Analogies

In each case in this article, we draw up a circuit, and then show that it is "equivalent" or "analogous" or isomorphic to some phenomenon or system. This means that for each electric quantity in the circuit, there is a corresponding quantity in the other system and vice versa. So after we filled in for example "Temperature" for "Voltage", we can treat the system using the machinery of circuit theory.

The principle of Minimum Dissipation

In principle, we can do all circuit analyses with just voltages and impedances. But sometimes it can be handy to introduce extra concepts, and here we want to introduce the concept of the generating functional. (A functional is a formula that produces a number from a set of variables.)  The main motivation for this is that in contemporary theoretical physics, the "Principle of Least Action" is considered to be very important. I want to see how this is related to circuit equivalents.

First we will derive an expression for the dissipation in a circuit, and see that the total dissipation is a funky functional.

The set of i equations for a circuit with i vertices:

    ∑ _j  (V_j - V_i)/R_ij = 0

By multiplying this by 2V_i , and summing over all i, juggling a bit with indices, we get:

    ∑_i,j  (V_j - V_i)² /R_ij =   0

The term (V_j - V_i)² /R_ij will be recognized as the dissipation by the resistor R_ij

The theorem we just derived says that the total dissipation in a circuit is zero . We can also see that if Voltage is a real number, and resistance is a positive real number, as is the case for "ordinary" circuits, then the dissipation in each component is positive. This implies that the only solution to a circuit without sources of energy is that all currents and all voltages are zero. So to do anything interesting, we have to put in some sources, or consider putting in as negative or imaginary resistance somewhere.

A cool property of the total dissipation is that you can recover all equations for the circuit by requiring that the total dissipation is in a minimum with respect to small variations of  V_i . Mathematically, this is saying all derivatives to V_i are zero:

  d/dV_i { [ ∑_i,j  ( V_j-V_i )² /R_ij ] }
    = 2  ∑ _j  ( V_j-V_i )/ R_ij = 0

Which is the original set of equations.

This is why the total dissipation can be considered the generating functional of the circuit. Deriving the equations from a generating functional is not really adding any new physics, but it is especially useful in seeing how different ways of describing things are equivalent.

We call this trick the principle of minimum dissipation . It has interesting analogs for the different kinds of circuits I will discus. For instance in one case it maps onto the principle of least Action . The principle of least Action is considered by some to be the most fundamental principle in theoretical physics. 

The "principle of minimum dissipation" is really a catchy but oversimplifying phrase. It is more accurate to say that all derivatives of the functional with respect to certain state variables are zero. For example, if we had said:

  S = ∑_i,j   I_ij²R_ij

and then minimize dissipation by varying I_ij instead of V_i, we would have obtained:

    I_ij = 0

Set up the analogy :
V(Voltage) <-> T (Temperature)
Q (Charge) <-> U (Energy)
I (Electric current) <-> W (Thermal Power)
P (Electrical dissipation)  <-> W ΔT   (see note)
C <-> ρC_p ΔVolume (Thermal capacitance of volume element)
R (Electrical resistance) <-> Δs² /( λ ΔVolume)  (Thermal resistance)

And draw the following circuit :

Figure 1: Electric circuit equivalent for time dependent heat conduction.

(Extra picture for search engine related reasons)
Applying Kirchhoff's current law :

    dT/dt = -1/(ρC_p)  div (w)  dT_i/dt = -1/C_i ∑_j W_ij

Applying Ohm's law :

    w = -(1/λ) grad (T)W_ij = -(T_i-T_j)/R_ij

Combining:

  dT/dt = λ/(ρC_p)∇²T dT_i/dt = 1/C_i ∑_j(T_i-T_j)/R_ij

Which is the equation for instationary heat conduction. 

Note on irreversibility and the generating functional:

The generating functional of this circuit is:

    S = ∑_edges (ΔT² / R)  + ∑ _Vertices (d/dt  ½ C T² )
    = ∑_edges (W ΔT)  + ∑ _Vertices ( T dU/dt)

This quantity looks a bit unfamiliar. We know that it should have something to do with irreversibility, because it is the analog of electrical dissipation, the irreversible conversion of electrical energy into heat. It would be nicer if the generating functional were Entropy generation.

Actually, we can do this by switching from Temperature (T) to a quantity that Zemansky called Negcitemp (N=-1/T). According to Zemansky (writer of well known thermodynamics textbooks), it sometimes makes sense to use Negcitemp (Negative Reci procal Temperature)  instead of temperature. For small temperature deviations around a nominal temperature, Negcitemp is just like a rescaled temperature. For larger deviations, things get non-linear, but C_p and λ are non-linear functions of T anyway. So let us assume that  C_p and λ are linear in Negcitemp, and get for the generating functional:

    S = ∑_edgesWΔN  + ∑_Vertices N dU/dt
    = -∑_edges W Δ(1/T)  - ∑_Vertices (1/T) dU/dt
    = -d/dt (Entropy)

So now we have the more familiar quantity of entropy generation rate as the generating functional, and indicator of irreversibility. This is no big deal in practice, but nice philosophically.

We can interpret the term as as a pseudovector (A^*) whose direction is normal to the (N-1) simplex built from all vertices except (A) and whose magnitude |A^*| is the (N-1) volume, multiplied by a factor 1/N!.

To derive it, we refer to the picture below, again using 3D, but implying generalisation to other dimensions.

