Potential Energy-Based Dissipation Relation

• Potential Energy-Based Dissipation is a framework that defines energy loss as the measurable difference between stored potential energy and observed kinetic or internal energy, supporting experimental and theoretical models.
• Experimental realizations, such as the powder-jet study, demonstrate that geometric factors like cavity radius directly impact jet speed and energy dissipation, validating a linear scaling law with drop height inputs.
• The concept extends to diverse systems—breaking waves, convection, magnetic colloids, and earthquake rupture—highlighting both the usefulness and limitations of interpreting dissipation in terms of stored potential or internal energy.

A potential energy-based dissipation relationship is a class of energy-accounting formulations in which dissipation is inferred from the conversion of an initially stored energy reservoir into kinetic, internal, or other downstream forms, with the dissipative loss identified as the part not recovered in the observable output. In a recent and unusually explicit realization, powder-jet formation from an impulsively driven granular free surface is described by a chain from drop-height gravitational potential energy to rebound-driven sliding, then to jet kinetic energy and ballistic rise, with dissipation accumulated during sliding along a concave cavity (Kobayash et al., 12 Apr 2026). Closely related constructions appear in breaking waves, confined thermal convection, driven colloidal systems, and earthquake rupture, although the literature also shows that the same phrase must be used carefully: in some fields the relevant sink is internal energy rather than potential energy, while in others the potential-energy reading is only an interpretation rather than an explicitly derived law (Liu et al., 2023, Cai et al., 17 Aug 2025, Tusch et al., 2014, Kammer et al., 2024, Yang et al., 2022).

1. Conceptual structure and canonical balance forms

At its most direct, the relationship has the form “input energy equals retained energy plus dissipation.” In the powder-jet model, the retained energy is the jet kinetic energy, and the dissipative term is an integrated sliding loss along a geometry-controlled path. The core balance is

$$\frac12 m_{\rm slide}V_{\rm slide}^2 - E_{\rm dis} = \frac12 m_{\rm jet}V_{\rm jet}^2, \qquad E_{\rm dis}=\epsilon \frac{\pi}{2}r\,m_{\rm slide}V_{\rm slide}^{\beta},$$

with $V_{\rm slide}=c_1 e\sqrt{2gH}$, so the upstream source is the gravitational input set by the drop height $H$ (Kobayash et al., 12 Apr 2026).

A broader but structurally similar form appears in thermal convection, where global viscous dissipation is linked to buoyancy production and then rewritten through a potential-energy budget: $$\langle \varepsilon_u\rangle = \langle wb\rangle = \Phi_b+\Phi_{i1}+\Phi_{i2}.$$ Here the dissipation is not assigned to a single local friction law; instead it is decomposed into surface potential-energy exchange, irreversible heat-transfer contribution, and a non-Oberbeck–Boussinesq contribution (Cai et al., 17 Aug 2025).

A third formulation, in wave breaking, defines conservative mechanical energy as

$E_m=E_k+E_p,$

with

$\epsilon_l=\frac{\Delta E_m}{\Delta t}.$

The paper then relates the dissipation rate per unit crest length to crest geometry through

$$\frac{\epsilon_l}{\rho g^{3/2}d^{5/2}}\propto \frac{H_b}{d}.$$

This does not state a literal “dissipation equals available potential energy” law, but it does tie dissipation to a geometric proxy for elevated water mass (Liu et al., 2023).

Taken together, these works suggest three recurring meanings of the expression. First, it can denote a literal partition of input potential energy into observable output plus loss. Second, it can denote a scaling in which a geometric measure such as crest height or cavity radius stands in for the amount of energy available to be dissipated. Third, in more abstract formulations, the “potential” may be a storage functional rather than a mechanical potential energy in the narrow sense (Holthusen et al., 30 Mar 2026, Altmann et al., 2024).

