Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 356 / 1 / d

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- Solution d

Solution

The only independent current is

J (t) = α P_{1} - β P_{2} = r e^{- (α + β) t} .

Hence the total transient entropy production is

\dot{S}_{tot} = J (t) lo g \frac{α P _{1} ( t )}{β P _{2} ( t )} \geq 0.

Its relaxational part may be written

\dot{S}_{rel} = J lo g \frac{P _{1} P _{2}^{ss}}{P _{2} P _{1}^{ss}}, (P_{1}^{ss}, P_{2}^{ss}) = \frac{1}{α + β} (β, α),

while the steady or housekeeping entropy production is zero. Both the current and total production vanish as

t \to \infty

.

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