Archive for the ‘John Baez’ Category
This post is part of a series
As stated in the Introduction, one of the motivations for this series is to work in parallel with John Baez’ series on network theory to highlight some applications of discrete calculus. In this post, I reformulate some of the material in Part 11 pertaining to Noether’s theorem.
The State-Time Graph
The directed graphs associated with discrete stochastic mechanics are described in the post The Discrete Master Equation, where the simple state-time graph example below was presented
Conceptually, the thing to keep in mind is that any transition from one state to another requires a time step. Therefore a transition from node to node is more precisely a transition from node to node .
On a state-time graph, a discrete 0-form can be written as
and a discrete 1-form can be written as
The Master Equation
The master equation for discrete stochastic mechanics can be expressed simply as
where is a discrete 0-form representing the state at all times with
and is a discrete 1-form representing transition probabilities with
for all . When expanded into components, the master equation becomes
Observables and Expectations
A general discrete 0-form on a state-time graph is defined over all states and all time. However, occasionally, we would like to consider a discrete 0-form defined over all states at a specific point in time. To facilitate this in a component-free manner, denote
so the identity can be expressed as
The discrete 0-form is a projection that projects a general discrete 0-form to a discrete 0-form defined only at time . For instance, given a discrete 0-form , let
In discrete stochastic mechanics, an observable is nothing more than a discrete 0-form
The expectation of an observable with respect to a state is given by
where was defined in a previous post. Note:
In preparation for the discrete Noether’s theorem, note that
For these commutators to vanish, we must have
This implies if and only if is constant on each connected component of the state-time graph.
In this section, we determine the conditions under which the expectation of an observable is constant in time, i.e.
for all . This is a fairly straightforward application of the discrete master equation, i.e.
indicating the condition we’re looking for is
In this section, we demonstrate that when both and are constant in time, this implies
which, in turn, implies . To do this, we first expand
The condition for this trace to vanish is the same as the condition for the commutators above to vanish, i.e.
Expanding the trace further results in
Summing over and when and are constants results in
while summing and in the third term results in
by definition of the transition 1-form. Consequently, when and are constants, it follows that
Finally, this implies if and only if and are constant in time.
This post is part of a series
The first equation is essentially the discrete calculus version of Ohm’s Law, where
is a discrete 1-form representing conductance,
is a discrete 0-form representing voltage, and
In components, this becomes
The second equation is a charge conservation law which simply says
is the sum of all currents into node and
is the sum of all currents out of node . This is more general than it may first appear. The reason is that directed graphs are naturally about spacetime, so the currents here are more like 4-dimensional currents of special relativity. The equation
is related to the corresponding Maxwell’s equation
where is the adjoint exterior derivative and is the 4-current 1-form
This also implies the discrete Ohm’s Law appearing above is 4-dimensional and actually a bit more general than the usual Ohm’s Law.
I’ve been thinking about this off and on since then as time allows, but questions seem to be growing exponentially.
For one, the equation
is curious because it implies that is a derivative, i.e.
Further, although by pure coincidence, in my paper with Urs, we introduced the graph operator
and showed that for any directed graph and any discrete 0-form that
Is it possible that and are related?
I think they are. This brings thoughts of spin networks and Penrose, but I’ll try to refrain from speculating too much beyond mentioning it.
If they were related, this would mean that the discrete Ohm’s Law above simplifies even further to
In components, the above becomes
This expresses an effective conductance in terms of the total number of directed edges connecting the two nodes in either direction, i.e.
If the ‘s appearing in the conductance 1-form are themselves effective conductances resulting from multiple more fundamental directed edges, then we do in fact have
Applications from here can go in any number of directions, so stay tuned!
This post is part of a series
In my last post, I noted that in following John Baez’ series, I’m finding the need to introduce operators that I haven’t previously used in any applications. In this post, I will introduce another. It turns out that we could get away without introducing this concept, but I think it helps motivate some things I will talk about later.