2. Geometry-controlled dissipation in powder jets

The clearest experimental realization is the powder-jet system studied in “Geometric control of powder jet dynamics and energy dissipation” (Kobayash et al., 12 Apr 2026). A vertical test tube partially filled with spherical glass beads is dropped from height $H\in[10,110]\,\mathrm{mm}$ onto a rigid floor. The powder layer has height $L=25\,\mathrm{mm}$, packing fraction $\phi\approx 0.55$, representative diameter $V_{\rm slide}=c_1 e\sqrt{2gH}$0, density $V_{\rm slide}=c_1 e\sqrt{2gH}$1, and is equilibrated at $V_{\rm slide}=c_1 e\sqrt{2gH}$2 RH and $V_{\rm slide}=c_1 e\sqrt{2gH}$3. The free surface is prepared with a concave cavity formed by a hemispherical tip of radius $V_{\rm slide}=c_1 e\sqrt{2gH}$4 or $V_{\rm slide}=c_1 e\sqrt{2gH}$5, with depth adjusted so that $V_{\rm slide}=c_1 e\sqrt{2gH}$6, making the cavities geometrically similar.

The impact kinematics begin with

$V_{\rm slide}=c_1 e\sqrt{2gH}$7

The measured jet speed is

$V_{\rm slide}=c_1 e\sqrt{2gH}$8

with $V_{\rm slide}=c_1 e\sqrt{2gH}$9 the jet-tip position above the powder surface, $H$0, and $H$1. Because the tip subsequently decelerates approximately at $H$2, the rise is treated as ballistic: $H$3

The central scaling law is

$H$4

written as

$H$5

For each concave radius, the coefficient $H$6 decreases linearly with $H$7. From direct fits of $H$8 versus $H$9,

$$\langle \varepsilon_u\rangle = \langle wb\rangle = \Phi_b+\Phi_{i1}+\Phi_{i2}.$$0

and from independently measured $$\langle \varepsilon_u\rangle = \langle wb\rangle = \Phi_b+\Phi_{i1}+\Phi_{i2}.$$1,

$$\langle \varepsilon_u\rangle = \langle wb\rangle = \Phi_b+\Phi_{i1}+\Phi_{i2}.$$2

At fixed $$\langle \varepsilon_u\rangle = \langle wb\rangle = \Phi_b+\Phi_{i1}+\Phi_{i2}.$$3,

$$\langle \varepsilon_u\rangle = \langle wb\rangle = \Phi_b+\Phi_{i1}+\Phi_{i2}.$$4

with coefficient of determination $$\langle \varepsilon_u\rangle = \langle wb\rangle = \Phi_b+\Phi_{i1}+\Phi_{i2}.$$5. Larger $$\langle \varepsilon_u\rangle = \langle wb\rangle = \Phi_b+\Phi_{i1}+\Phi_{i2}.$$6 also produces broader jets, with reduced velocity and maximum height.

The minimal mechanical model makes the dissipation mechanism explicit. A uniform sliding layer of thickness $$\langle \varepsilon_u\rangle = \langle wb\rangle = \Phi_b+\Phi_{i1}+\Phi_{i2}.$$7 has mass

$$\langle \varepsilon_u\rangle = \langle wb\rangle = \Phi_b+\Phi_{i1}+\Phi_{i2}.$$8

the jet mass is

$$\langle \varepsilon_u\rangle = \langle wb\rangle = \Phi_b+\Phi_{i1}+\Phi_{i2}.$$9

and the mass ratio is

$E_m=E_k+E_p,$0

The characteristic sliding speed is

$E_m=E_k+E_p,$1

Dissipation is modeled phenomenologically as

$E_m=E_k+E_p,$2

with characteristic path length proportional to $E_m=E_k+E_p,$3. The resulting jet-speed law is

$E_m=E_k+E_p,$4

Because the data show $E_m=E_k+E_p,$5, the study infers

$E_m=E_k+E_p,$6

Then

$E_m=E_k+E_p,$7

Using $E_m=E_k+E_p,$8 and $E_m=E_k+E_p,$9, the authors estimate

$\epsilon_l=\frac{\Delta E_m}{\Delta t}.$0

This formulation is notable because the geometry enters only through sliding distance, yet it predicts the observed linear decrease of both the energy-conversion coefficient $\epsilon_l=\frac{\Delta E_m}{\Delta t}.$1 and the fixed-$\epsilon_l=\frac{\Delta E_m}{\Delta t}.$2 jet height with $\epsilon_l=\frac{\Delta E_m}{\Delta t}.$3. The study therefore proposes the extracted pair $\epsilon_l=\frac{\Delta E_m}{\Delta t}.$4, or simply the slope of $\epsilon_l=\frac{\Delta E_m}{\Delta t}.$5, as a quantitative metric of dissipation and flowability for comparisons involving humidity, particle size, and particle shape.

