Susceptibility

In statistical mechanics, a susceptibility is a linear-response coefficient: it measures how much an ensemble average of an observable changes when one perturbs the Hamiltonian by a small conjugate field.

Definition (response to a field)

Suppose a field $h$ couples linearly to an extensive observable $M$ (e.g. magnetization) through

with equilibrium described by the canonical ensemble at inverse temperature $\beta$. Let $V$ denote volume (or number of sites). The susceptibility per unit volume is

This is the prototypical example; more generally, susceptibilities form a matrix of derivatives relating intensive “fields” to extensive “responses,” mirroring the thermodynamic distinction between intensive and extensive variables.

Partition-function and fluctuation formulas

Let $Z(\beta,h)$ be the canonical partition function for $H_h$. Then

so the susceptibility can be written as a second derivative:

Using standard identities (see fluctuation formulas from $\log Z$ ), one obtains the fluctuation–response relation

where the variance is the ensemble variance of $M$.

More generally, if $h$ couples to $X$ and one measures the response of $A$, then

expressed via the ensemble covariance .

Correlation-function representation

If $M$ is a spatial sum of local degrees of freedom, for example $M=\sum_x s_x$, then translation invariance gives

so

Here $\langle s_x s_0\rangle_c$ is the connected two-point correlation . This identity makes the link to the correlation length transparent: if connected correlations decay slowly in space, the spatial sum is large, and the susceptibility becomes large.

Physical interpretation

• Large susceptibility means the system is easily polarized/ordered by a small field; microscopically, this corresponds to large fluctuations (large $\mathrm{Var}(M)$) and/or long-range correlations.
• Near continuous phase transitions, the susceptibility often grows strongly or diverges in the thermodynamic limit , reflecting the growth of correlated domains measured by the correlation length.