In Situ and In Isolation

The clerk observed that it was only once a year.  "A poor excuse for picking a man's pocket every twenty-fifth of December!  But I suppose you must have the whole day. Be here all the earlier next morning."

                                                                 Charles Dickens

In many situations a physical entity exists in a certain context, and measurements can be performed across the entity in situ, but those measurements return a different value than we would get if we measured across the entity in isolation. This is because the measurement in situ is measuring not just the entity but also the surroundings in parallel. A simple illustration of this is provided by elementary circuit analysis. Consider a set of five resistors connected as shown below.

The upper case symbols R_j represent the in situ resistances measured across each resistor while connected to the overall circuit as shown, and the lower case symbols r_j represent the resistances that would be measured across the individual resistors in isolation. Using the ordinary series-parallel relations we have

and similarly we have

These latter two relations imply

Also, solving (1) for 1/r⁵ and substituting into equations (2), and subtracting one result from the other, we get

Making use of (3) and (4), we get

Likewise we can combine the series-parallel relations

with equation (1) to give the results

Then using the values of r₁ to r₄ from equations (5) and (7), the value of r₅ is given from equation (1) by

For a numerical example, suppose we measure the in situ resistances

in units of ohms. Equations (5) give the individual resistances

When computing the values of r₃ and r₄ from equations (7) we note that R₅ + (R₄ – R₃) = 0 in this example, so the value of r₃ is infinite, meaning it is actually an open circuit. With infinite r₃ we can see from the right hand equation (6) that r₄ = R₄. Thus, making use of (8), we have

Other examples of situations in which the measures of entities may be different in situ than in isolation include the properties of dressed electrons in physics (accounting for interactions with the ambient quantum fields), and revenue streams in economics. In the latter example, the after-tax income from a given revenue stream can be different if the stream is considered in isolation versus in combination with other streams, since the rate of taxation can be affected by the total revenue.

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