3. Representative realizations in fluids, colloids, and fracture

Several other systems realize the same idea in different energetic languages.

System	Storage or geometric quantity	Dissipation relation
Breaking waves	$\epsilon_l=\frac{\Delta E_m}{\Delta t}.$6, crest height $\epsilon_l=\frac{\Delta E_m}{\Delta t}.$7	$\epsilon_l=\frac{\Delta E_m}{\Delta t}.$8
Thermal convection	buoyancy potential $\epsilon_l=\frac{\Delta E_m}{\Delta t}.$9	$$\frac{\epsilon_l}{\rho g^{3/2}d^{5/2}}\propto \frac{H_b}{d}.$$0
Magnetic colloids	interaction potential $$\frac{\epsilon_l}{\rho g^{3/2}d^{5/2}}\propto \frac{H_b}{d}.$$1	$$\frac{\epsilon_l}{\rho g^{3/2}d^{5/2}}\propto \frac{H_b}{d}.$$2
Earthquakes	stored elastic energy $$\frac{\epsilon_l}{\rho g^{3/2}d^{5/2}}\propto \frac{H_b}{d}.$$3	$$\frac{\epsilon_l}{\rho g^{3/2}d^{5/2}}\propto \frac{H_b}{d}.$$4, with $$\frac{\epsilon_l}{\rho g^{3/2}d^{5/2}}\propto \frac{H_b}{d}.$$5 locally

In breaking-wave DNS, the water-phase energies are

$$\frac{\epsilon_l}{\rho g^{3/2}d^{5/2}}\propto \frac{H_b}{d}.$$6

The dissipation rate during an active breaking interval is defined as

$$\frac{\epsilon_l}{\rho g^{3/2}d^{5/2}}\propto \frac{H_b}{d}.$$7

Dimensional analysis and inertial scaling give

$$\frac{\epsilon_l}{\rho g^{3/2}d^{5/2}}\propto \frac{H_b}{d}.$$8

and for plunging breakers

$$\frac{\epsilon_l}{\rho g^{3/2}d^{5/2}}\propto \frac{H_b}{d}.$$9

The paper reports a “good linear dependence” between dissipation and $H\in[10,110]\,\mathrm{mm}$0 (Liu et al., 2023).

In confined thermal convection, the mechanical-energy framework yields a direct global relation between kinetic dissipation and potential-energy conversion: $H\in[10,110]\,\mathrm{mm}$1 For Oberbeck–Boussinesq Rayleigh–Bénard convection, the exact identity is

$H\in[10,110]\,\mathrm{mm}$2

whereas in the generalized framework the unified scaling is

$H\in[10,110]\,\mathrm{mm}$3

with

$H\in[10,110]\,\mathrm{mm}$4

Here the transition is controlled by whether plumes reach a characteristic height scaling as $H\in[10,110]\,\mathrm{mm}$5 (Cai et al., 17 Aug 2025).

In the magnetic colloidal doublet, dissipation is obtained from a calibrated interaction potential

$H\in[10,110]\,\mathrm{mm}$6

through the trajectory-level work

$H\in[10,110]\,\mathrm{mm}$7

Because the protocol is cyclic and symmetric,

$H\in[10,110]\,\mathrm{mm}$8

For $H\in[10,110]\,\mathrm{mm}$9, the measured mean work is

$L=25\,\mathrm{mm}$0

In this case the potential-energy-based estimate is direct: the driven interaction potential determines the injected work, and the mean cycle work equals the mean dissipation (Tusch et al., 2014).