In all previous applications, the important algebra was a noncommutative graded differential algebra. The grading means that the degree of elements add when you multiply them together. For example, the product of two nodes (degree 0) is a node (degree 0+0), the product of a node (degree 0) and a directed edge (degree 1) is a directed edge (degree 0+1), and the product of a directed edge (degree 1) with another directed edge is a directed surface (degree 1+1).
Note the algebra of nodes is a commutative sub-algebra of the full noncommutative graded algebra.
There is another related commutative edge algebra with corresponding edge product.
The edge product is similar to the product of nodes in that it is a projection given by
It is a projection because for an arbitrary discrete 1-form
The product of two discrete 1-forms is
I have not yet come across an application where the full edge algebra is needed. When the product does arise, one of the discrete 1-forms is usual the coboundary of a discrete 0-form, i.e.
When this is the case, the edge product can be expressed as a (graded) commutator in the noncommutative graded algebra, i.e.
An example of this will be seen when we examine electrical circuits.
This post is part of a series
As stated in the Introduction to this series, one of my goals is to follow along with John Baez’ series and reformulate things in the language of discrete calculus. Along the way, I’m coming across operations that I haven’t used in any of my prior applications of discrete calculus to mathematical finance and field theories. For instance, in the The Discrete Master Equation, I introduced a boundary operator
Although, I hope the reason I call this a boundary operator is obvious, it would be more precise to call this something like graph divergence. To see why, consider the boundary of an arbitrary discrete 1-form
A hint of sloppy notation has already crept in here, but we can see that the boundary of a discrete 1-form at a node is the sum of coefficients flowing into node minus the sum of coefficients flowing out of node . This is what you would expect of a divergence operator, but divergence depends on a metric. This operator does not, hence it is topological in nature. It is tempting to call this a topological divergence, but I think graph divergence is a better choice for reasons to be seen later.
One reason the above notation is a bit sloppy is because in the summations, we should really keep track of what directed edges are actually present in the directed graph. Until now, simply setting
if there is no directed edge from node to node was sufficient. Not anymore.
Also, for applications I’ve used discrete calculus so far, there has always only been a single directed edge connecting any two nodes. When applying discrete calculus to electrical circuits, as John has started doing in his series, we obviously would like to consider elements that are in parallel.
I tend to get hung up on notation and have thought about the best way to deal with this. My solution is not perfect and I’m open to suggestions, but what I settled on is to introduce a summation not only over nodes, but also over directed edges connected those nodes. Here it is for an arbitrary discrete 1-form
where is the set of all directed edges from node to node . I’m not 100% enamored, but is handy for performing calculations and doesn’t make me think too much.
For example, with this new notation, the boundary operator is much clearer
As before, this says the graph divergence of at the node is the sum of all coefficients flowing into node minus the sum of all coefficients flowing out of node . Moreover, for any node there can be one or more (or zero) directed edges from into .
I’ve enjoyed applying discrete calculus to various problems since Urs Schreiber and I wrote our paper together back in 2004
Shortly after that, I wrote an informal paper applying the theory to finance in
From there I wrote up some private notes laying the foundations for applying a higher-dimensional version of discrete calculus to interest rate models. However, life intervened, I went to work on Wall Street followed by various career twists leading me to Hong Kong where I am today. The research has laid fairly dormant since then.
I started picking this up again recently when my friend, John Baez, effectively changed careers and started the Azimuth Project. In particular, I’ve recently developed a discrete Burgers equation with corresponding discrete Cole-Hopf transformation, which is summarized – including numerical simulation results – on the Azimuth Forum here:
Motivated by these results, I started looking at a reformulation of the Navier-Stokes equation in
This is still a work-in-progress, but sorting this out is a necessary step to writing down the discrete Navier-Stokes equation.
Even more recently, John began a series of very interesting Azimuth Blog posts on network theory. I knew that network theory and discrete calculus should link up together naturally, but it took a while to see the connection. It finally clicked one night as I laid in bed half asleep in one of those rare “Eureka!” moments. I wrote up the details in
There is much more to be said about the connection between network theory and discrete calculus. I intend to write a series of subsequent posts in parallel to John’s highlighting how his work with Brendan Fong can be presented in terms of discrete calculus.