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Quantum field theories (QFTs) at finite densities of matter generically involve complex actions. Standard Monte Carlo simulations based upon importance sampling, which have been producing quantitative first principle results in particle physics for almost forty years, cannot be applied in this case. Various strategies to overcome this so-called sign problem or complex action problem were proposed during the last thirty years. We here review the sign problem in lattice field theories, focusing on two more recent methods: dualization to worldline type of representations and the density-of-states approach.

The current three-dimensional averaging mathematical model of flow, also known as the Reynolds averaged Navier–Stokes equations or Reynolds equations, was developed based on the idea of Reynolds in 1895. This model is given by the classical averaging of velocity and pressure parameters from the three-dimensional Navier–Stokes equations. However, by doing this, these averaging parameters obtained by this classical approach are not generalized in comparison to ones estimated by the dual approach. This paper proposes a dual approach to establishing the three-dimensional flow equation. The model setup is more complicated than the classical model in terms of integration because the procedure can be repeated several times. In this paper, the authors perform twice: (1) first, integration of the velocity and pressure parameters from time $t$ to $t+T_m$, with time $T_m<T$, where $T$ is the repeated period of parameters; (2) second, integration from time t to $t+T$. Fluctuating quantities such as velocity and pressure in turbulent flow, over time, are simulated using trigonometric Fourier series. The three-dimensional flow model obtained from this dual approach could provide more accurate results than those given by the Reynolds averaged Navier–Stokes equations.