In earthquake rupture, stored elastic energy $L=25\,\mathrm{mm}$1 plays the role of the upstream reservoir. The static energy release rate is

$L=25\,\mathrm{mm}$2

and the moving-tip balance in linear elastic fracture mechanics is

$L=25\,\mathrm{mm}$3

This is a potential-energy-based dissipation relationship only for the near-tip part of the process. The paper stresses that frictional heat

$L=25\,\mathrm{mm}$4

and other tail processes are not automatically part of $L=25\,\mathrm{mm}$5, and that breakdown work is not generally identical to fracture energy (Kammer et al., 2024).

4. Generalizations to free-energy, internal-energy, and dissipation-potential formalisms

A different strand of the literature extends the idea beyond gravitational or elastic potential energy and treats dissipation through paired scalar potentials. In “A Convex Route to Thermomechanics: Learning Internal Energy and Dissipation,” the constitutive structure is built from

$L=25\,\mathrm{mm}$6

with the state laws

$L=25\,\mathrm{mm}$7

For the implemented class of standard dissipative solids this reduces to

$L=25\,\mathrm{mm}$8

The relevant “potential” is therefore internal energy rather than mechanical potential energy, and dissipation is encoded by a convex dissipation potential (Holthusen et al., 30 Mar 2026).

The general energy-based framework of “A novel energy-based modeling framework” uses a storage function $L=25\,\mathrm{mm}$9 and proves

$\phi\approx 0.55$0

Potential energy contributes through gradients of $\phi\approx 0.55$1, but dissipation is generated by the symmetric positive semidefinite operator $\phi\approx 0.55$2, not by potential energy alone (Altmann et al., 2024).

An even closer structural analogy is developed in “Port-Hamiltonian Systems with Dissipation Potential,” where conventional damping matrices are replaced by convex scalar dissipation potentials $\phi\approx 0.55$3 and $\phi\approx 0.55$4. The power balance becomes

$\phi\approx 0.55$5

This restores what the paper calls a variational symmetry between stored and dissipated energy, but it remains a storage-and-dissipation formalism rather than a literal law of potential-energy loss (Jia et al., 13 May 2026).

A classical precursor is Rayleigh’s dissipation function, where conservative and dissipative forces are generated by different scalar functions: $\phi\approx 0.55$6 The paper “Rayleigh's dissipation function at work” emphasizes that $\phi\approx 0.55$7 is not potential energy, not generally part of the Lagrangian, and not generally equal to dissipated power, even though it is formally analogous to a potential in velocity space (Minguzzi, 2014).

A final variation appears in fractional wave equations. For hereditary constitutive laws, a priori energy estimates yield dissipation through positive memory terms. For non-local fractional wave equations, however, energy is conserved once potential energy is reinterpreted as a non-local quadratic form such as

$\phi\approx 0.55$8

Here the constitutive law changes the correct notion of potential energy rather than creating a dissipative term (Zorica et al., 2019).

5. Limits, misconceptions, and explicit non-equivalences

Several papers explicitly warn against overextending the potential-energy label. In collisionless plasma turbulence, the preferred dissipation estimate is not electric work and not a potential-energy law, but the pressure–strain interaction

$\phi\approx 0.55$9

The exact species internal-energy equation is

$V_{\rm slide}=c_1 e\sqrt{2gH}$00

and the paper argues that this directly tracks internal-energy increase and temperature enhancement, whereas $V_{\rm slide}=c_1 e\sqrt{2gH}$01 tracks electromagnetic-energy loss (Yang et al., 2022).

A different controversy appears in stratified-fluid energetics. “APE dissipation is a form of Joule heating. It is irreversible, not reversible” argues that available potential energy dissipation $V_{\rm slide}=c_1 e\sqrt{2gH}$02 is not a reversible conversion

$V_{\rm slide}=c_1 e\sqrt{2gH}$03

but a heating term that enters irreversible entropy production in the same way as viscous dissipation. The paper’s revised pathway is

$V_{\rm slide}=c_1 e\sqrt{2gH}$04

with $V_{\rm slide}=c_1 e\sqrt{2gH}$05 the dead internal energy and $V_{\rm slide}=c_1 e\sqrt{2gH}$06 the exergy of stratification (Tailleux, 2018). This directly contradicts the interpretation that all mixing-induced dissipation can be represented as recoverable background gravitational potential energy.

