Process, stationarity and quasipotential

Costs and weight stay separate. Measure-weighted volume belongs to statistical weight. The effective free-energy-like cost is its negative logarithm; volume and free energy are therefore not literally the same quantity. Pareto and proven cost bounds belong to the efficiency comparison. Closure without costs is a separate consistency result, not a proof of efficiency.

The identity is called exact regrouping. It holds for a disjoint partition with the measure correctly included. An exact marginalization can likewise preserve the weights of the retained variables exactly, even though the omitted variables cannot be reconstructed. That does not make the reduced description fully information-preserving.

The formulation “genuine coarse-graining produces entropy and the identity does not hold for it” is too general. Changing a description is not in itself a physical act of dissipation. In an equilibrium distribution, two hidden equilibrium states can be merged into one visible label without this giving rise to heat production. An approximation or Markov closure can, however, lose predictions; effective entropy balances must be derived separately. That is why we keep both exact regrouping and the broader dynamical gluing rule.

When a model claims that π is the stationary distribution of L, the following must hold:

with the specified boundary conditions. This requirement is exact, but it does not determine L uniquely. Away from stationarity, the distribution and the dynamics must respect the full evolution equation. If S is only a design or performance cost, it need not be a stationary potential: that interpretation must not be added without a model hypothesis.

For a finite irreducible Markov chain with links allowed in both directions and positive rates, the cycle criterion reads:

Ordinary detailed balance then holds precisely when all cycle affinities are zero. Stationarity alone allows nonzero cycle currents. Time reversal of, for example, momentum variables requires the formulation appropriate to that case. See Joshi, Theorem 4.8 and Schnakenberg's network formulation.

Choose the convention explicitly:

Under suitable regularity and positive diffusion, small-noise theory describes rare paths via the rate functional:

The quasipotential relative to a given attractor is obtained by minimizing this cost over paths and durations starting from that attractor. This is a minimal fluctuation cost, not automatically physical heat, human effort or a new universal resistance. For the applications and their conditions, see Bouchet and Reygner, §2.

With a single suitable attracting minimum, a = I and b = −∇S, the quasipotential in this convention is S−Smin. With suitable confinement and boundary conditions, the Gibbs distribution exp(−S/ε)/Z is even exactly invariant.

Another common convention uses √ε instead of √(2ε). In that notation the exponent is exp(−2S/ε) and the quasipotential 2(S−Smin). That difference is a noise normalization, but it must not disappear while ε keeps the same name. Direct example: dX = −Xdt + √ε dW has stationary variance ε/2; with √(2ε) it is ε.

With several attractors, the global stationary rate function also requires the structure of transitions between them. A single local quasipotential from one attractor is not simply the global stationary exponent. Nor are prefactors and measure factors determined by the exponent alone.

Valid coupling: when a family π_ε ∝ exp(−S/ε)μ_ε is meant to describe the stationary weights, the measure carries no extra exponential ε-dependence, and the required stationary large-deviation limit exists, S−min S must agree with that global rate function. At finite ε, prefactors and the full equation L*π = 0 remain necessary. Small-noise theory does not yet choose a unique generator.

Take, in two ordinary position coordinates:

The drift is not a Euclidean gradient when ω ≠ 0. Yet for every ω the same Gaussian distribution is invariant:

A direct check of the stationary covariance gives:

The stationary probability current is:

It has zero divergence, but for ω ≠ 0 it is not zero everywhere. The quasipotential remains S. The corresponding Hamilton–Jacobi identity also holds:

Thus “S is the quasipotential” does not imply “the process is gradient-like and all affinities are zero”. These are different hypotheses. This example is the simplest concrete relative of the older EEML-K1 structure with conservative and dissipative parts. Non-reversible processes with a preserved Gibbs measure have also been studied explicitly by Rey-Bellet and Spiliopoulos.

If Ω_C is the physically admissible space for the full process, the process must respect it. For a transition kernel this means, where applicable:

This does not automatically hold for Ω_E,C. Reaching a target and then losing it can be physically allowed. The reaching-and-maintaining example in Part III shows this explicitly. E must be a separately proven invariant or an explicitly absorbing criterion before the effect set is declared invariant.

For an open subsystem, the environment or a suitable time-dependent constraint must be included. An apparent violation of the subsystem bookkeeping is not automatically a violation of conservation in the full system.

Likewise, a transition between abstract model classes is not automatically a transition in spatial topology. The Geroch/Tipler route may therefore not be used as a general requirement “every transition between closure classes requires a singularity or quantum mechanics”. Which topological theorem is relevant depends on the geometric object and on the theorem's hypotheses; that extra identification is missing from the abstract EEML architecture.

Ricci flow. Perelman gives a gradient interpretation with an additional function f and a suitable treatment of diffeomorphisms. Along the coupled flow, his functional satisfies:

In that convention it is therefore not a decreasing positive resistance. Moreover, the deterministic geometric flow does not yet specify a noise measure, stationary ensemble or Freidlin–Wentzell problem on the space of metrics. “Perelman's F is exactly the quasipotential” is not shown by this. See Perelman, §1.

Self-organised localisation. The cited proposal contains inflation, stochastic fields, phase transitions and a volume-weighted Fokker–Planck description. The volume growth is extra structure; this is not simply an ordinary probability-conserving generator with L*π = 0. It can be investigated as a separate model route, not as an already proven EEML bridge. See Giudice, McCullough and You.

H1 is split.

• H1a: the independently determined S yields, in the designated small-noise regime, the correct stationary rate function.
• H1b: the process is reversible under the specified time reversal; for the ordinary finite Markov description, all cycle affinities are zero.

The rotating Gaussian example satisfies H1a and violates H1b. Cycle currents test H1b, not automatically H1a. A driven biological experiment that nevertheless yields no visible current is not by definition a faulty measurement: there may be resolution limits, hidden currents or genuinely different conditions. The test outcome must be able to surprise.

H2 as supplied is not simply Prigogine's principle. Under suitable near-equilibrium conditions, that principle characterizes a stationary state for given kinetics and external constraints. It does not state in general that nature chooses the least dissipative generator out of all possible ones. The precise Markov derivation and its approximate character can be found in Maes and Netočný.

The check example uses a three-state ring with rates a = 3c and b = c. Then the stationary probabilities stay equal and the cycle affinity 3 ln 3 stays constant, but:

Making c small lowers the entropy production per unit time, while transport also slows down. Fixing only the drive does not prevent this trivial reduction. A selection hypothesis about generators must therefore also fix the effect, time frame, kinetic constraints and resources. That ties in exactly with the older EEML rule of equal effect.

Course and coherence. Specify the process law, initial preparation and environment. When memory is visible, use a sufficiently extended state or an explicitly history-dependent law. The process respects the applicable hard physical constraints; a target set is not automatically invariant. When statistical weights are given as a stationary distribution, they must satisfy L*π = 0 together with the process law. Where a stationary small-noise limit exists, the leading exponent must also agree with the global rate function determined by the dynamics, with the same noise normalization and measure convention. Detailed balance remains a separate, testable property. Which extra rule EEML imposes on the physically realized dynamics remains open.

This is a substantive connection between the layers. It preserves more of the old EEML than a universal gradient requirement: both relaxation and ongoing circulation can be investigated, and static agreement can no longer pass for a complete process explanation.