Phase coexistence implies nondifferentiability of the pressure

Consider a lattice system in finite volume $\Lambda$ with an external field $h$ coupled linearly to an extensive order parameter $A_\Lambda$: $H_{\Lambda,h}=H_{\Lambda,0}-h\,A_\Lambda$. Let $Z_\Lambda(\beta,h)$ be the partition function and $p_\Lambda(\beta,h)=\frac{1}{|\Lambda|}\log Z_\Lambda(\beta,h)$ the finite-volume pressure .

Assume the thermodynamic limit pressure exists:

with the limit taken along a standard van Hove sequence.

Statement

The function $h\mapsto p(\beta,h)$ is convex. Moreover:

• If $p(\beta,h)$ is differentiable at a given $h$, then every infinite-volume Gibbs measure $\mu$ at parameters $(\beta,h)$ has the same order-parameter density

  $$a(\mu)=\lim_{\Lambda\uparrow\mathbb{Z}^d}\frac{1}{|\Lambda|}\,\mathbb{E}_\mu[A_\Lambda],$$

  and it is fixed by the derivative:

  $$a(\mu)=\frac{1}{\beta}\,\partial_h p(\beta,h).$$

• Conversely, if there exist two Gibbs measures $\mu_1,\mu_2$ at the same $(\beta,h)$ with

  $$a(\mu_1)\neq a(\mu_2),$$

  then $p(\beta,h)$ is not differentiable at $h$; equivalently, the left and right derivatives satisfy

  $$\partial_h^- p(\beta,h) < \partial_h^+ p(\beta,h).$$

  This is a first-order transition / phase coexistence signature in the sense of Gibbs-phase transitions .

In the important special case $A_\Lambda=M_\Lambda$ (magnetization), this connects coexistence of distinct magnetized phases to a cusp in $h\mapsto p(\beta,h)$.

Key hypotheses

• Linear field coupling $-hA_\Lambda$.
• Existence of the thermodynamic limit pressure $p(\beta,h)$.
• Existence of at least two infinite-volume Gibbs measures at the same $(\beta,h)$ with different $a(\mu)$ (for the “coexistence $\Rightarrow$ nondifferentiability” direction).

Conclusions

• Differentiability of $p(\beta,h)$ at $h$ rules out coexistence of Gibbs states with different densities of the conjugate order parameter.
• Coexistence forces a non-single-valued “slope” at $h$ (a nontrivial subgradient), hence nondifferentiability.
• In convex-analysis language (see convex functions ), multiple coexisting values of $a(\mu)$ correspond to a nontrivial subdifferential of $p(\beta,\cdot)$ at $h$.

Proof idea / significance

In finite volume, differentiate $\log Z_\Lambda(\beta,h)$ with respect to $h$ to obtain identities of the form $\partial_h p_\Lambda(\beta,h)=\frac{\beta}{|\Lambda|}\mathbb{E}[A_\Lambda]$; this is a special case of derivatives as connected correlations (and yields fluctuation formulas such as susceptibility = variance for magnetization when differentiating again). Passing to the thermodynamic limit, convexity guarantees existence of one-sided derivatives. If distinct Gibbs measures at the same $h$ realize distinct limiting densities $a(\mu)$, then $p(\beta,\cdot)$ must have more than one supporting slope at $h$, i.e. it cannot be differentiable there.