Thermodynamic Geometry Insights

• Thermodynamic geometry is a mathematical framework that endows thermodynamic state spaces with a Riemannian metric, enabling coordinate-invariant analysis of fluctuations and dissipation.
• It quantifies minimal-dissipation protocols via optimal control paths (geodesics) that minimize work, offering practical guidelines for traversing phase transitions.
• The approach bridges microscopic behaviors and macroscopic phenomena, using scalar curvature and metric properties to signal stability and critical fluctuations.

Thermodynamic geometry is the mathematical framework that endows the space of thermodynamic states or control parameters with a Riemannian structure, allowing physical properties such as fluctuations, dissipation, stability, and phase transitions to be rigorously analyzed through geometric invariants. This approach generalizes classical thermodynamic and statistical characterizations, offering coordinate-invariant, quantitative criteria for microscopic and macroscopic behavior—including phase transitions, optimal control protocols, stability boundaries, and correlations of fluctuations.

1. Foundational Principles: Metric Structure and Thermodynamic Length

The hallmark of thermodynamic geometry is the construction of a Riemannian metric $g_{\mu\nu}$ on the space of control parameters $\lambda^\mu$, typically defined by equilibrium correlation and linear-response functions. In the linear-response regime, the excess work (“availability”) dissipated under a protocol %%%%2%%%% over time $T$ is given by the quadratic form: $$A = \int_0^T g_{\mu\nu}(\lambda(t))\,\dot\lambda^\mu(t)\,\dot\lambda^\nu(t)\,dt$$ where

$$g_{\mu\nu}(\lambda) = \beta \int_0^\infty \langle \delta X_\mu(t)\,\delta X_\nu(0)\rangle_{\mathrm{eq}}\,dt = \beta\,T_{\mu\nu}(\lambda)\,\partial_\mu\partial_\nu\ln Z(\lambda)$$

with $\beta = 1/T$ and $X_\mu$ the thermodynamic variables conjugate to $\lambda^\mu$ (Basri et al., 1 Dec 2025).

The associated “thermodynamic length” of a path $\lambda(s)$, $s\in[0,1]$, is defined as: $$L[\lambda] = \int_0^1 \sqrt{g_{\mu\nu}(\lambda(s))\,\dot\lambda^\mu(s)\,\dot\lambda^\nu(s)}\,ds$$ Through a Cauchy–Schwarz inequality, the dissipated work is bounded below by $A \geq L^2/T$, attaining equality for constant-speed geodesics. The minimization of $L$ for fixed $T$ yields optimal, minimal-dissipation protocols as geodesics of $g$.

2. Divergence Structure near Second-Order Phase Transitions

Approaching critical points (e.g., $(T_c,h_c)$), the equilibrium response functions and relaxation times diverge, leading to divergence of $g_{\mu\nu}$. In statistical models described by Widom scaling, the singular part of the free-energy density $\phi$ scales as $\phi(t,h_r)\sim t^b f(h_r/t^\Delta)$, where $t = (T-T_c)/T_c$, $h_r = (h-h_c)/h_c$, and $f$ is a scaling function (Basri et al., 1 Dec 2025).

Key scaling relations:

• Static susceptibilities:

$$\frac{\partial^2\phi}{\partial t^2} \sim t^{b-2},\quad \frac{\partial^2\phi}{\partial t\partial h_r} \sim t^{b-\Delta-1},\quad \frac{\partial^2\phi}{\partial h_r^2} \sim t^{b-2\Delta}$$

• Relaxation time: $\tau \sim t^{-bz/d}$, for $h_r/t^\Delta = O(1)$, with $z$ and $d$ the dynamical and spatial exponents.

The metric divergence leads to possible breakdowns of the geometric picture, including non-existence or incompleteness of geodesics.

3. Criteria for Finite-Length Crossing of Critical Points

Despite the divergence of $g_{\mu\nu}$ at criticality, the integrated thermodynamic length $L$ may remain finite if the divergence is sufficiently mild. For trajectories approaching the critical point as $\lambda(s)\sim s^\gamma$, the line element scales as $$g_{\mu\nu}\dot\lambda^\mu\dot\lambda^\nu \sim s^{-\alpha}$$, with

$$\alpha = 2 - b(1 - z/d) \quad \text{for } h_r/t^\Delta=O(1),$$

and the length integral converges near $s=0$ if $\alpha<2$, i.e., $b(1-z/d)>0$. Similarly, along $h_r/t^\Delta\to\infty$, the exponent is $\alpha = 2 - p(1-z/d)$, requiring $p(1-z/d)>0$. These conditions delimit universality classes in which a finite-length crossing is possible.

