Walker diffusion method for solution of ohmic circuit problems

TL;DR

This paper introduces a probabilistic Walker diffusion method for solving complex ohmic circuit problems, offering an alternative to traditional linear algebra approaches, demonstrated on a fractal-like circuit.

Contribution

A novel probabilistic Walker diffusion approach for solving complex ohmic circuits, especially those with intricate connectivity like fractal structures.

Findings

The method effectively solves circuits with complex connectivity.

It provides an alternative to matrix-based solutions.

Demonstrated on a Sierpinski triangle circuit.

Abstract

A probabilistic method is derived for solution of ohmic circuit problems. It is compared to the standard approach, which is construction and solution of a set of coupled, linear equations manifesting Kirchhoff's laws. An example is made of an electrical circuit that has the complicated connectivity of a bond-and-node Sierpinski triangle, which would be tedious to solve by matrix methods.

Tables

Table 1. Table 1: Calculated resistance R α ​ β subscript 𝑅 𝛼 𝛽 R_{\alpha\beta} after N 𝑁 N walks between source node and sink node. Note that a different sequence of values would occur for a different initial random “seed”. The true value R α ​ β ≈ 3.08642 subscript 𝑅 𝛼 𝛽 3.08642 R_{\alpha\beta}\approx 3.08642 .

	$N$				$R_{\alpha \beta }$
	${10}^5$				$3.05995$
	${10}^6$				$3.07489$
	${10}^7$				$3.08236$
	${10}^8$				$3.08610$

Equations

$$\mathbf{J}(\mathbf{r})=-D(\mathbf{r})\nabla \rho (\mathbf{r})$$

$$J_{i\to j}=-{\sigma }_{ij}\left({\varphi }_j-{\varphi }_i\right)/r_{ij}$$

$$P_{i\to j}=\frac{g_{ij}}{{\sum }_k{g}_{ik}}$$

$$T_i=\frac{\tau }{{\pi }_i}\propto \frac{1}{{\sum }_k{g}_{ik}}.$$

$${\rho }_i=\frac{t_i}{⟨t_k⟩}$$

$${\varphi }_i=\left({\rho }_i/{\rho }_{\alpha }\right)△V=\left(t_i/t_{\alpha }\right)△V$$

$$\sum _k{I}_{ik}=0$$

$$\sum _k{g}_{ik}\left({\varphi }_i-{\varphi }_k\right)=0$$

$${\varphi }_i-\sum _k{P}_{i\to k}{\varphi }_k=0$$

$$J_{i\to j}^{(w)}=g_{ij}\left({\rho }_i-{\rho }_j\right),$$

Full text

The walker diffusion current density $\mathbf{J}(\mathbf{r})$ at location $\mathbf{r}$ due to the driving force $-\nabla\rho(\mathbf{r})$ (the walker density gradient) is

[TABLE]

where $D(\mathbf{r})$ is the local walker diffusion coefficient. This rendering of Fick’s first law for particle diffusion resembles Ohm’s law for the electrical current density $J_{i\rightarrow j}$ from node $i$ to node $j$ due to the electrical potential difference $\phi_{j}-\phi_{i}$:

[TABLE]

where $\sigma_{ij}$ is the conductivity of the bond connecting the nodes and $r_{ij}$ is the bond length. Clearly the local walker diffusion coefficient $D(\mathbf{r})$ and the local walker density $\rho(\mathbf{r})$ in Fick’s law correspond to the conductance $\sigma_{ij}/r_{ij}\equiv g_{ij}$ and the electrical potential $\phi_{i}$, respectively, in Ohm’s law. This motivates the development of a walker diffusion method for solution of the set of Ohm’s law equations that represents an electrical circuit.

