On the page, a new way to think about irreversible processes emerges from the geometry of the spaces where physics plays out. Instead of adding a new equation to describe entropy, the authors tap a tiny but powerful detail: the shape of the space itself, encoded in something called a Poisson structure. It’s as if the road you travel on already has hidden hills and valleys that steer you toward disorder, even when you start with energy conserved. This is not a techno trick but a way of reading the deep math that underpins motion, heat, and the arrow of time.
Written by Erwin Luesink at the University of Amsterdam’s Korteweg–de Vries Institute of Mathematics, the paper builds a bridge between abstract geometry and tangible physics. The core idea is striking in its economy: take the reversible lanes of Hamiltonian mechanics and gently deform them in a way that preserves energy but lets entropy rise. The engine behind this idea is a mathematical object called a 2 cocycle in the second Lichnerowicz–Poisson cohomology. When that cohomology isn’t trivial, you can deform the Poisson bracket in a way that creates a new symmetric structure. Pair that with the original Hamiltonian, and you get flows that move along energy-conserving paths while entropy creeps upward, in harmony with the second law of thermodynamics.
Irreversible dynamics from Poisson geometry
Geometrically, the reversible backbone of physics sits in the Poisson bracket, a bilinear operation that tells you how observables talk to each other. In many systems, this bracket is degenerate, meaning some quantities never change under motion. Those conserved quantities, or Casimirs, are the seeds of entropy in the new framework. The trick is not to bulldoze the geometry with a metric but to layer a symmetric bracket on top of it. When you combine the Hamiltonian with this symmetric piece, you obtain a dynamics that preserves energy yet produces entropy. The result is a concrete realization of ideas that have haunted nonequilibrium thermodynamics for a century, but now anchored in pure geometry.
The heart of the construction is a deformation of the Poisson structure. If you take the original Poisson bivector π and add a small piece εa, you get πε = π + εa. The crux is that a must be a Poisson 2-cocycle, i.e., it lies in the kernel of the Poisson differential dπ but is not a trivial coboundary. When such a nontrivial cocycle exists, you can build a symmetric 2-bracket from a 4-tensor that encodes the deformation. The upshot: you now have a mathematical gadget that respects energy and also fuels entropy production. In practice, this means you can describe irreversible dynamics as a natural outgrowth of the geometry that already governs the reversible part of the theory.
Crucially, the authors emphasize that this does not require an external geometry or a chosen metric. The symmetric bracket is born from the Poisson structure itself and its deformation theory. That makes the framework elegant and intrinsic: entropy production arises from how the space is curved by a cocycle, not from an added Euclidean distance. It’s a reminder that time’s arrow might be woven into the fabric of the phase space as much as into the equations that run on it.
Two concrete playgrounds SE(2) and Galilei
One arena is the dual of the special Euclidean group in the plane, se(2) star. This object isn’t exotic math; it’s the language of planar rigid bodies and many control problems. On se(2) star, the authors identify a natural Casimir, the squared norm of the linear momentum, C = p1^2 + p2^2. This Casimir partitions the space into coadjoint orbits that look like circles or planes—the geometric leaves where reversible motion unfolds under the Lie–Poisson bracket.
The game changer comes when you allow a nontrivial second Poisson cohomology class to enter. This 2-cocycle cannot be written as a mere coboundary, so it provides a genuine deformation direction. The entropy is chosen to be a Casimir of the original bracket, say S = 1/2 (p1^2 + p2^2). By extending se(2) star to a centrally extended affine version and keeping the original Hamiltonian playing by its own rules, the dissipative partner is built from the cocycle. In effect, energy H keeps flowing along the reversible channels, while the symmetric part drags the system across leaves of the foliation toward higher entropy. A small extra coordinate c traces a mass-like parameter in the extension, and a tidy affine structure emerges from which the dissipative dynamics are cleanly extracted.
In parallel, the story unfolds for the Galilei group, SGal(3). Here the algebra encodes the familiar symmetries of nonrelativistic physics: rotations, boosts, space and time translations. The authors explore a central extension known as the Bargmann algebra, which brings mass into the stage as a genuine central charge. The affine Lie–Poisson structure on the dual of this extended algebra carries three Casimirs that mirror mass, a kinetic-energy-like invariant, and a Pauli-Lubanski style relation between angular and linear momentum. The upshot is again a playground where one Poisson structure governs energy conservation and another, introduced through a nontrivial cocycle, fuels entropy production. The irreversible dynamics is then generated by a Gibbs-like function F = H + S, with S built from natural invariants like the squared momentum and the cross product of momentum with the mass-weighted angular momentum.