Concrete implications:

• $z/d>1 \rightarrow$ infinite thermodynamic length (e.g., 2D Ising, Potts)
• $z/d < 1 \rightarrow$ finite-length crossing possible (e.g., 3D Ising with $z/d\approx0.68$) However, even with finite $L$, the curvature (Riemann tensor) of the thermodynamic manifold generically diverges at the critical point.

4. Numerical Computation of Minimal-Dissipation Paths

When $g_{\mu\nu}$ is only available numerically (e.g., from mean-field or simulation data), geodesic distances and optimal protocols can be constructed via discrete gridding and fast-marching or level-set methods. In practice, for the antiferromagnetic Ising system:

• Discretize the $(\beta,h)$ plane.
• Solve the mean-field equations at each grid point for magnetizations.
• Compute $g_{\mu\nu}(\beta,h)$ from the linear-response integrals.
• Use the Fast Marching Method to obtain the minimal distance field $D(\beta,h)$.
• Recover the geodesic by steepest descent from target to source in $-∇D$.

Four regime types appear: (i) Both endpoints disordered: optimal path runs along infinite temperature direction (metric rank-deficient). (ii) Endpoints in different phases: cross the critical line rapidly, then minimize via disordered-phase directions. (iii) Both endpoints in ordered phase with direct shorter geodesic: stay in ordered manifold. (iv) Shortest route crosses the transition twice: excursions into disordered regions are “cheap” in metric length.

These computations confirm that, when $L$ is finite, the true minimum-dissipation protocol can exploit phase transitions even if start and end points share a phase (Basri et al., 1 Dec 2025).

5. Geometric Interpretation of Fluctuations, Stability, and Correlation Lengths

Thermodynamic geometry generalizes the equilibrium fluctuation theory by relating metric components to variances and covariances of thermodynamic variables; in the entropy representation: $$g_{ij} = -\frac{\partial^2 S}{\partial X^i \partial X^j}$$ On the state manifold, local heat capacities correspond to $g_{ii}$, with global stability requiring positive-definiteness of $g_{ij}$ and $\det g>0$ (Bellucci et al., 2010).

The scalar curvature $R$ encodes the strength and nature of correlations:

• $\xi \sim |R|^{-1/2}$ (2D): divergence of $R$ marks phase transitions.
• Sign of $R$ distinguishes attractive ($R<0$) vs. repulsive ($R>0$) regimes.

At spinodal or critical points, divergence of $R$ signals instability and onset of critical fluctuations.

6. Riemannian Structure, Geodesic Equations, and Broader Implications

The optimal control problem is reduced to solving the geodesic equations: $$\frac{d^2\lambda^\sigma}{ds^2} + \Gamma^\sigma_{\mu\nu}(\lambda)\,\frac{d\lambda^\mu}{ds}\,\frac{d\lambda^\nu}{ds} = 0$$ where the Christoffel symbols are constructed from $g_{\mu\nu}$. This formalism is applicable across classical fluids, spin systems, quantum phase transitions, and computational protocols in finite-time thermodynamics (Bellucci et al., 2010, Basri et al., 1 Dec 2025).

Thermodynamic geometry thereby provides:

• Quantitative criteria for stability and criticality via curvature and determinant conditions.
• Unification of fluctuation theory, optimal dissipation, and linear response.
• Connection between phase transition theory and transport-based optimal control.

Extensions to quantum systems utilize contact and fiber-bundle structures, where the divergence of geodesic length toward rank-deficient quantum states geometrizes the third law of thermodynamics (Tejero et al., 13 Nov 2025).

7. Summary Table: Key Features of Thermodynamic Geometry

Feature	Mathematical Formulation	Physical Interpretation
Metric tensor $g_{\mu\nu}$	Equilibrium covariances, linear response	Fluctuations, response, dissipation
Thermodynamic length $L$	Integral along protocol geodesic	Minimal work, protocol optimality
Scalar curvature $R$	Determinant/Hessian of $g_{\mu\nu}$	Correlation strength, phase transition
Geodesics	Euler–Lagrange eqs, Christoffel symbols	Minimum-dissipation protocol trajectories
Divergence at criticality	$g_{\mu\nu}$, $R$ blow up	Breakdown of stability, fluctuation scaling

In total, thermodynamic geometry rigorously connects equilibrium fluctuation phenomena, optimal nonequilibrium control, and phase-transition theory in a unified, coordinate-invariant framework. Geometric criteria characterize when dissipation can be minimized across phase transitions and elucidate the role of critical exponents and universality. Numerical and analytical methods enable explicit protocol construction and stability diagnostics even in regimes with divergent fluctuation behavior (Basri et al., 1 Dec 2025, Bellucci et al., 2010).