---
title: Potential Energy-Based Dissipation Relation
url: https://www.emergentmind.com/topics/potential-energy-based-dissipation-relationship
type: topic
---

# Potential Energy-Based Dissipation Relation

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 [2604.10709]. 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 [2302.01699][2508.12289][1404.4712][2403.06916][2202.02409].

## 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 \(H\) [2604.10709].

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 [2508.12289].

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 [2302.01699].

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 [2603.28707][2406.12391].

## 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” [2604.10709]. 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 \(45\,\mu\mathrm{m}\), density \(2.5\,\mathrm{g/cm^3}\), and is equilibrated at \(70\%\) RH and \(23^\circ\mathrm{C}\). The free surface is prepared with a concave cavity formed by a hemispherical tip of radius \(r=5,\,7.5,\,10.5,\) or \(12\,\mathrm{mm}\), with depth adjusted so that \(h=r\), making the cavities geometrically similar.

The impact kinematics begin with
\[
V_{\rm in}=-\sqrt{2gH},\qquad
V_{\rm out}=e\sqrt{2gH}.
\]
The measured jet speed is
\[
V_{\rm jet}=\frac{z(\tau+\Delta t)-z(\tau)}{\Delta t},
\]
with \(z(t)\) the jet-tip position above the powder surface, \(\tau=5\,\mathrm{ms}\), and \(\Delta t=5\,\mathrm{ms}\). Because the tip subsequently decelerates approximately at \(-g\), the rise is treated as ballistic:
\[
\frac12 m_{\rm tip}V_{\rm jet}^2=m_{\rm tip}gL_{\rm max},
\qquad
L_{\rm max}=\frac{V_{\rm jet}^2}{2g}.
\]

The central scaling law is
\[
V_{\rm jet}^2\propto H,
\]
written as
\[
V_{\rm jet}^2=2gCH,
\qquad
L_{\rm max}=CH.
\]
For each concave radius, the coefficient \(C\) decreases linearly with \(r\). From direct fits of \(V_{\rm jet}^2\) versus \(H\),
\[
C=16.4-8.64\times 10^2\,r,
\]
and from independently measured \(L_{\rm max}\),
\[
C=10.7-3.23\times 10^2\,r.
\]
At fixed \(H=10\,\mathrm{mm}\),
\[
L_{\rm max}=0.107-3.23\,r \qquad \text{(m)},
\]
with coefficient of determination \(>0.994\). Larger \(r\) 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 \(\xi\) has mass
\[
m_{\rm slide}=\int_S \rho\,dS\cdot \xi,
\]
the jet mass is
\[
m_{\rm jet}=\int_{\rm jet}\rho\,dV,
\]
and the mass ratio is
\[
\alpha\equiv \frac{m_{\rm slide}}{m_{\rm jet}}.
\]
The characteristic sliding speed is
\[
V_{\rm slide}=c_1V_{\rm out}=c_1e\sqrt{2gH}.
\]
Dissipation is modeled phenomenologically as
\[
E_{\rm dis}=\epsilon \frac{\pi}{2}r\,m_{\rm slide}V_{\rm slide}^{\beta},
\]
with characteristic path length proportional to \((\pi/2)r\). The resulting jet-speed law is
\[
V_{\rm jet}^2
=
\alpha c_1^2e^2(2gH)
-
\epsilon\pi\alpha c_1^\beta e^\beta (2gH)^{\beta/2}r.
\]

Because the data show \(V_{\rm jet}^2\propto H\), the study infers
\[
\beta=2.
\]
Then
\[
V_{\rm jet}^2=(2gH)\,\alpha c_1^2e^2(1-\epsilon\pi r),
\qquad
L_{\rm max}=H\,\alpha c_1^2e^2(1-\epsilon\pi r).
\]
Using \(c_1\sim 2\) and \(e\sim 0.8\), the authors estimate
\[
\alpha=3.9,\qquad \epsilon=9.6.
\]

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 \(C\) and the fixed-\(H\) jet height with \(r\). The study therefore proposes the extracted pair \((\epsilon,\beta)\), or simply the slope of \(C(r)\), 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 | \(E_p\), crest height \(H_b\) | \(\epsilon_l/(\rho g^{3/2}d^{5/2})\propto H_b/d\) |
| Thermal convection | buoyancy potential \(bz\) | \(\langle \varepsilon_u\rangle=\Phi_b+\Phi_{i1}+\Phi_{i2}\) |
| Magnetic colloids | interaction potential \(U(r,\theta,B)\) | \(W=\int_0^\tau \dot B\,\partial_B U\,dt\) |
| Earthquakes | stored elastic energy \(\Pi\) | \(G_0=-d\Pi/dl\), with \(G=\Gamma_{\mathrm{tot}}\) locally |