What makes the Galilei example so illuminating is its resonance with familiar physics. The mass in the Bargmann extension is not just a bookkeeping device; it is the central charge that ensures Galilean symmetry survives quantum-esque thinking and thermodynamic consistency. The mathematics shows that by weaving a cocycle into the symmetry story, you can support entropy production without sacrificing the sanctity of energy conservation. It is a neat bridge between symmetry, geometry, and thermodynamics that feels almost tactile in its clarity.
Why this matters across physics and engineering
Beyond the elegance of the math, the paper connects to a long-standing quest to model irreversible processes in a way that respects conservation laws. The framework the authors lay out sits squarely at the crossroads of two established formalisms: metriplectic mechanics and GENERIC. Both aim to fuse reversible Hamiltonian dynamics with a dissipative, entropy-producing partner, but they usually require a particular metric or extra structure. Here the symmetric bracket emerges purely from the existing Poisson geometry and a nontrivial cocycle. That means you can, in principle, describe dissipation without importing an external geometric gadget, which is philosophically appealing and practically powerful.
This has tangible implications for control theory, robotics, and engineering disciplines where symmetry groups like SE(2) repeatedly appear. It offers a principled way to design or analyze systems that need to shed energy into disorder while still obeying energy conservation in a broader sense. In continuum physics, the move hints at a path to modeling irreversible phenomena in fluids, plasmas, and complex media without leaving the comfort zone of Hamiltonian reasoning. The key is to identify an appropriate Poisson structure with a nontrivial H2π(M); the authors show that such structures are not rare in the kinds of systems scientists actually study.
On the quantum and statistical mechanics side, the Galilei/Bargmann discussion lands squarely in a domain where symmetry, mass, and thermodynamics collide. The mass appears as a central charge, and the Casimirs of the extended algebra carry physical meaning that echoes in quantum representations and partition functions. That is not just a curiosity; it suggests a consistent geometric route to integrating symmetry with dissipation in a way that respects both energy conservation and entropy production—an alignment that many physical theories strive for but rarely achieve in a single, clean framework.
What remains and future horizons
The paper surveys finite dimensional triumphs and then looks toward the horizon of infinite dimensions. The authors argue that the core condition for their construction, the nontrivial second Lichnerowicz–Poisson cohomology, is often satisfied in continuum models. Yet translating the finite blueprint to the world of fields and fluids is nontrivial. Infinite-dimensional cohomology is trickier, and the algebraic machinery becomes more delicate when densities and functional-analytic issues enter the scene. Still, the road is laid out clearly: identify a Poisson structure with a Casimir that can serve as entropy, find a nontrivial cocycle that is not a coboundary, and build the symmetric bracket from the cocycle to drive irreversible dynamics while respecting energy.
The immediate next steps lie in concrete continuum models and PDEs. Can this framework illuminate irreversible magnetohydrodynamics, geophysical flows, or reaction-diffusion systems with strong symmetry? How do these geometric dissipation mechanisms interplay with numerical methods and stability analyses for large-scale simulations? The authors acknowledge these questions and sketch a roadmap where the math informs algorithm design and physics alike, knitting together a universal language for dissipation that remains faithful to conservation at its core.
In short, the work from the University of Amsterdam shows that irreversible dynamics can be carved out of the same geometric rock that enshrines reversible dynamics. The message is not that entropy is a trick we add on top, but that it can be built into the very fabric of the phase space when we let the right cohomology speak. If the cosmos favors symmetry and structure, it may also couple them to disorder in a disciplined, mathematically precise way. That is a provocative, hopeful claim, and it invites both mathematicians and physicists to think harder about how time’s arrow might be encoded in geometry rather than appended to equations.
Institutional note The study is conducted within the Korteweg–de Vries Institute for Mathematics at the University of Amsterdam, with Erwin Luesink as the lead author. The work showcases a novel geometric route to irreversible dynamics that could ripple through classical mechanics, control theory, and beyond.