We expand the volume into a base |B^*| and height (h_b), and relate this to the angle (θ):

Write (∂h_b /∂θ) in terms of A* and B*:

:

   

Finally, we reach QED:

The N-simplex equation allows us to get the resistance values from the inverse Cayley Menger matrix:

Acoustic fields

Acoustic fields works like heat conduction, but now the resistors are replaced by induction coils. Acoustic fields are reversible, they have no resistors. The requirement that the total dissipation is zero no longer requires that there is a source of energy to have non-zero solutions, because the argument depended on the resistances being always positive; they are now imaginary. These sourceless non-zero solutions are of course: waves!

Set up the analogy :

V (Voltage) <-> p (pressure)
Q (Charge) <-> (Mass)
I (Current) <-> ρvΔA (mass flux)
P (Electrical power) <-> P (Acoustic power)
C (Capacitance) <->ρΔVolume/(κp₀)
L<-> (Δs² ) /ΔVolume  

Figure 2: Electric circuit equivalent for acoustic fields.
(Extra picture for search engine related reasons)

Applying Kirchhoff's current law :

    dp/dt = -κp₀/ρdiv(ρvA)  dp_i/dt = -1/C_i ∑_j(ρvΔA)_ij

Applying Ohm's law :

    d(ρv)/dt = - grad(p)  d(ρvΔA)_ij/dt = -1/L_ij ∑_ij(p_i-p_j)

Combining:

 d²p/dt² = κp₀/ρ ∇²p   d²p_i/dt² = 1/C_i ∑_j(p_i-p_j)/L_ij

Which is the acoustic wave equation, with c² = κp₀/ρ .

Animated GIF of acoustic circuit. The magnitude of the velocity is animated as change in inductor width, the magnitude of the pressure is animated as the size of the bars attached to the capacitors.

Generating functional :

 S = d/dt ∑_edges ( ½mv² ) + d/dt ∑_vertices ( ½Cp² )
     = d/dt (total energy stored in components)

To retrieve the field equations from the generating functional, we have to write it in terms of p_i:

     S = d/dt ∑_vertices ( ½C p² ) + dt ∑_edges ( ½Δp² )

This last equation looks a bit awkward. Perhaps a nicer (and more popular) approach for a generating functional to use space time diagrams, in which the the principle of minimum dissipation is replaced by the principle of least Action.

The Navier-Stokes equation

The acoustic equations can be modified to also include terms to account for the transport or advection of inertia, and for viscosity. This leads to the Navier Stokes equation, which describes fluid dynamics. Fluid dynamics is non-linear, and has funky features like pseudo unpredictability.

The central idea for transforming the linear circuit theory to the non-linear stuff like Navier Stokes, is what I call a bucket  The idea is shown below.

Figure 3: Principle of bucket discretisation of advection.

After each time step, the fluid will be displaced relative to our cell structure. So we have to redivide the fluid among the cells each time step. We do this by interchanging buckets. It can be seen from the drawing that the interchanged bucket size is vA dt. The buckets carry with them all information of the fluid,  i.e. all dynamical variables such as v, p, T, etc.
It is possible to view this process as a coordinate transformation from material coordinates, which are attached to the fluid, to spatial coordinates, which are fixed in space.

Suppose at time t, we had a cell (i), which has buckets leaving to a set of neighbours (j) with volume velocities v_ij A_ij, and had incoming buckets from a set of cells (k) with volume velocities v_ki A_ki . By bookkeeping an arbitrary dynamic variable (φ) in the cell , we get:

Which is the discrete version of the advection term in fluid dynamics:

    dφ/dt = -(v.∇)φ + ...

The central non-linearity comes from the fact that v itself  is also advected:

    dv/dt = -(v.∇)v + ...

A good thing about the buckets is that it automatically takes care of some nasty subtleties regarding discretisation schemes which can easily cause numerical instability. (For example, the direction of the flow influences the way we treat a neighbour) The simulation software  I made using the bucket idea turned out to be very robust.

We will symbolize the advection by a bucket drawn at each edge. We then get a diagram for the compressible Navier Stokes equation.

Figure 4: Electric circuit equivalent of the Navier Stokes equation for compressible fluid dynamics.

The Navier Stokes equation can be further refined by including viscosity. (Use a mesh resistance, which works like mesh inductance )

I just want to show this cool picture of a Von Karman Vortex street, made with a simulation based on the modified acoustic equivalent. 

Figure 5: Simulation result of Navier Stokes: A Von Karman vortex street

Navier stokes on an arbitrary trianglular net

When we try to implement the Navier Stokes equaiton on an arbitrary triangular network, we encounter an additional difficulty. We can use the previously found methods to find all the impedances, and use the bucket formula to transport properties from vertex to vertex. But what is the momentum that we should assign to a vertex?

If we make use of explicit coordinates (x,y,z), he answer is that we multiply the mass flux (the analogue of current), by the vector associated with the edge:  This converts the (N-1)-chunk of mass flux to a vector of N-chunks: The momentum. The momentum on each edge is shared by the 2 vertices it connects. We divide it between the 2 vertices, weighted by the volumes of the vertices.

The Klein Gordon Equation

An interesting network is that below, which turns out to be an equivalent of the Klein Gordon equation. 

The Klein Gordon equation is the relativistic wave equation for spin zero particles. The network is drawn for one dimension (x), and with two layers, UP an DOWN. We derive:
   

Figure 6: Electric circuit equivalent of the Klein Gordon equation, with 2 possible modes.

The equation splits into two superposed modes, the symmetric ( UP+DOWN) and (UP-DOWN). The two modes both obey the Klein Gordon Equation. The symmetric mode has mass zero, and the anti symmetric mode has mass (L_y C)^-1/2 . Jos Bergervoet has suggested a simpler circuit for the Klein Gordon Equation:

Figure 7: Electric circuit equivalent of the Klein Gordon equation, with 1 possible mode.

