G ≡ U - τσ + pV

where U is the internal energy, τ is the temperature, σ is the entropy, p is pressure, V is volume.

Therefore,

dG = dU - τdσ - σdτ + pdV + Vdp

Now: "Consider a system in thermal contact with a heat reservoir R

_{1}at temperature τ and in mechanical contact with a pressure reservoir R

_{2}that maintains the pressure p, but cannot exchange heat." (Kittel p.262)

In this system S, dτ=0 and dp=0, so

dG

_{S}＝ dU

_{S}- τdσ

_{S}+ pdV

_{S}

In equilibrium, this equals zero for each set of (τ,p).

In monetary economics, excess demand for money can be derived from equation of exchange as follows:

G ≡ PY - MV

where P is the price, Y is the output, M is the money, V is the velocity of money.

Therefore,

dG = dPY + PdY - dMV - MdV

Now, consider a mediating system in financial contact with a central bank R

_{1}that maintains the money M and in transactional contact with a real economy R

_{2}that maintains the output Y.

In this system S, dM=0 and dY=0, so

dG

_{S}＝ YdP

_{S}- MdV

_{S}

In equilibrium, this equals zero for each set of (M,Y).

This analogy is the idea behind the previous post.

From this analogy, you can derive the monetary economics version of Clausius-Clapeyron relation. Note that in this analogy p(pressure) corresponds to Y(output), V(volume) corresponds to P(price), τ(temperature) corresponds to M(money), and σ(entropy) corresponds to V(velocity of money).

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