Susceptibility equals variance of magnetization

Let $\mu_{\Lambda,\beta,h}$ be a finite-volume Gibbs measure on a finite region $\Lambda$, with inverse temperature $\beta>0$ and external magnetic field $h$ coupled linearly to the total magnetization $M_\Lambda(\sigma)=\sum_{x\in\Lambda}\sigma_x$ (e.g. for the Ising model ). Let

• $Z_\Lambda(\beta,h)$ be the partition function ,
• $p_\Lambda(\beta,h)=\frac{1}{|\Lambda|}\log Z_\Lambda(\beta,h)$ be the (finite-volume) pressure ,
• $m_\Lambda(\beta,h)=\frac{1}{|\Lambda|}\,\mathbb{E}_{\mu_{\Lambda,\beta,h}}[M_\Lambda]$ be the magnetization density,
• $\chi_\Lambda(\beta,h)=\partial_h m_\Lambda(\beta,h)$ be the (finite-volume) susceptibility per site.

Statement

Assume the Hamiltonian depends on $h$ only through a term $-h\,M_\Lambda$ (linear field coupling). Then $\chi_\Lambda(\beta,h)$ exists and

Equivalently,

Key hypotheses

• Finite region $\Lambda$ and a well-defined finite-volume Gibbs measure .
• External field enters the energy as $H_{\Lambda,h}=H_{\Lambda,0}-h\,M_\Lambda$ (linear coupling).
• Expectations and variances in the ensemble are finite (automatic for bounded spins).

Conclusions

• $\chi_\Lambda(\beta,h)\ge 0$ since it is proportional to a variance.
• Susceptibility is a fluctuation: large magnetization fluctuations imply large response to $h$.
• This is a concrete instance of the fluctuation–response principle .

Proof idea / significance

Differentiate $\log Z_\Lambda(\beta,h)$ with respect to $h$. The first derivative gives $\beta\,\mathbb{E}[M_\Lambda]$, and the second derivative produces the covariance (variance) term. The identity underlies many response formulas and connects critical behavior to diverging fluctuations.