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Hydrostatic generating functionals and derivative-expansion boundaries — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

Zeroth-order hydrostatic data do not determine bulk viscosity

Physical scope

Conditional comparison of two constitutive model instances sharing one hydrostatic pressure functional; no solution existence, empirical response, or universal transport claim.

Example

Suppose an application supplies a shared thermodynamic state map on which the pressure in the static hydrostatic functional equals the equilibrium pressure used by the local dissipative constitutive branch. At a common admissible local input with nonzero expansion $θ$ , fix any $ζ_{*} > 0$ in pascal-seconds and compare two model instances that agree on all hydrostatic data and use constant bulk-viscosity coefficients $ζ_{1} = ζ_{*}$ and $ζ_{2} = 2 ζ_{*}$ . The bulk-viscosity coefficient here is the coefficient $ζ$ multiplying the expansion in $Π = - ζ c θ$ ; $θ = \nabla_{a} u^{a}$ is the expansion of the stated observer field. Their adopted hydrostatic functionals are identical, while their bulk-pressure corrections differ by $Π_{2} - Π_{1} = - ζ_{*} c θ \neq 0$ . This difference has units of pressure. Thus hydrostatic data alone do not fix the bulk viscosity. This is a conditional comparison of constitutive maps; it asserts neither a solution of the balance equations nor an empirical response.

Example references

Restricted hydrostatic effective functional

Restricted gradient constitutive dissipative model

Nabamita Banerjee et al., Constraints on Fluid Dynamics from Equilibrium Partition Functions

Pavel Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories

Facts & Assumptions

Given: Common hydrostatic data and the same pressure function in both model instances; a common admissible local constitutive input with $θ \neq 0$ ; and a constant $ζ_{*} > 0$ in $P a s$ .

[F1]

Restricted hydrostatic effective functional: the adopted $W_{h s}$ depends on the static data and pressure function $p (T, μ)$ , and contains no bulk-viscosity coefficient.

[F2]

Restricted gradient constitutive dissipative model: the constitutive formula includes the bulk-pressure correction $Π = - ζ c θ$ .

[F3]

Restricted gradient constitutive dissipative model: $θ = \nabla_{a} u^{a}$ is the expansion of the smooth future unit observer field.

[F4]

Restricted gradient constitutive dissipative model: $θ$ has inverse-length units, $c > 0$ is in metres per second, $ζ$ is in pascal-seconds, and $Π$ is in pascals.

[A1]

The state-map identification between the pressure in the hydrostatic functional and the equilibrium pressure of the dissipative branch is an additional application-specific assumption for this comparison.

[A2]

Both constitutive maps are evaluated at the same admissible local input with $θ \neq 0$ ; no global solution of the balance equations is assumed.

Verification

1.1givenF1A1algebra

The two instances have the same $T_{0}, μ_{0}, h, N, Φ, q$ and $p (T, μ)$ by [A1]. The formula in [F1] therefore has exactly the same integrand and integration measure for both instances; changing $ζ$ does not alter $W_{h s}$ .

1.2givenF2F3F4A2algebra

The constants $ζ_{1} = ζ_{*}$ and $ζ_{2} = 2 ζ_{*}$ are $C^{1}$ and nonnegative because $ζ_{*} > 0$ . By [F2], at the common input in [A2] their corrections satisfy $Π_{2} - Π_{1} = - (2 ζ_{*}) c θ - (- ζ_{*} c θ) = - ζ_{*} c θ$ , which is nonzero since $ζ_{*}, c > 0$ and $θ \neq 0$ ; $θ$ has the definition in [F3] and the units are pascals by [F4].

2.1step 1.1step 1.2A1A2∎

Thus the same hydrostatic functional is compatible with two distinct bulk-pressure responses under the declared constitutive assumptions, so the functional alone does not identify $ζ$ . This conclusion is conditional on the stated model identification and local input; it is not a claim about a measured fluid or an actual solution.