The equation for this circuit is: 

   
It only has one mode (particle species) as opposed to the previous circuit, which had 2 modes.

Electrostatic fields

Set up the analogy :

V <-> V;
Δ_xV (Voltage difference in x-direction) <-> - E_x Δx (Chunk of Electric field)
Q (charge) <-> Q (charge)
I (current) <-> d(DΔA)/dt (rate of change of dielectric displacement through surface)
C_x (Capacitance placed in x-direction) <-> ε ΔVolume/Δx² (Storage container of Electrostatic field energy)

Figure 8: Electric circuit equivalent for electrostatic fields

Kirchhoff's current law:

    d/dt (div(D)) = 0  0 =  ∑_j(d/dt (DΔA))_ij

Ohm's law:

    D = -grad(V)/ε (DΔA)_ij = -1/C_ij (EΔs)_ij = -1/C_ij ∑_ij(V_i-V_j)

Combined, this gives:

    d/dt (∇²V/ε ) = 0   d/dt(-1/C_ij ∑_ij(V_i-V_j)) = 0

Usually, you say that at t=0, the divergence of the field is equal to the charge density. You then get:

    ∇²V = ρ/ε 

Kirchhoff's voltage law gives:

    d/dt (curl(E)) = 0 

The Generating functional is:

    S = d/dt ∑(E.D ΔVolume)

    E.D is the field energy density.

To retrieve the field equations from the generating functional, we have to write it in terms of V_i:

     S = d/dt ∑_ij ( ½C_ij (V_i-V_j)² )

Comment on E versus D
It can sometimes seem a bit irritating to have 2 different quantities associated with electric fields (E and D ). In principle, this can be avoided, just like it can be avoided to use currents by always writing them as a voltage difference divided by a resistance. But I think it is important to distinguish between 1-chunks like E_x Δx and (N-1)-chunks like D_x ΔVolume/Δx . This distinction is analogous to the distinction between electric potential and electric current, a distinction that we would surely want to be aware of when we repair household electra.

Comment on field energy
This representation may seem somewhat artificial, the vacuum is supposed to be empty, and not contain any capacitors. However, the vacuum does contain electrostatic energy, which is stored locally in the vacuum. This energy is the same energy that is stored in the imaginary capacitors. So they are not that abstract as it seems: the energy is really there.

Putting in conductors
You just put resistors in parallel to the capacitors. Interestingly, short-circuiting capacitors increases the capacitance of a geometry, thereby also also decreasing the effective speed of light through the geometry. This can be seen easiest in one dimension. Suppose a number of capacitors are connected in a chain. From Kirchhoff's voltage law it follows that when impedances are in series, you get the effective impedance of the chain by adding the individual impedances. This means that the for the effective capacitance (Ceff): 1/ Ceff = 1/C1 + 1/C2 + 1/C3 + ... If we short-circuit some capacitors in the chain, the reciprocal of the effective capacitance gets smaller, so the effective capacitance itself gets bigger.

The Maxwell equations

The Maxwell equations describe both electric and magnetic phenomena, and their interaction. So this is stuff that you need to understand if you want to understand nature. To put Maxwell into a circuit diagram, you start with the diagram for electrostatic fields . Then, we have to think how we can put in the magnetic field. We think naturally of inductors, as they seem to be the magnetic counterparts of capacitors. But it is a bit tricky. We know that the vacuum does not conduct electricity, so we can't put any inductors in parallel with the capacitors. We could try putting them in series. But that would mean that there is only magnetic energy when a current is flowing through the inductors, and therefor also through the capacitors. But this would charge them up indefinitely, and produce infinite electrostatic fields. The clue comes from the observation that an inductor is not 'elementary', when you look at its geometry. It consists of a coil, a spiral of wire. The elementary object is a single loop. After a considerable struggle with this idea, I realized that a proper treatment requires a new concept, the mesh inductance. This is an inductance associated with a loop rather than an edge. 

Generalizing a circuit to an n-complex
This idea is part of  a cool generalization of a circuit, called a n-complex, or cell complex. A conventional circuit can be thought of as a 1-complex. The idea will be brought along from the following list:

0- Complex : a set of loose Vertices (points) or 0-chunks
1- Complex: (=Concentional circuit): Edges (or 1-chunks) that connect Vertices
2- Complex: Faces or 2-chunks that connect Edges
3-Complex: Solids or 3-chunks that connect Faces
N-Complex: N_chunks that connect (N-1) chuncks

So how do we generalize Kirchhoff's laws and Ohms law? We first need the concepts boundary and the coboundary operators. Boundary and co-boundary operators are just mathematical formalizations of what we intuitively understand right away from the diagrams. Roughly, the boundary of an n -cell is the set of [n-1] cells that form its boundary. The (co)boundary operator will also take care of some minus signs book keeping, associated with the orientation choices of the positive directions.

Suppose we have N vertices and M edges. Then the Boundary operator for Edges can be thought of as an ( N X M) matrix({ a_ij}), that has entry a_ij=0 if the vertex (i) is not connected to the edge (j),  a_ij = -1 if it is the source of the chosen arrow on the edge, and  a_ij = +1 if it is the destination of the chosen arrow on the edge. The arrows can be chosen arbitrarily, but once chosen, we should of course keep them fixed.
The coboundary of the set of k-chunks gives the set of k +1 chunks that has the k-chunk as a part of its boundary, once again taking care of all minus signs and arrow orientations etc. The Coboundary matrix for the Vertices is simply the transpose of the Boundary matrix for the Edges.

Reformulating a conventional circuit (1-complex) into our new jargon
Now we are armed to formulate ordinary circuits in a new jargon, which will be useful when we start to generalize further.