Earthquake energetics supplies a third limit case. The paper on earthquakes stresses that only tip-localized dissipation belongs in the fracture-energy term

$V_{\rm slide}=c_1 e\sqrt{2gH}$07

while residual frictional heating, prolonged slip-tail processes, and distributed off-fault damage belong to the broader event-scale budget. Accordingly,

$V_{\rm slide}=c_1 e\sqrt{2gH}$08

holds only under restrictive localization assumptions, and otherwise

$V_{\rm slide}=c_1 e\sqrt{2gH}$09

in the absence of additional dissipative channels (Kammer et al., 2024).

Even in breaking-wave DNS, the strongest potential-energy reading remains partly interpretive. The paper defines and tracks

$V_{\rm slide}=c_1 e\sqrt{2gH}$10

and finds $V_{\rm slide}=c_1 e\sqrt{2gH}$11 through inertial scaling. However, it does not derive an explicit law of the form $V_{\rm slide}=c_1 e\sqrt{2gH}$12; the interpretation of $V_{\rm slide}=c_1 e\sqrt{2gH}$13 as a proxy for available gravitational potential energy is therefore an informed reading rather than an author-stated identity (Liu et al., 2023).

6. Significance, metrics, and current directions

The main practical significance of these relationships is that they convert difficult microscopic dissipation mechanisms into measurable macroscopic balances. In the powder-jet problem, the slope of $V_{\rm slide}=c_1 e\sqrt{2gH}$14 and the inferred pair $V_{\rm slide}=c_1 e\sqrt{2gH}$15 are proposed as quantitative metrics for comparing powder-specific interactions such as humidity, particle size, and particle shape (Kobayash et al., 12 Apr 2026). In convection, the relation

$V_{\rm slide}=c_1 e\sqrt{2gH}$16

provides the missing kinetic-dissipation backbone for extending Grossmann–Lohse theory to RBC, HC, VC, and NOB cases within a unified framework (Cai et al., 17 Aug 2025).

A complementary experimental direction is to infer dissipation from exact mechanical-energy conversion laws rather than from final heating. In a harmonically trapped driven superfluid, the perturbed harmonic-potential theorem yields the decomposition

$V_{\rm slide}=c_1 e\sqrt{2gH}$17

and the working 1D extraction formula

$V_{\rm slide}=c_1 e\sqrt{2gH}$18

This is not a potential-energy-only law, but it is an exact energy-conversion law from center-of-mass mechanical energy into internal energy, obtained without assuming thermalization (Tanghe et al., 21 Aug 2025).

In nonequilibrium reaction–diffusion systems, the analogous quantity is chemical free-energy dissipation rather than mechanical potential energy. The Turing-pattern study defines the dominant cycle affinity

$V_{\rm slide}=c_1 e\sqrt{2gH}$19

shows that pattern onset requires

$V_{\rm slide}=c_1 e\sqrt{2gH}$20

and derives an accuracy–dissipation tradeoff in which positional error decreases with excess dissipation and robustness grows asymptotically as

$V_{\rm slide}=c_1 e\sqrt{2gH}$21

This broadens the topic from mechanical potential energy to nonequilibrium thermodynamic driving (Zhang et al., 2022).

Across these systems, a consistent conclusion emerges. A potential energy-based dissipation relationship is not a single universal equation but a family of rigorously structured balances. In the strongest cases, it identifies dissipation as the missing part of an energy conversion chain whose source is a stored potential-energy reservoir. In more general settings, it supplies a geometric or constitutive proxy for the energy available to be degraded. Its usefulness depends on careful identification of the relevant storage quantity, the observable energy sink, and the scale at which “dissipation” is being defined.