Notes on micro-continua exhibiting quantum effects
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APPLIED MATHEMATICS AND MECHANICS (ENGLISH EDITION) https://doi.org/10.1007/s10483-020-2678-6
Notes on micro-continua exhibiting quantum effects∗ Heng XIAO† College of Mechanics and Construction Engineering, MOE Lab of Disaster Forecast and Control in Engineering, Jinan University, Guangzhou 510632, China (Received Jul. 27, 2020 / Revised Aug. 10, 2020)
1
Uniqueness concerning exact linearization
odinger equation governing the quantum effects has been shown Most recently[1–2] , the Schr¨ to be derivable as exact linearization from the following nonlinear field equations governing the dynamic responses of the newly discovered quantum-continua[1–2] : ∂ρ + ∇ · (ρu) = 0, ∂t ∂u (1) ∇q − ρ∇V = mρ + u · (∇u) , ∂t 2 q = 1 ~ ρ∇2 (ln ρ). 4m As such, exact closed-form solutions are available for the normalized density field ρ, the velocity field u, the internal stress field q, etc. However, there arise the issues concerning how such solutions and whether any other solutions can be derived from Eq. (1). These issues have been left open before and will be treated here with short notes and corrigenda. First, from Eq. (1)3 with Planck’s constant ~ and the mass m, it may be deduced that ∇q =
1 ~2 2 ρ∇(|∇(ln ρ)| + 2∇2 (ln ρ)). 8m
Therefore, Eq. (1)2 may be recast as follows: 1 ~2 ∂u 1 2 2 + u · (∇u) = ∇ (|∇(ln ρ)| + 2∇ (ln ρ)) − V . ∂t 8 m2 m
(2)
(3)
Then, since it follows from Eq. (3) that the acceleration field is irrotational, an irrotational velocity field, u = ∇Θ, is accordingly sought as a solution to Eq. (3) with ∂Θ 1 1 ~2 1 2 2 + |∇Θ| = (|∇(ln ρ)| + 2∇2 (ln ρ)) − V. 2 ∂t 2 8m m
(4)
∗ Citation: XIAO, H. Notes on micro-continua exhibiting quantum effects. Applied Mathematics and Mechanics (English Edition) (2020) https://doi.org/10.1007/s10483-020-2678-6 † Corresponding author, E-mail: [email protected] Project supported by the National Natural Science Foundation of China (No. 11372172) and the Start-up Fund from Jinan University, Guangzhou, China c
Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2020
2
Heng XIAO
Given ρ and V , from the Cauchy-Kovalevskaya theorem, it is inferred that Eq. (4) with an initial value provides a unique solution for Θ. Again, from this theorem, it is deduced that u = ∇Θ as a solution to Eq. (3) is unique. Hence, Eq. (1) can be equivalently reduced to ∂ρ + ∇ · (ρ∇Θ) = 0, ∂t 2 ∂Θ + 1 |∇Θ|2 = 1 ~ (|∇(ln ρ)|2 + 2∇2 (ln ρ)) − 1 V ∂t 2 8 m2 m
(5)
with u = ∇Θ and Eq. (1)3 . Given the potential field V , the above system governing both ρ and Θ can further be reduced to Eqs. (24) and (25) in Ref. [2] via the following transformation with two field variables R and S: S ~ ρ = R2 + S 2 , Θ = arctan . (6) m R Thus, the first three of Eq. (19) in Ref. [2] are derived as all possible solutions to Eq. (1) with the complex field variable Ψ = R + iS therein governed by the Schr¨odinger equation.
2
Realist approach to quantum effects
With the discovery of the quantum-continua[1–2] , it turns out that the quantum
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