Step 0.  We define a voltage (V_i) on each Vertex.

Step 1. We let the Coboundary operator act on the Vertices (as the discrete analog of the differential Grad operator) producing a set of 1-chunks:

    Coboundary (V_i) = ΔV_ij         (Step 1)  

This is familiar, we just take the voltage difference across each edge.

Step 2.  We apply Ohm's law to to map our 1-chunks to twisted  N-1 Chunks:

    I_ij = ΔV_ij/R_ij       (Step 2, or Ohms law)  

In 3 dimensions, a chunk of current will scale with area. In N dimensions, this generalizes to a N-1 dimensional subspace. Such a subspace will generally have an arrow associated with it. In the case of a surface, we think of the normal vector of the surface. 

So why is the chunk called "twisted"? This is because its arrow direction is always inherited from the voltage difference, rather than from its own geometry. Another way to see this is that the spatial information contained in the resistance value  R_ij is stripped of its arrow; it is always positive. So  I_ij always has the same arrow as ΔV_ij.   And, if you take the product I_ij ΔV_ij you get an N-chunk of generating functional, which has an always-positve volume associated with it, in contrast to an oriented volume that non-twisted chunks would produce. When we study the Dirac equation, we will put step 2 and 3 together to form the twisted coboundary .

Step 3: Generalize Kirchhoff's current law.

    Coboundary(I_ij) = 0    (Step3, or the generalized Kirchhoff's current law)

It may seem a bit strange at first that we use the coboundary rather than the boundary. After all, vertices are the boundary of edges. But in step 2 we made currents N-1 chunks, and the coboundary of N-1 chunks should be a set of N chunks. These N chunks are just the dual of the 0-chunks on the vertices. See figure 9 for an illustration.

Once again, there is an analog differential operator, this time the div operator. Kirchhoff's current law is always about incoming fluxes that have to add up to zero.

Figure 9: Structure of the laws of electric circuits in terms of coboundaries.

Combining Step 1,2 and 3, we find the set complete of equations for a 1-complex:

    Coboundary(Ohm(Coboundary(Vertices)))=0

What about Kirchhoffs voltage law? We already have a complete mathematical description of the circuit, so the voltage law can be viewed as an alternative formulation. It reads:

    Coboundary (ΔV_ij) = 0

or combining with a previous formula:.

    Coboundary (Coboundary (V_i)) = 0

This can be derived directly from the general theorem that the Coboundary of a Coboundary is zero. (Also the Boundary of a Boundary is zero ). These 2 statements are important fundamental laws. They can be visualized if you play around a bit with circuits and arrows, perhaps writing out the boundary matrix.

Note: The div, curl and grad operators are all instances of Cartan's exterior derivative (d). Thus, the Coboundary operator is the discrete analog of Cartan's exterior derivative. Ohm's law is the discrete analog of the Hodge star operator , multiplied by a material constant.

Formulating the Maxwell circuit as a 2-complex
For Maxwell the electric field on an edge can no longer always be expressed as a gradient of a potential. This is typical of a 2-complex. So we do not start by defining a potential of vertices,  but 1 dimension higher: on the Edges. This is the diagram:

Figure 11: Electric circuit equivalent of the Maxwell equations

(Extra picture for search engine related reasons)

With the  analogy :

V <-> V (Voltage)
Δ_xV <-> E_xΔx (Chunk of Electric field)
Q <-> Q (charge)
J_z (Mesh current) <-> H_zΔz (Chunk of Magnetic field)
C_x <-> ε ΔVolume /Δx² (Storage container of Electrical field energy)
L_z <->μ Δz² /ΔVolume (Storage container of Magnetic field energy)

Step 0: Each edge has a 1-chunk EΔs associate with it, that we call an E-chunk. 

Step 1: Take the coboundary of the set of E-chunks. This will give you a set of loops. Note also that there are many different loops that we might want to choose, that all traverse the circuit. We could even in principle choose loops that go round a track 10 times. But the only physically relevant loops are those that we give a finite mesh impedance. In our Maxwell diagram, only the loops that are inside the faces of the cubes have finite impedance and are used. It will be convenient for later to ignore loops that will not get a finite impedance.  

Anyway, after taking the coboundary of the E-chunks, we will have performed the discrete analog of curl(E). 

    curl(E) = -d/dt B  ∑_{along_loop} EΔs = -d/dt(BΔA) 

We will use this as a definition of B, or magnetic induction. The d(BΔA)/dt are 2-chunks, that we will call B-chunks.

Step 2: Apply Ohms law, but now use the mesh inductance to map the  B-chunks which are 2-chunks onto twisted N-2 chunks, which we define as H. In 3 dimensions, H comes in twisted 1-chunks of HΔs, or vectors associated with a loop. The vector will be recognized as the normal vector of the loop. The equation is the discrete analog of:

        H =  (1/μ) B  BΔA = 1/L_mesh(HΔs)

Step 3. Take the Coboundary of the H-chunks.

    curl(H) = dD/dt ∑_{along_loop} HΔs = d/dt(DΔA) 

Step 4. Once more apply Ohms law, but now over the capacitances at each edge, we get the discrete analog of:

    E = (1/ε) D EΔs = 1/C(DΔA) 

Summarizing, we have the Maxwell equations:

        curl (E) = -dB/dt
        H = (1/μ) B
        curl(H) = dD/dt
        E = (1/ε) D

Application of the generalized Kirchhoffs Voltage law by taking the coboundary of the coboundary of E and H :

        d/dt (div(D)) = 0
        d/dt (div(B)) = 0

It is generally axiomized that at t=0, we have:

        div D =  ρ
        div B = 0

Note that we can have magnetic energy without having to charge the capacitors. For example, a constant magnetic field would correspond to identical mesh current in each loop. This means that the net edge currents are zero, so the capacitors are not being charged. The magnetic energy is stored inside the mesh inductance. Once again, this energy is real in the sense that it is locally present in the vacuum.

