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Differentiate the normalized average, including the derivative of its denominator. The enthalpy fluctuation formula follows fromSince $d\beta /dT=-1/(k_B{T}^2)$,Thus a larger equilibrium spread of enthalpy corresponds to a larger response to heating at fixed pressure. The variance is nonnegative, so this finite-system equilibrium description has $C_P\ge 0$. The relation connects a response coefficient to fluctuations of the appropriate exchanged quantity; at fixed volume the corresponding quantity is internal energy.

Away from singular regimes, assume enthalpy and heat capacity at constant pressure are extensive: $⟨H⟩=Nh$ and $C_P=N{c}_P$, with nonzero intensive $h$. The enthalpy fluctuation formula then givesAbsolute fluctuations grow as $\sqrt{N}$, while the mean grows as $N$. In the thermodynamic limit, enthalpy per particle therefore concentrates around its mean; it is relative, rather than absolute, fluctuations that vanish. This is the usual relative fluctuations of an extensive observable scaling. It assumes ordinary extensive heat capacity and a nonzero mean with a stated energy reference; a critical divergence or an artificially zero mean requires separate analysis.