Helmholtz decomposition and potential functions for n-dimensional analytic vector fields

Making Sense of Neural Networks in the Light of Evolutionary Optimization

“…This equation can be rewritten in terms of the flux J defined as in the following way: If the probability distribution over time reaches a stationary (unchanging) state, then the probability density in such a state P stat ( x ) is determined by the condition: where As shown in the Appendix, equation (11) implies that J stat can be written in general as where A i ( x ) are rows of an anti-symmetric matrix 𝔸( x ) called a rotation potential, and ROT is a multidimensional generalization of the common three-dimensional curl operator. 33 Therefore, the dynamics of the whole system [namely, the functions forming the vector f ( x )] can be parameterized, as follows from equations (12) and (13), in terms of P stat ( x ) and 𝔸( x ): It is convenient to introduce a potential energy function u ( x ) by analogy with the Boltzmann distribution in statistical physics: [recall that σ 2 is an analog of the temperature in units of energy; the partition function is absorbed here into an additive constant in the definition of u ( x )]. Then equation (14) transforms to Consider its expansion in terms of σ 2 as a small parameter.…”

“…where Ai(x) are rows of an anti-symmetric matrix 𝔸(𝒙) called a rotation potential, and ROT is a multidimensional generalization of the common three-dimensional curl operator. 33 Therefore, the dynamics of the whole system [namely, the functions forming the vector f(x)] can be parameterized, as follows from equations ( 12) and ( 13), in terms of Pstat(x) and 𝔸(𝒙):…”

“…A generalization of the Helmholtz decomposition in a space with an arbitrary number of dimensions (not necessarily 3) leads to the statement that a d -dimensional (with d = M + N to agree with the main text) vector field J ( x ) can be represented as a sum of two terms: 33 where the first term ™∇Φ( x ) is an irrotational gradient field, which can be written as the negative gradient of the gradient potential Φ( x ) (unlike the cited paper, we use the physical convention, with the negative sign before the gradient), and r ( x ) is a solenoidal (divergence-free) rotation field. A sufficient (but not necessary) condition for equation (44) to hold is that J ∈ C 2 (ℝ d , ℝ d ) and decays faster than | x | - δ for | x | → ∞ with δ > 2.…”

Variable thermal conductivity in context of Green‐Naghdi theory of thermo‐microstretch solids

“…These functions are used in many applications of mathematical physics to substitute a sufficiently smooth vector field into an irrotational (curl-free) gradient field and a solenoidal (divergence-free) rotation field [49]. Potential functions allow the decoupling of the equations of motion and help to reduce the order of differential equation.…”

“…Let us introduce the potential function $$\psi _1$$ and $$\psi _2$$ as $$$$\begin{align} &u_1=\frac{\partial \psi _1}{\partial x}+\frac{\partial \psi _2}{\partial y},\quad u_3=\frac{\partial \psi _1}{\partial y}-\frac{\partial \psi _2}{\partial x}. \end{align}$$$$These functions are used in many applications of mathematical physics to substitute a sufficiently smooth vector field into an irrotational (curl‐free) gradient field and a solenoidal (divergence‐free) rotation field [49]. Potential functions allow the decoupling of the equations of motion and help to reduce the order of differential equation.…”