The Maxwell equations can be combined to form the electromagnetic wave equation:

   
The model presented for the Maxwell equations  could be seen as an aether model . In the link, it is argued that this does not violate relativity.

Putting in conductors
This is the same as with electrostatic fields, you just put resistors in parallel to the capacitors.

Putting in compact components
Sometimes components much smaller than a wavelength can influence the field. This is especially the case with resonators. They can resonate at a frequency much lower than the frequency that is associated with c/s . (s is a typical dimension of the system) To put in these components, you just add them to the circuit. You don't have to create the whole geometry, you can just put a big physical capacitance across a small 'vacuum' capacitor, which will then become negligible. Likewise you can put in coils, not a spiraled conductors but as single circuit elements. Then you can start to calculate how this physical circuit would interact with the vacuum.

Visualizing the dynamics:
To visualize an electromagnetic wave, you can picture a line of capacitors being charged at time=0. Along this line you would have a constant E-field, pointing along the line. This causes a voltage difference across neighbouring parallel lines of capacitors. This causes a current to flow, discharging the first line of capacitors, and charging the neighbouring ones. But this current corresponds to mesh currents. So as the neighbouring E-field is being built up, some of the energy is being transferred to magnetic energy in the meshes. By the time that the fields of the neighbouring lines are equal to the field of the original line, there is no capacitive driving force to displace more charge. But now there is inductive driving force, which acts like an inertia. The transport of charge continues, now against the direction of E. This is similar to a mass/spring system, where the mass will move against the force of the spring, once it has gained momentum.
In the meanwhile capacitive energy is being transferred to neighbours-of neighbours of the original line. So the energy spreads out into space. Unlike with heat conduction, the process is reversible. The energy is not dissipated, but is pumped back and forth from its magnetic form to its electric form.

Animated GIF of a Maxwell circuit. The magnitude of the magnetic field is animated as rate of rotation of the mesh inductors, the magnitude of electric field is animated as the size of the colored bars attached to the capacitors.

The Poynting vector of the electromagnetic circuit is a Cut function: It is assigned to a edge-loop pair. We multiply the 1-chunk of E with the (N-2)-chunk of H to get a (N-1) chunk of power flux (E x H). Note that the Poynting vector in 3D space is represented by more than 3 components in the circuit, which makes it seem unlike a vector. This is because the Cut function becomes only a vector after being "contracted with a cut": If you specify the cut (the analog of a surface), the cut function gives you the flux across the cut at each point.

The Poynting vector as a cut function.

The Generating functional for the Maxwell circuit is:

   S = d/dt ∑_edges (E.D ΔVolume )+ d/dt ∑ _meshes (H.B ΔVolume)

    = d/dt ( total energy stored in components)

To retrieve the field equations from the generating functional, we have to write it in terms E only (A form with H instead of E is also possible):

  S = d/dt ∑_edges (½ C_edge (EΔs)² ) +dt ∑_meshes (∑_loopEΔs )²

But this is more elegantly done using space time diagrams , in which the the principle of minimum dissipation is replaced by the principle of least Action.

The Schrödinger equation

Another important equation in physics is the Schrodinger equation. (It is actually an approximation of the Klein Gordon equation.) It describes the quantum mechanical wave function of a particle in a potential field ( V ).

   
The Schrödinger equation looks almost the same as the heat conduction equation . We need to put in the potential V, and to take care of i , the square root of –1. To represent the potential (V), we add resistors (r) to ground potential.

The analogy becomes:
V <->Ψ
1/r <-> V ΔVolume
1/Rx <-> ħ²/(2m)  ΔVolume /Δx²
C <-> iħ ΔVolume

Figure 12: Electric circuit equivalent of the Schrodinger equation, with imaginary-valued capacitance.

These together yield the Schrodinger equation, but we had to choose an imaginary capacitance. This is no problem mathematically, we can just do all calculations as we did with real numbers. But it is perhaps a concession to visualizability. A major consequence of choosing imaginary capacitance is that the solutions are now of the type:

    Ψ ~ exp(ikx) exp(-iωt)

rather than

    Ψ ~ exp(ikx) exp(-t/τ)

A subtle but important difference: It means we don’t get exponential decay with time into thermal equilibrium as with heat conduction but we get everlasting oscillations which conserve |Ψ|². 

Another approach is to try to write out  Ψ into real numbers  Ψ = ( X + iY ). We then obtain equations for X and Y that are of the form:

    d²X /dt² = d⁴X /dx⁴

This equation is like the equation for waves in a bending beam. You can make a kind of beam construction using springs and bars. This has led to a mechanical discrete analog of the Schrodinger equation with springs and rods, that sometimes pops up in literature. I don't know if it can be built using electrical components.

Space-time circuits

So far, we have considered discrete space, but time has till now been considered continuous. Interestingly, it is possible to construct a model that has space and time discretized in the same way. I like this, because according to the theory of relativity, space and time should be deeply related. 

The trick is to put negative resistance in the time direction. This sign is related to the negative sign of the time component of the metric of space-time.

As an example, we will create the acoustic wave equation in terms of a space-time circuit.
We will use the so-called Velocity potential as the analogue of Voltage.

