Diffusion with finite-helicity field tensor: A mechanism of generating heterogeneity

Topological constraints on a dynamical system often manifest themselves as breaking of the Hamiltonian structure; well-known examples are nonholonomic constraints on Lagrangian mechanics. The statistical mechanics under such topological constraints is the subject of this study. Conventional arguments based on phase spaces, Jacobi identity, invariant measure, or the H theorem are no longer applicable since all these notions stem from the symplectic geometry underlying canonical Hamiltonian systems. Remembering that Hamiltonian systems are endowed with field tensors (canonical 2-forms) that have zero helicity, our mission is to extend the scope toward the class of systems governed by finite-helicity field tensors. Here, we introduce a class of field tensors that are characterized by Beltrami vectors. We prove an $H$ theorem for this Beltrami class. The most general class of energy-conserving systems are non-Beltrami, for which we identify the “field charge” that prevents the entropy to maximize, resulting in creation of heterogeneous distributions. The essence of the theory can be delineated by classifying three-dimensional dynamics. We then generalize to arbitrary (finite) dimensions.

Figure 1

Classification of 3D conservative dynamics. The right column shows the equilibrium distribution function $f^{eq}$ of an ensemble of particles obeying (1) when $\mathbf{\nabla }H$ is a white noise process (see Sec. 3). The square brackets in the last column indicate that the dependence of $f^{eq}$ on $w$ and $\mathfrak{B}$ is not necessarily a functional one.

Figure 2

(a) Numerical integration of the Euler rotation equation. The orbit is the intersection of the surfaces $C$ and $H_0$. (b) Numerical integration of (19). If the Hamiltonian is perturbed $\mathbf{\nabla }H=\mathbf{\nabla }{H}_0+\mathbf{\Gamma }$, the particle explores the surface $C$.

Figure 3

Numerical integration of (1) for $\mathit{w}=(cosz-siny,-sinz,cosy)$. (a) The orbit explores the energy surface $H_0$ and falls toward a sink. (b) If the Hamiltonian is perturbed $\mathbf{\nabla }H=\mathbf{\nabla }{H}_0+\mathbf{\Gamma }$, there are no integral surfaces.

Figure 4

Calculated equilibrium probability distribution $f$ in the $\left(x,y\right)$ plane at $z=0$ with constant Poisson operator $\mathit{w}={\partial }_z$.

Figure 5

(a) Magnetic field strength (22) in the $\left(x,y\right)$ plane. (b) Calculated equilibrium probability distribution $f$ in the $\left(x,y\right)$ plane at $z=0$ with Poisson operator (21).

Figure 6

(a) Magnetic field strength (25) in the $\left(x,y\right)$ plane. (b) Spatial profile of ${\lambda }^{-1}$ in the $\left(x,y\right)$ plane. (c) Calculated equilibrium probability distribution $f$ in the $\left(x,y\right)$ plane at $z=0$ with Poisson operator (24). The scale at the right of (b) and (c) refers to plot (c).

Figure 7

Calculated equilibrium probability distribution $f$ in the $\left(x,y\right)$ plane at $z=0$ with Beltrami operator (26).

Figure 8

Calculated equilibrium probability distribution $f$ in the $\left(x,y\right)$ plane with antisymmetric operator (27). Each plot number $i$ corresponds to the instant $t=i\mathrm{\Delta }t$, where $\mathrm{\Delta }t$ is a fixed time interval.

Figure 9

(a) Magnetic field strength (30) in the $\left(x,y\right)$ plane. (b) Calculated equilibrium probability distribution $f$ in the $\left(x,y\right)$ plane at $z=0$ with antisymmetric operator (29).

Figure 10

(a) Plot of $\widehat{\mathfrak{B}}$ for $\widehat{\mathit{w}}$ given by Eq. (31). (b) Calculated equilibrium probability distribution $f$ in the $\left(x,y\right)$ plane at $z=0$ with antisymmetric operator (31).

Figure 11

Time evolution of the probability distribution $f$ in the $\left(x,z\right)$ plane at $y=0$. Each plot number $i$ corresponds to the instant $t=i\mathrm{\Delta }t$, where $\mathrm{\Delta }t$ is a fixed time interval.

Figure 12

(a) The hierarchical structure of antisymmetric operators. Each box is named by the corresponding operator. (b) The hierarchical structure of antisymmetric operators for $n=3$. Notice that measure preserving operators do not appear because they degenerate to Poisson operators when $n=3$. Specifically, the measure preserving condition $\mathbf{\nabla }\times \left(g\mathit{w}\right)=\mathit{0}$ reduces to the integrability condition for $\mathit{w}$ [see (3)]. Similarly, the symplectic operator does not appear because canonical pairs cannot be defined in odd dimensions.

Figure 13

Relation between operator properties and self-organizing behavior. When available, the relevant $H$ function is shown. The red and blue lines indicate the set in of geometric sources of heterogeneity. In a canonical system (symplectic operator) the only source of heterogeneity is energy. In noncanonical systems (Poisson operators), heterogeneity can arise by Casimir invariants. For Poisson operators, $H$ is determined by the Jacobian $g_c$ sending $dV$ to phase space variables as $dV=g_{c}d\mathit{p}d\mathit{q}$. Measure preserving operators are not endowed with phase space and $H$ is built on the invariant measure $d{V}_{IM}=g_{IM}^{-1}dV$. Beltrami operators do not possess an invariant measure and $H$ is given on the coordinates $d{V}_{\mathfrak{B}}=g_{\mathfrak{B}}^{-1}dV$ where the field charge $\mathfrak{B}$ vanishes. Antisymmetric operators exhibit a new kind of self-organization caused by the nonvanishing of $\mathfrak{B}$. When $\widehat{\mathit{b}}=\mathbf{\nabla }\zeta$, $g_{\zeta }=w{e}^{\zeta }$. Notice that foliation may arise also in non-Hamiltonian systems if the kernel of the antisymmetric operator admits an integrable part.