribution

It derives and compares SPH-based motion equations for superfluid helium-4 from classical and quantum perspectives, demonstrating their equivalence under specific thermodynamic conditions.

Findings

Equivalence of classical and quantum equations when internal energy per particle is zero.

Quantum pressure effects can be incorporated if quantum pressure gradient balances mutual friction.

Successful simulation of vortex lattice formation in superfluid helium-4.

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 Schrodinger 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 nonnegligible 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 cms-1 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.