Set up the analogy :
V (Voltage) <-> φ   (Velocity potential)
I_x (Electric current in x-direction) <->(ρvΔAΔt)_x (Mass displacement in x-direction)
I_t (Electric current in t-direction) <-> p (ρ/(κp₀)) ΔVolume (Pressure times spatial volume)
P (Electrical dissipation)  <-> S (action) 
1/R_x (Electrical conductivity in x-direction) <-> ρΔVolume Δt /Δx²
1/R_t (Electrical conductivity in t-direction) <-> -(ρ/(κp₀)) ρΔVolume Δt /Δt²

    Note that (ρ/(κp₀))  = c², the speed of sound squared.

Figure 14: Electric circuit equivalent of the scalar wave equation discretized in both space and time.

The velocity potential (φ) is defined such that:

 v =  -grad φ
  p/ρ = -dφ/dt

Velocity and pressure live in the circuit as voltage differences across edges (i.e. as 1-chunks):

  v_xΔx = -Δ_xφ
  p/ρΔt = -Δ_tφ

Write out Kirchhoff's current law at a vertex (using Ohm's law to get the currents):

    (ρvΔAΔt)_x[x,t] -  (ρvΔAΔt)_x[x-Δx,t] + (ρ/(κp₀))p[x,t] ΔVolume - (ρ/(κp₀))p[x,t-Δt] ΔVolume = 0

 Divide by ρΔVolumeΔt (assumed constant for the moment) and rearrange:

   ( p[x,t] - p[x-Δx,t] )/Δt = - (1/(κp₀)) ( v_x[x,t] - v_x[x,t-Δt] )/Δx

Which is the discrete analog of:

     dp/dt = -(1/(κp₀)) div v

Next, write out Kirchhoff's voltage law around a loop:

    v_x[x,t]Δx + p/ρ[x+Δx,t]Δt -  v_x[x,t+Δt]Δx - p/ρ[x,t]Δt = 0

This time, divide by Δx*Δt/ρ, and rearrange:

   ( ρv_x[x,t+Δt] - ρv_x[x,t] )/Δt   =  -( p[x+Δx,t] - p[x,t])/Δx 

Which is the discrete analog of:

     dρv/dt = -grad p

So we once more have the acoustic wave equation, but now in space-time form.

There is now no longer a role for the inductors and capacitors, the only component is the resistor.

The Generating functional is now
   
    S = ∑_edges ΔV² /R
        =∑_x-edges (ρv_x²) ΔVolume Δt   - ∑ _{t -edges} (p²/(κp₀))  ΔVolume Δt

        =∑_edges (dφ /dx_μ)(dφ /dx^_μ) ΔVolume Δt

The generating functional is now Action instead of dissipation. The "dissipation" in this analogue has of course no longer anything to do with energy loss.
Action is a fundamental quantity, perhaps even more fundamental than energy. In a sense, it is energy density integrated over space and time.
According to quantum mechanics, there is a fundamental chunk of action, equal to ħ. More on that in the future.

Note that the use of negative resistances in the time direction means that the total action (<->dissipation) in the circuit is zero.

We can also put the Klein Gordon equation in space-time form, by connecting a resistance to ground potential to each vertex. The mass term is represented by a current to ground.

Space-time circuit for the Maxwell equations

In the diagram below, we apply the idea of a space-time circuit to the Maxwell equations. It all works out nicely, and we obtain the relativistic formulation of the Maxwell equations in terms of the 4-vector potental (A) and field tensor (F).

Figure 15: Electric circuit equivalent of the Maxwell equation discretized in both space and time. To depict it in 3D, we draw only 2 dimensions of space.

The capacitors and mesh inductors are replaced by mesh resistances. Like in the scalar case, the dissipation in these resistors is reinterpreted as Action. Again following the scalar case, the mesh resistances which have a time component (R_tx, R_ty, R_tz) are negative, so that the total action in the circuit is zero even with non-zero currents.
We do not use a scalar potential φ, but a vector potential A, a 1-chunk of which (A_μΔr_μ) is defined on each edge. In (3+1) space-time dimensions, there are 4 components of A , and 3+3 components F. The 3+3 components of the electric field and the magnetic field are now contained in the 6 mesh F-chunks F_μνΔr_μΔr_ν.

Lets remind ourselves of the relation between F and A, and their more familiar friends E and B:

  F_xy = dA_x/dy - dA_y/dx = B_z 
  F_zx = dA_z/dx - dA_x/dz = B_y 
  F_yz = dA_y/dz - dA_z/dy = B_x

  F_xt = dA_x/dt - dA_t/dx = E_x
  F_yt = dA_y/dt - dA_t/dy = E_y
  F_zt = dA_z/dt - dA_t/dz = E_z

or, using 4-index notation:

  F_μν = dA_μ/dr_ν - dA_ν/dr_μ      F_μνΔr_μΔr_ν =  (dA_μ/dr_ν - dA_ν/dr_μ)Δr_μΔr_ν

 The mesh resistances R_μν:

  R_μν  =  ε_μν (Δr_μΔr_ν)² / (ΔVolume Δt) 

with ε_μν =

The Generating functional is now the Action of the electromagnetic field:

  S = ∑_meshes (Δ_μνA)² /R_μν 

Note the compactness of the relativistic formulation of the Maxwell equations.

If we impose Kirchhoffs current law on A, we get the discrete version of the Lorentz gauge condition d_μA_μ = 0.

Curved Space time

With electric networks, you can implement any metric of space time. If dx, dy, dz and dt vary from place to place, as in curved space, you can just adapt the impedance values accordingly. Furthermore, there is a 1-to-1 correpondence between the metric tensor in a simplexial chunk of space time and the resistors on its edges, as we saw in the section on the N-simplex equation. You could reinterpret the changes values as being caused by a variable ε and μ constants of the vacuum. There are even people who tried to construct a gravity theory on this principle, for example:
http://arxiv.org/abs/gr-qc/9909037  
A link brought to my attention by Gordon D. Pusch.