The key criterion on walker behavior is that for a system at equilibrium (that is, no walker flux) all walker densities $\left\{\rho_{i}\right\}$ must be constant and identical. Thus the probability $p_{i\rightarrow j}$ that a walker at node $i$ moves to the connected node $j$ on its next move attempt must satisfy the relation $p_{i\rightarrow j}=p_{j\rightarrow i}$, which implies $p_{i\rightarrow j}\propto g_{ij}$.

The equilibrium condition is achieved by the “variable residence time” algorithm obtained as follows: On average, an attempted move by the walker at node $i$ is successful with probability $\pi_{i}=\sum p_{i\rightarrow k}\propto\sum g_{ik}$ where the sums are taken over all connected nodes $k$. [Note that the walker is a “blind ant”: the probability of success increases with the number of connected nodes $k$.] Then the time interval associated with a successful move is, on average, $T_{i}=\tau/\pi_{i}$ where $\tau$ is the time interval associated with an attempted move. That successful move is made to connected node $j$ (rather than to a different connected node) with probability $P_{i\rightarrow j}=p_{i\rightarrow j}/\sum p_{i\rightarrow k}=g_{ij}/\sum g_{ik}$. Thus the actual behavior of the walker is well approximated by a sequence of moves in which the destination of each move from a node $i$ is determined randomly by the set of probabilities $\left\{P_{i\rightarrow j}\right\}$, where

[TABLE]

and the time interval over which the move occurs is

[TABLE]

The sums in these expressions are taken over all nodes $k$ that are connected to node $i$. As the walker moves over the system of nodes, the time interval $T_{i}$ associated with a visit to node $i$ is accrued to the “residence time” $t_{i}$. Then the (normalized) walker density $\rho_{i}$ at node $i$ is

[TABLE]

where the average value $\left\langle t_{k}\right\rangle$ is taken over all nodes $k$ comprising the circuit.

The application of a potential difference $\triangle V$ between two nodes of the circuit (causing an electrical current to flow from one to the other) corresponds to the selection of one node (designated by the subscript $\alpha$) to be a walker source and the other (designated by the subscript $\beta$) to be a walker sink. A large number of walkers, placed at the source node, diffuse over the circuit in the manner described above until they visit the sink node (where no residence time is accrued so that $\rho_{\beta}=0$). The residence times at the nodes then give the steady-state distribution $\left\{\rho_{i}\right\}$, from which the electrical potentials are obtained according to

[TABLE]

where $\rho_{\alpha}$ is the walker density at the source node.

The conventional method to obtain the potentials (and currents $I$) in an ohmic electrical circuit is by solving a set of linear equations, each of which corresponds to Kirchhoff’s current law for one of the nodes of the circuit. By conservation of charge, the sum of currents to and from a node must be zero. Thus in the case of node $i$,

[TABLE]

where $I_{ik}$ is the current through the bond connecting nodes $i$ and $k$, and the sum is over all nodes $k$ connected to the node $i$. Equation (7) for node $i$ is equivalently

[TABLE]

which simplifies to

[TABLE]

where the sum is over all nodes $k$ connected to node $i$. Note however that equations for the source and sink nodes are, instead, $\phi_{\alpha}=\triangle V$ and $\phi_{\beta}=0$. The set of equations (of number equal to the number $n$ of nodes in the circuit) is then solved for the set of potentials $\left\{\phi_{i}\right\}$, from which the currents are obtained.

Solution is accomplished most conveniently by putting the relations in matrix form: $\mathbf{A}\mathbf{x}=\mathbf{b}$ where $\mathbf{x}$ is a column vector comprised of the (unknown) potentials $\phi_{i}$ and $\mathbf{A}$ is an $n\times n$ matrix whose entries are constructed from the set $\left\{g_{ij}\right\}$ of conductances. The tedious task is filling out the matrix $\mathbf{A}$ to reproduce the set of equations, but this approach can use standard matrix solvers and gives “exact” values for the potentials.

