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Geometric Thermodynamics of Cycles: Curvature and Local Thermodynamic Response

Eric R. Bittner

Abstract

Classical thermodynamics contains familiar geometric relations associated with cyclic processes, most notably the identification of mechanical work with the area enclosed by a trajectory in the $(P,V)$ plane. We show that the area laws for work and reversible heat arise as projections of a single canonical two--form defined on the equilibrium thermodynamic manifold, providing a unified description of thermodynamic cycles in both the $(P,V)$ and $(T,S)$ representations. The same structure yields a direct link between cycle geometry and thermodynamic response: the work generated by infinitesimal cycles is set locally by the mixed curvature $U_{SV}$ of the equilibrium energy surface, which can be expressed in terms of measurable susceptibilities. This identifies thermodynamic work as a local geometric field over state space rather than solely a global property of cyclic processes. More broadly, the framework connects classical cycle geometry to stochastic thermodynamic trajectories, providing a geometric interpretation of nonequilibrium work relations such as the Jarzynski equality.

Geometric Thermodynamics of Cycles: Curvature and Local Thermodynamic Response

Abstract

Classical thermodynamics contains familiar geometric relations associated with cyclic processes, most notably the identification of mechanical work with the area enclosed by a trajectory in the plane. We show that the area laws for work and reversible heat arise as projections of a single canonical two--form defined on the equilibrium thermodynamic manifold, providing a unified description of thermodynamic cycles in both the and representations. The same structure yields a direct link between cycle geometry and thermodynamic response: the work generated by infinitesimal cycles is set locally by the mixed curvature of the equilibrium energy surface, which can be expressed in terms of measurable susceptibilities. This identifies thermodynamic work as a local geometric field over state space rather than solely a global property of cyclic processes. More broadly, the framework connects classical cycle geometry to stochastic thermodynamic trajectories, providing a geometric interpretation of nonequilibrium work relations such as the Jarzynski equality.
Paper Structure (9 sections, 1 theorem, 60 equations, 2 figures)

This paper contains 9 sections, 1 theorem, 60 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Theoretical Framework
  3. Contact and Symplectic Structure of Thermodynamic Cycles
  4. Local Geometry of Heat Flow
  5. Geometric Origin of Thermodynamic Area Laws
  6. Fluctuating Thermodynamic Trajectories
  7. Thermodynamic Response and Local Cycle Geometry
  8. Conclusions
  9. Mapping a Quasistatic Cycle from $(P,V)$ to $(T,S)$

Key Result

Theorem 1

Let $(\mathcal{T},\alpha)$ be the thermodynamic contact manifold of a simple compressible system with contact form Let $\iota:\mathcal{E} \hookrightarrow \mathcal{T}$ denote the inclusion of the equilibrium Legendre submanifold $\mathcal{E}$ satisfying $\alpha = 0$. Then In the energy representation $U=U(S,V)$, where denotes the mixed second derivative of the energy surface. For any reversible

Figures (2)

  • Figure 1: Local mapping of a quasistatic cycle from the $(P,V)$ plane to the $(T,S)$ plane for an ideal gas. (a) An elliptical cycle in the $(P,V)$ plane centered at $(V_0,P_0)$ with semi-axes $a$ and $b$. (b) The corresponding image of the same cycle in the $(T,S)$ plane under the exact ideal-gas mapping. The black points indicate the reference state $(V_0,P_0)$ and its image $(S(P_0,V_0),T_0)$. The nonlinear transformation from $(P,V)$ to $(T,S)$ does not preserve geometric centers, so the image of the center state need not coincide with the geometric center of the mapped curve. Parameters used are $P_0=2$, $V_0=3$, $a=0.4$, $b=0.3$, $n=1$, $C_V=3/2$, and units are chosen such that $R=1$.
  • Figure 2: Geometric structure of thermodynamics in the energy representation. The equilibrium states form a surface $U(S,V)$, whose tangent plane encodes the First Law. Thermodynamic cycles correspond to closed curves on this surface, and their projections onto lower-dimensional planes define the geometric structure of work and heat. The projected area on the $SV$ plane corresponds to the canonical two-form $\Omega = -U_{SV} dS \wedge dV$, which gives the geometric origin of thermodynamic work relations.

Theorems & Definitions (2)

  • Theorem 1: Canonical thermodynamic two--form
  • proof