 
The Dirac equation and cell algebra

It has annoyed me for some time that I did not have an electric circuit equivalent for the Dirac equation. We have diagrams for scalar (spin 0) particles, ( Klein Gordon equation) and for Vector (spin 1) particles ( Maxwell equation ). The Dirac equation describes spin ½ particles, which are the building blocks of matter: Electrons, protons and neutrons. And lots of other less well known particles. Important, but what is so difficult about it? Before going in to this, I want to say that the circuit diagram for the Dirac equation reveals a nice algebraic structure that underlies the theory of electric circuit equivalents of fields. This structure is what I will call "Cell algebra". 

The core weirdness of the Dirac equation is that it is about spinors . Spinors are unlike any objects that we see in our macroscopic world around us. For example, if you rotate a spinor by 360 degrees, it will have been multiplied by -1, rather than remain unchanged like "ordinary" objects. Using imprecise language, you could say that spinors are sort of square roots of vectors. Cell algebra will allow us to see exactly how spinors are the square roots of vectors.

In a circuit, each Vertex can be given a coordinate. Suppose we built our circuit with real wires and components, took it off the drawing board, rotated it, and put it back on our drawing board. Then all coordinates of the components relative to our drawing board would be changed. Normally, we would expect any Voltages defined on a vertex to simply be transported with its vertex to its new coordinate. Also, the currents on each edge would be transported to their new location. Because each edge can be associated with a difference in 2 coordinates (those of it bounding vertices), we could associate a vector with it, and this vector would be changed to the new vector defined by the 2 new bounding coordinates. In the jargon of modern physics: "Voltages transform as scalars" and "currents transform as vectors". Or in the case of space-time circuits, "currents transform as 4-vectors".

But:
Spinors do not transform like vectors or scalars. So if we we took a "Dirac" circuit, rotated it, and put it back on our drawing board, we would suddenly find all Voltages and currents changed. They would of course be changed according to nice rules, but still..

This weirdness has really nothing to do with circuits, but it is inherent in any description of spinor fields. If you rotate space, you always have to change the values of your spinors according to special rules. You cannot reconstruct spinors from ordinary geometric objects in such a way that you get the correct spinors after rotating the geometry without some external arbitrariness.

So at first it may seem impossible for spinors to live in a circuit diagram. But remember that the spinor components are not directly observable (neither are absolute potential and vector potential). Interestingly, we will see that those quantities constructed from them that are directly observable, actually transform like ordinary vectors and scalars. So the idea will be that although we could build our Dirac equivalent with real components and wires, we have to pretend that we cannot directly observe the currents and voltages. Or more precisely, even if we can observe them, an internal observer living in the circuit could not. 

It turns out that we can express the theory of circuits and its generalization the theory of cell complexes in an elegant way using operators that I will call Cell operators.
A Cell operator is the square root of a unit lattice translation operator. Huh? Let me explain.

Suppose we have a of vertex. Then the Cell operator Ξ_x maps it to the edge on its incrementing side:

 Figure: The action of the Cell  operator Ξ_x on a vertex.

Applying the operator a second time, we map the edge to a vertex, which is the incrementing neighbor of the first vertex.

Figure: The action of the Ξ_x operator on an edge. 

Now we see why Ξ is the square root of the unit lattice translation operator. Applying it twice gives the next door neighbor of the same type, while applying it only once gives an object of a different type, one that links to the neighbor.

Now look what happens when we combine Cell operators of different directions. Starting from a vertex, we get an x-edge by applying Ξ_x. We then get an xy-face by applying Ξ_y. In the right side diagram below we do the same, but in reverse order:

Figure: The successive action of the Ξ_x and Ξ_y operators on a vertex, producing a face whose sign depends on the order of operation.

Crucially, if we reverse the order of these 2 operations, we get the same face, but with a minus sign:

    Ξ_xΞ_{y =} -Ξ_xΞ_y

Now that we intuitively see what the idea is, we can formally define cell algebra:

    Ξ_μΞ_{ν +} Ξ_νΞ_μ = 2 δ_μν Z_μ

With Z_μ the unit lattice translation operator in direction μ, which is equal to Ξ_μ². This squeezes all you need to know in one formula.
This is rather like Clifford algebra or geometric algebra , except for the appearance of the lattice translation operator. But the translation operator becomes infinitely close to identity as the lattice becomes infinitely fine. Thus, cell algebra is a discrete version of geometric algebra, which carries over into geometric algebra in the continuum limit. Actually, there is one other difference: In a special case of geometric algebra called space-time algebra, there is a minus sign in square of the time-like operator. This is to introduce the Minkowski metric. In our case, the metric information will be all in Ohm's law, while the anticommutators defining cell algebra are the same for both space and time.

