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This is a thread to discuss any topics related to this amazing book:
Re: Section 15, Dirichlet operators and electrical circuits.
Gloss: This shows how a Hamiltonian framework can be applied to a network of resistors. Let the nodes in the circuit be x1,...,xn. Between xi and xi is a resistor, with conductance (= reciprocal of resistance) cij. Form the symmetric matrix H which has cij in each non-diagonal entry, i.e., Hij = Hji = cij, and let Hii be minus the sum of all the other values in the ith row (or ith column, same thing here). By construction, H is both self-adjoint and infinitesimal stochastic. Such a matrix is called a Dirichlet operator. Because H is self-adjoint, it is a valid quantum mechanical operator, and because it is infinitesmal stochastic, it is a valid stochastic mechanical operator. So it is in an overlapping territory between the two theories.
Let V be a vector in R^n, representing the voltage at each of the points x1,...,xn. The book shows that <V, H(V)> equals the power consumed by circuit!
Here I will add a few points, to this charming topic that has been introduced.
(1) H(V) itself has a physical interpretation, which is the vector I of currents that is induced by the voltage vector across the resistor network. The signs are oriented so Ij is the net flow of current into the node xj.
It just requires a simple calculation to show this. To get some practice, let's do it for an example with three nodes x1, x2, x3, where there is a 1 ohm resistor connecting each pair of points.
Then H = ((-2, 1, 1), (1, -2, 1), (1, 1, -2)).
Let the voltage vector be (a,b,c).
H(a,b,c) = (-2a + b + c, a - 2b + c, a + b - 2c) = (b-a + c-a, a-b + c-b, a-c + b-c) = (inflow into a, inflow into b, inflow into c) = current I
(2) As a conclusion, it follows that <V, H(V)> = <V, I> = power consumed by the network.
(3) To give a full physical interpretation of the Hamiltonian dynamic on this network of resistors, we should connect a capacitor between each node and a ground point. Give them all the same unit capacitance. This is what will make the current H(V)(j) into node j actually produce a rising voltage at V(j) -- this condition is needed to fulfill the equation dV / dt = H(V).
(4) As was pointed out in the text, any constant voltage vector K = (a,a,a,...) will satisfy H(K) = 0. Interpretation: There is no current when all voltages are equal.
(5) Intuitively, we see that any given initial state of the network, given by a voltage vector V, the network will asymptotically charge/discharge to an equilibrium state where, for each connected component of the graph, all voltages are the same.
For each connected component C, the final voltage is easily calculated, by dividing the total charge in C in the network by the number of nodes in C:
FinalVoltage = Sum(V(j)) / n = mean(V(C)), for all j in C.
(6) An eigenvector is a voltage vector V that heads in a "straight line" towards the equilibrium vector. A non-eigenvector does some "turning" as it heads towards the equilibrium vector.