 ppauper
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### point vortices in heat conduction/finance

OK: the steady state heat conduction equation is Laplace's equation a steady system of point vortices satisfies this equation except at the vortices themselvesso....what's the interpretation of point vortices in 2D steady heat conduction ? a steady infinite heat source at isolated points ? or something else ?and we can go from Black-Scholes to heat conduction and vice versa, so what's the interpretation in finance ?any takers ? Posts: 23951
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### point vortices in heat conduction/finance

First, unless you can create Escher's impossible staircase in heat gradients (T_a > T_b > T_c > T_a), you won't get vortices. Second, I thought Laplace's equation only applied to irrotational flows. rcohen
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### point vortices in heat conduction/finance

QuoteOriginally posted by: ppauperOK: the steady state heat conduction equation is Laplace's equation a steady system of point vortices satisfies this equation except at the vortices themselvesso....what's the interpretation of point vortices in 2D steady heat conduction ? a steady infinite heat source at isolated points ? or something else ?and we can go from Black-Scholes to heat conduction and vice versa, so what's the interpretation in finance ?any takers ?ppauper,I don't know what the equivalent of a vortex or a point heat/flow source is in finance. However, what I can add to your comments is that a vortex is generally associated with fluid flow. I'm not aware of anything in heat conduction that's analogous to a vortex. In classical fluid mechanics, there are two distinct uses for Laplace's equation. One is in reference to the stream function and the other to the potential function. The stream function is derived from the continuity equation for incompressible flows. The potential function, on the other hand, relates to inviscid, irrotational flows. In irrotational flows, both satisfy Laplace's equation.It is the potential function that you're probably referring to in relation to heat flow in heat conduction problems. In this case, the point flow source in fluid mechanics would be exactly equivalent to a point heat source in heat conduction. It is the "isotherm," or the line of constant temperature in heat conduction.
Last edited by rcohen on May 3rd, 2007, 10:00 pm, edited 1 time in total. ppauper
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### point vortices in heat conduction/finance

QuoteOriginally posted by: Traden4AlphaSecond, I thought Laplace's equation only applied to irrotational flows.I don't want to get side-tracked from my original question, but as rcohen said, if you've got a two-dimensional incompressible fluid, you can introduce a streamfunction via .That is what I'm talking about.Point vortices satisfy except at the point itself, and the streamfunction is infinite (or maybe negative infinity) at the vortex itself ppauper
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### point vortices in heat conduction/finance

QuoteOriginally posted by: rcohenI don't know what the equivalent of a vortex or a point heat/flow source is in finance. However, what I can add to your comments is that a vortex is generally associated with fluid flow. I'm not aware of anything in heat conduction that's analogous to a vortex. (snip)It is the potential function that you're probably referring to in relation to heat flow in heat conduction problemsNo, I wasn't talking about the potential.(but if I recall my hydrodynamics, isn't the potential for a system of sources and sinks the sameas the streamfunction for a system of vortices ? that's a side issue that's not that important so if I've mis-spoken please no one go off on a tangent....)I was saying:(1) steady state heat conduction in 2D satisfies the 2D Laplace equation (2) given a solution to the 2D Laplace equation, we can interpret it in the context of heat flow(3) the streamfunction for point vortices satisfy the 2D Laplace equation except at the points themselves. If we take this solution to the 2D Laplace equation, how do we interpret it in terms of heat conduction(4) once we know (3), apply it to Black-Scholes by transforming the steady BS equation into the steady heat equation EBal
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### point vortices in heat conduction/finance

BS is one-dimensional, isn't it? ppauper
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### point vortices in heat conduction/finance

QuoteOriginally posted by: EBalBS is one-dimensional, isn't it?but if you have 2 stocks or such like ? EBal
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### point vortices in heat conduction/finance

It's been a long time since I last touched this, but I believe that In 2D any analytic function of is a solution of the Laplace equation. Logarithm of z is the solution which corresponds to both vortex and sink for potential incompressible 2 dimensional flow, because both vorticity and potential satisfy the same Laplace equation. For stationary heat equation you could use both real (log) and imaginary (angle) parts of the above solution but one of them has a discontinuity along a curve.Let's say you have two stocks following uncorrelated GBM and for simplicity interest rate is zero. Then you are asking about stationary solutions of 2D Laplace equation in terms of . What is the meaning of the logarithmic solution in terms of these variables. I don't know it has any meaning at all. Why are you asking? unkpath
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### point vortices in heat conduction/finance

I think you may be confused here, but the Laplace equation you are talking about has nothing to do with diffusion. You are talking about incompressible potential flow in 2D, so do not confuse that with the heat operator, at least physically it is different. That said, you could still try to apply the point vortex bag of tricks, but I am not sure that this is the way to go.Also, in which case is the time-independent BS equation relevant? Isn't it the case only for perpetual claims, which are relevant, but not that relevant. QuoteOriginally posted by: ppauperQuoteOriginally posted by: EBalBS is one-dimensional, isn't it?but if you have 2 stocks or such like ? ppauper
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### point vortices in heat conduction/finance

QuoteOriginally posted by: EBal What is the meaning of the logarithmic solution in terms of these variables. I don't know it has any meaning at all. Why are you asking?I'm just curious and wondering whether something interesting could come out of it ppauper
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### point vortices in heat conduction/finance

QuoteOriginally posted by: unkpathI think you may be confused here, but the Laplace equation you are talking about has nothing to do with diffusion. You are talking about incompressible potential flow in 2D, so do not confuse that with the heat operator, at least physically it is different. no, I'm not confused in the least.I'm saying that we have a solution to Laplace's equation from one physical situation.I'm asking what the interpretation of that solution is in a very different context where Laplace's solution applies.That seems to be very clear, but folks seem to be having difficulty understanding what I'm asking.QuoteAlso, in which case is the time-independent BS equation relevant? Isn't it the case only for perpetual claimsit certainly applies to (non-bermudan) perpetual options.  