The power of cell algebra is that it allows us to define the whole cell complex with everything in it in purely algebraic terms. We can start with a single vertex, and create all other vertices by successive applications of the translations operators, which are even powers of Ξ operators. From there, we can create edges, faces, solids, again by application of Cell operators. Let us just list all object types in 4 dimensions, comparing them to the continuous versions from geometric algebra:

 scalar                                                               vertices V
vectors  (γ_t, γ_x, γ_y, γ_z)                                        edges  (Ξ_t, Ξ_x, Ξ_y, Ξ_z)V
bi-vectors (γ_tγ_x, γ_tγ_y, γ_tγ_z, γ_xγ_y, γ_yγ_z, γ_zγ_x)      faces (Ξ_tΞ_x, Ξ_tΞ_y, Ξ_tΞ_z, Ξ_xΞ_y, Ξ_yΞ_z, Ξ_zΞ_x)V                                                              
pseudo vectors (γ_xγ_yγ_z, γ_tγ_yγ_z, γ_tγ_xγ_z, γ_tγ_xγ_y) 3-volumes (Ξ_xΞ_yΞ_z, Ξ_tΞ_yΞ_z, Ξ_tΞ_xΞ_z, Ξ_tΞ_xΞ_y)V                                                          
pseudo scalar  γ_tγ_xγ_yγ_z                                     4-cell Ξ_tΞ_xΞ_yΞ_zV                                

We can define the boundary and coboundary operators on an arbitrary object (M) by using cell algebra. The object (M) is some combination of cell operators.

Define the link operator:

  link operator = (Ξ - Ξ^-1) = (Ξ_x+Ξ_y+Ξ_z+Ξ_t-Ξ_x^-1-Ξ_y^-1-Ξ_z^-1-Ξ_t^-1).

The link operator takes a particular k-chunk , and spits out all components, which will be both (k+1)-chunks and (k-1)-chunks that are linked to that k-chunk. It should be fairly straightforeward to visualize.

An important property of the link operator is that:

    [link operator]²   =  (Z_x + Z_x^-1 - 2)  + (Z_y + Z_y^-1 - 2) + (Z_z + Z_z^-1 - 2) + (Z_t + Z_t^-1 - 2)

Except for a minus sign, this is just the discrete version of the differential operator for the wave equation:

  ∂² = (∂/∂x)² + (∂/∂y)²+ (∂/∂z)²- (∂/∂t)²

    [Dirac Boundary]²   = R_α((Z_x - 1)/R_αx +( Z_x^-1 - 1)/R_αx^-1  + (Z_y - 1)/R_αy + (Z_y^-1 - 1)/R_αy^-1 + ... )

Notation: R_αx means the impedance of the object that connects the object α with iths neighbor in the decrementing x-direction. The resistances act as weights, carrying the information of discretisation step sizes and metric.

Now we can write down the space time circuit diagram for the massless Dirac equation.. There are impedances defined on each component. These are a bit messy to try to draw in a 2 dimensional  figure, so we leave them out. The maths is easier than the visualization.

Figure: Cell structure of the electric circuit equivalent for the Dirac Equation. There should be impedances on each component. These are left out to keep the drawing tidy. (It is maybe not practical to actually try to visualize all this)

We can assign a field quantity (chunk) to each object, and all we have to do is take the Dirac boundary of this potential: 

    γ_μ∂^μ Ψ = 0   [Dirac Boundary] Ψ = 0
Because the Dirac boundary is the square root of the 'd Alembert operator, this must be the Dirac equation.

Now we must establish the relation with the continuous version on a component by component basis. In the usual 3+1 dimensions, we need 4 complex-valued spinor components, which we split into 8 real-valued components. Because we have 16 discrete multivector components, we need only half of them to describe the Dirac equation. We chose only the even-graded ones.
We could have chosen the potential on the odd-graded chunks instead of the even graded chunks. This would have worked equally well. We can map the 2 cases to each other by multiplying with for example Ξ_t.

To see the link with complex numbers, note that various combinations of Cell operators are related to i, the square root of minus one. For example:

  Ξ_xy² = (Ξ_xΞ_y)² = - Ξ_x²Ξ_y² = i² * Z_xZ_y

The translation (Z_xZ_y) again becomes very small (close to identity) if the discretization is very fine.

Take the following relationship:

This is the massless Dirac equation. The actual 1-to-1 correspondence on a component basis becomes apparent by writing out both equations in full. We juggled a with with various forms of i to more clearly show the relation between the continuous and discrete cases. 

To put in mass, we can do a trick that we also used for the Klein Gordon equation. That is, we make 2 copies of the circuit, which we interconnect. We then decompose into symmetric and anti symmetric modes. The antisymmetric modes will have mass, the symmetric modes will remain massless. It it fun to speculate that these 2 modes are just the electron with its neutrino...

Interpretation.

Now that we have a representation of the Dirac equation, we should try and visualize a few solutions. This is under construction...

We also need to explain why the Dirac field transforms as spinors.
We know that translation operators Z(x,y,z) should transform as vectors. But the Ξ are square roots of translation operators:

  Z(x,y,z)  =  Ξ_x^2x  Ξ_y^2y  Ξ_z^2z   =  π_μ  Ξ_μ^2μ   =  π_μ  Ξ_μ^{A_μν  2ν}  =  π_μ  ( Ξ_μ^{A½_μν  ν} )²

it seems reasonable that the square root of a translation operator should transform using the square root of the rotation operator.

But I still have to figure this out a bit better.

Some links

Eric Forgy has drawn my attention to some links and literature on related subjects. A good start is here:
http://math.unm.edu/~stanly/mimetic.html
I missed some of this literature previously, because different key words are used. For example I had never heard of Hodge star, co-boundaries, etc. (My excuse is that I am an engineer, normally working on very different things) A keyword is "mimetic" which means that a discrete system mimics a continuum. The keyword "cell method" refers to a discretisation method that is much like a circuit mathematically, but uses different symbolism. I have tried to learn the lessons from some of this literature, and incorporate it into this page.

Another place where I learned a lot is this newsgroup:

sci.physics.reseach (Google archives all messages)

And a new group, focussing on discrete physics:

sci.physics.discrete (Google archives all messages) 

Further work on electric circuits.