Minimum entropy production is not a principle but a theorem with a fence.
It is sometimes said that systems evolve so as to minimise entropy production. This paper writes that this is not a law of nature but a theorem holding only inside a fence, and counts the fence. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED. Onsager’s reciprocal relations (1931), Prigogine’s theorem of minimum entropy production and the Brusselator are all standard.
First, and this is the core. The gap between the steady state and the minimum is exactly the antisymmetric part of the Onsager matrix. In a system with two forces, hold one fixed by boundary conditions and leave the other free. Under the linear laws the entropy production is a quadratic form in the forces, and making it stationary in the free force places that point, relative to the fixed force, at minus the sum of the off-diagonal coefficients over twice the diagonal one. The steady-state condition, that the second flux vanish, places it instead at minus one off-diagonal coefficient over the diagonal one. Taking the difference leaves a quantity proportional to the difference of the off-diagonal coefficients, that is, to the antisymmetric part (confirmed here symbolically). Onsager symmetry is therefore not decoration: the point of minimum entropy production and the steady state are two different conditions, and they coincide because that symmetry holds. Where the symmetry fails, so does the theorem: in a magnetic field or a rotating frame the reciprocal relation acquires a sign change, the plain symmetry does not hold, and there the minimum is not the steady state. Paper 17 asked for an address, and the address of this theorem is the symmetric part of the Onsager matrix, the antisymmetric part appearing unchanged as the gap.
Second, that it is a minimum is supplied by the second law. The Hessian of the entropy production is exactly twice the symmetric Onsager matrix (confirmed here). The second law requires the entropy production to be non-negative for every force, which is precisely the statement that the Onsager matrix is positive semi-definite, so the Hessian is too and the stationary point is a minimum rather than a maximum. What deserves notice is that being stationary and being minimal are different statements: the first comes from a symmetry and the second from the second law. Paper 136 said that the second law is a corollary of a stronger equality, whereas here, conversely, the second law is deciding a property of a different statement. One law does different work in different places.
Third, far from equilibrium it fails outright. The reaction system known as the Brusselator was integrated numerically. Its steady state is known to become unstable, and with the parameter A set to one it does so once B exceeds one plus A squared, that is two. Indeed at B equal to 1.7 the system settles to the steady state and the late-time range of X is zero, whereas at B equal to 2.5 the range is 2.018677 and at 3.0 it is 3.381648, the system running on a limit cycle for ever in both cases. Nothing is being minimised, because the state does not converge at all. So the statement that systems evolve so as to minimise entropy production is not right in general. It is not a law of nature but a theorem with a fence, and the fence has four bars: that the linear laws hold, that Onsager symmetry holds, that the phenomenological coefficients are constant, and that the boundary conditions do not depend on time. Drop any one and the conclusion goes.
Fourth, Prigogine’s two famous results live in mutually exclusive regimes. Minimum entropy production belongs to the linear regime near equilibrium, where there is a stable steady state and no structure arises. Dissipative structures belong far from equilibrium, where oscillations, patterns and instabilities appear. Both are Prigogine’s, and no single system exhibits both at once: where the minimum principle holds there are no dissipative structures, and where dissipative structures appear the minimum principle has already failed. Paper 125 called the roll on one name attached to two theorems, Noether’s first and second, and here too one name is attached to two results, but the shape differs. Noether’s two both hold in an overlapping domain, while Prigogine’s two do not overlap at all: two results of one person has more than one type. And no general extremal principle is known for the far-from-equilibrium regime. Prigogine’s own general evolution criterion is not a potential, and a decreasing quantity can be defined without there being a determined state it is minimised at. The expectation that something must be being minimised has no support there.
IMPORTANT (scope): a census of established standard material (Layer 2). Every work cited is standard. What is not claimed: any new mathematical theorem (there is none) / any new physical law (there is none) / that Prigogine’s theorem is wrong (it is not; it is right inside its fence) / that the theory of dissipative structures is wrong (it is not; it concerns a different regime). What is added: making explicit that the gap is the antisymmetric part, writing that the second law is what makes the extremum a minimum, exhibiting the failure numerically, and writing that the two results occupy exclusive regimes. What remains: the phrase that a system evolves so as to minimise wears the face of a law, while in fact it is a theorem holding inside four assumptions. And removing the quietest of them, Onsager symmetry, moves the minimum straight off the steady state. The size of the gap turned out to be the size of the symmetry that was removed.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). Minimum Entropy Production Is Not a Principle but a Theorem with a Fence [Paper 143]. Zenodo.
DOI: 10.5281/zenodo.22082367
ORCID: 0009-0006-8587-3450