In contrast, the walker diffusion method (WDM) is easily implemented in a very simple computer code, and requires as input only a description of the circuit. That is most conveniently given by an $n\times n$ array where the entry $(i,j)$ is the conductance $g_{ij}$. (Of course when two nodes are not connected, the corresponding entry is [math].) As this is a probabilistic method, the set $\left\{\phi_{i}\right\}$ of electrical potentials approaches the true values as the number of walkers released at the source node increases.

An example of a complicated bond-and-node circuit is shown in Fig. (1). Whatever the distribution of values $\left\{g_{ij}\right\}$, a single walker (or multiple walkers) diffusing over the closed system produces $\rho_{i}\rightarrow 1$ at all nodes.

Note that this circuit has the form of a Sierpinski triangle (AKA gasket or sieve) at iteration $3$. When a potential difference $\triangle V$ is applied across two of the three corner nodes (apex vertices), an analytical value for the equivalent resistance $R_{\alpha\beta}$ can be obtained by use of the triangle-star ($\triangle\rightarrow Y$) transformation (key-1, ). In the case that all bonds in the Sierpinski triangle have conductance $g_{ij}=1$, the resistance $R_{\alpha\beta}=(2/3)(5/3)^{3}\approx 3.08642$.

The WDM obtains $R_{\alpha\beta}$ in the following way: The walker flux emitted at the source node is $J_{\alpha}^{(w)}=\sum_{k}g{}_{\alpha k}\left(\rho_{\alpha}-\rho_{k}\right)$ which corresponds to the electrical current $I_{\alpha}=\sum_{k}g_{\alpha k}\left(\phi_{\alpha}-\phi_{k}\right)$, where the sum (in both expressions) is over all nodes $k$ connected to the source node $\alpha$. Then the equivalent resistance $R_{\alpha\beta}=\triangle V/I_{\alpha}=\rho_{\alpha}/J_{\alpha}^{(w)}$.

Table I gives calculated values of $R_{\alpha\beta}$ for the particular case mentioned above, showing that the true value is approached as the number $N$ of walkers released at the source node increases.

Note that another way to obtain the set $\left\{\phi_{i}\right\}$ (and so the electrical currents $\left\{I_{i\rightarrow j}\right\}$ as well) is via the relation

[TABLE]

where $J_{i\rightarrow j}^{(w)}$ is the walker flux between connected nodes $i$ and $j$. A large number of walks between source node $\alpha$ and sink node $\beta$ produce the fluxes $\left\{J_{i\rightarrow j}^{(w)}\right\}$. Then by use of Eq. (10) the walker densities $\left\{\rho_{i}\right\}$ are calculated one at a time beginning with $\rho_{\beta}=0$ and working up to $\rho_{\alpha}$. Finally the $\left\{\rho_{i}\right\}$ are converted to $\left\{\phi_{i}\right\}$ by multiplying the former by $\triangle V/\rho_{\alpha}$ according to Eq. (6).

Concluding remarks

The walker diffusion method utilizes the “variable residence time” algorithm for walker diffusion over a bond-and-node undirected network, where the node degree may vary dramatically over the network (so the network is not necessarily a regular grid). [The degree of node $i$ is the number of nodes $k$ to which it is connected.]

It should be evident that the WDM is not a particle model of a physical process: a randomly diffusing walker does not resemble an electron responding to an applied electric field. Rather, the eponymous walker diffuses over the nodal circuit according to particular rules, thereby “solving” the system of local Ohm’s law equations associated with the set of nodes.

Despite the attractive physical arrangement of the nodes in Fig. (1), it is only their connectivity that defines the circuit. Thus there is no length scale, or Euclidean dimension within which the circuit is embedded. As a consequence there can be no diffusion coefficient $D_{w}$ that describes the behavior of the walker. This is in contrast to the object that is a bond-and-node Sierpinski triangle embedded in 2D space, considered in Ref. (key-2, ).

Acknowledgements.

I thank Professor Robert “Bob” Smith (Department of Geological Sciences) for arranging my access to the resources of the University of Idaho Library (Moscow, Idaho).