Assuming that the heat bath's internal energy is much larger than the energy of S (E ≫ E i), we can Taylor-expand Ω Ω B to first order in E i and use the thermodynamic relation ∂ ∂ S B / ∂ ∂ E = 1 / T , where here S B , T are the entropy and temperature of the bath respectively: k ln ⁡ ⁡ p i = k ln ⁡ ⁡ Ω Ω B ( E − − E i ) − − k ln ⁡ ⁡ Ω Ω ( S , B ) ( E ) ≈ ≈ − − ∂ ∂ ( k ln ⁡ ⁡ Ω Ω B ( E ) ) ∂ ∂ E E i + k ln ⁡ ⁡ Ω Ω B ( E ) − − k ln ⁡ ⁡ Ω Ω ( S , B ) ( E ) ≈ ≈ − − ∂ ∂ S B ∂ ∂ E E i + k ln ⁡ ⁡ Ω Ω B ( E ) Ω Ω ( S , B ) ( E ) ≈ ≈ − − E i T + k ln ⁡ ⁡ Ω Ω B ( E ) Ω Ω ( S , B ) ( E )