Quantum Gravity 317.0: Thermodynamic Laws and Probable Causation/Influence (PI)
- From: OsherD <mdoctorow@xxxxxxxxx>
- Date: Fri, 27 Mar 2009 11:15:03 -0700 (PDT)
From Osher Doctorow
From the Fundamental thermodynamic Relation (1st and 2nd Laws ofThermodynamics combined), we get that if a system has p different
kinds of particles and the volume V is its only external variable,
then:
1) dU = TdS - pdV - sum ui dNi, T = temperature, p = pressure, S =
entropy, Ni = particle number, sum from i = 1 to p, U = internal
energy.
Here the influence of pressure (a force per area variable) on internal
energy U from the viewpoint of Causal Probability/Influence (PI) can
be seen from the negative sign on pdV (negative signs indicating
Causes, positive signs Effects), although this can be differently
interpreted in various scenarios.
Also, from:
2) DS(U) = T, where DS is the partial derivative with respect to S
holding V and {Ni} constant
we see that in terms of Birkhoff Causation (closely related to PI), T
(temperature) is the Cause of Internal Energy with the other variables
held fixed.
Similarly, from:
3) DV(U) = -p (S, {Ni} fixed)
we see that pressure is the Cause of internal energy with the other
variables held fixed.
It also turns out that:
4) DNi(U) = ui (S, V, {Ni for j not equal to i fixed}, ui chemical
potentials
the chemical potentials are the Cause of particle numbers.
By the way, we also can obtain:
5) DV(S) = p/T (U, {Ni} fixed)
so that pressure divided by temperature Cause the Entropy with the
other variables held fixed. Here force and energy seem to interact.
See Wikipedia's "Thermodynamic Equations".
Osher Doctorow
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