David Bowman wrote:
* * *
.. . . Suppose we connect two similar
such systems together such that the total potential energy is given by
U = A_1*(x_1)^2 + A_2*(x_2)^2. Now assume that the initial state is one with
x_1 = X and x_2 = 0. Assume also that the systems are connected in such a
way that there is a conservation constraint connecting x_1 and x_2 such that
x_1 + x_2 must be a constant. Including this constraint (x_1 + x_2 = X) into
the equation for U gives: U = A_1*(x_1)^2 + A_2*(X - x_1)^2. . . .
.. . . If the system is not conservative, but rather dissipative, then
the parameter x_1 will continue to move in such a way as to generate entropy
in both the system and its surroundings (as per the 2nd law). This
neccessitates that the system's energy will spread out among as many
degrees of freedom as possible. This will eventually result in the system's
macroscopic potential energy being dissipated as much as possible (consistent
with the assumed x_1 + x_2 conservation law/constraint) into all the
microscopic degrees of freedom that interact with the system's
macroscopically displaced state and each other. The final equilibrium will
effectively minimize the macroscopic U function above. Now it is
straightforward to minimize the above formula for U over the possible values
.. . .
David Bowman
dbowman@gtc.georgetown.ky.us
* * *
This is very interesting and should be compared with another (perhaps
more common ) approach:
David has used:
1) Energy Function: Q1^2/(2*C1) + Q2^2/(2*C2)
2) Initial State: Q1=Q ; Q2=0
3) Conservation of Charge: Q1 + Q2 = Q
4) He has then determined the final state by minimizing the energy function.
The other approach referred to would use 1), 2) and 3) as above,
but instead of 4), above, it would use:
4') Equality of potential difference in the final state: Q1/C1 = Q2/C2
Given 1), 2) and 3), 4) and 4') imply each other.
The "other" approach is what is usually seen (it requires no calculus!).
The equality of the two approaches is an insight which can be carried over
to mathematiclly equivalent phenomena, such as the totally inelastic
collision of two particles. (See my article "Mandated Energy Dissipation -
e pluribus unum" in AJP, Oct(?), 1996.