In breaking-wave DNS, the water-phase energies are
\[
E_k=\frac12\int_V \rho |\boldsymbol{u}\cdot\boldsymbol{u}|\,dV,
\qquad
E_p=\int_V \rho gy\,dV-E_{p0},
\qquad
E_m=E_k+E_p.
\]
The dissipation rate during an active breaking interval is defined as
\[
\epsilon_l=\frac{\Delta E_m}{\Delta t},
\qquad
\Delta t=\frac{1}{2f}.
\]
Dimensional analysis and inertial scaling give
\[
\frac{\epsilon_l}{\rho g^{3/2}d^{5/2}}\propto \frac{H_b}{d},
\]
and for plunging breakers
\[
\frac{\epsilon_l}{\rho g^{3/2}d^{5/2}}
=
\chi\left(\frac{H_b}{d}-0.76\right).
\]
The paper reports a “good linear dependence” between dissipation and \(H_b/d\) [2302.01699].

In confined thermal convection, the mechanical-energy framework yields a direct global relation between kinetic dissipation and potential-energy conversion:
\[
\langle \varepsilon_u\rangle
=
-\Phi_z
=
\Phi_b+\Phi_{i1}+\Phi_{i2}.
\]
For Oberbeck–Boussinesq Rayleigh–Bénard convection, the exact identity is
\[
\langle \varepsilon_u\rangle=(Nu-1)Ra^{-1/2}Pr^{-1/2},
\]
whereas in the generalized framework the unified scaling is
\[
\langle \varepsilon_u\rangle\sim Nu^\eta Ra^{-1/2}Pr^{-1/2},
\]
with
\[
\eta=
\begin{cases}
1, & \text{NOB-SHC (RBC, VC)},\\
0, & \text{NOB-CHC and OB-HC}.
\end{cases}
\]
Here the transition is controlled by whether plumes reach a characteristic height scaling as \(\sim H\) [2508.12289].

In the magnetic colloidal doublet, dissipation is obtained from a calibrated interaction potential
\[
U(r,\theta,B)=U_{dip}(B,r,\theta)+U_{mag}(B)+U_{el}(r),
\]
through the trajectory-level work
\[
W(\tau)=\int_0^\tau dt\,\dot B(t)\,\partial_B U({\bf r}(t),B(t)).
\]
Because the protocol is cyclic and symmetric,
\[
\Delta F=0,
\qquad
W_{diss}=W.
\]
For \(\tau=2\,\mathrm{s}\), the measured mean work is
\[
\langle W\rangle = 3.3\pm 0.2\,k_B T.
\]
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 [1404.4712].

In earthquake rupture, stored elastic energy \(\Pi\) plays the role of the upstream reservoir. The static energy release rate is
\[
G_0=-\,d\Pi/dl,
\]
and the moving-tip balance in linear elastic fracture mechanics is
\[
G=\Gamma_{\mathrm{tot}},
\qquad
G\approx \left(1-\frac{v_r}{c_R}\right)G_0.
\]
This is a potential-energy-based dissipation relationship only for the near-tip part of the process. The paper stresses that frictional heat
\[
H=\int_0^D \tau_r\,d\delta
\]
and other tail processes are not automatically part of \(\Gamma_{\mathrm{tot}}\), and that breakdown work is not generally identical to fracture energy [2403.06916].

## 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
\[
e=e(\ten{F},s),
\qquad
\phi=\phi(\dot{\ten{F}},\dot s,\ten g;\ten F,s),
\]
with the state laws
\[
\ten{P}-\frac{\partial e}{\partial \ten{F}}=\frac{\partial \phi}{\partial \dot{\ten{F}}},
\qquad
T-\frac{\partial e}{\partial s}=\frac{\partial \phi}{\partial \dot s},
\qquad
\ten q=\frac{\partial \phi}{\partial \ten g}.
\]
For the implemented class of standard dissipative solids this reduces to
\[
\ten P=\frac{\partial e}{\partial \ten F},
\qquad
T=\frac{\partial e}{\partial s},
\qquad
\ten q=\frac{\partial \phi}{\partial \ten g}.
\]
The relevant “potential” is therefore internal energy rather than mechanical potential energy, and dissipation is encoded by a convex dissipation potential [2603.28707].

The general energy-based framework of “A novel energy-based modeling framework” uses a storage function \(H\) and proves
\[
\dot H
=
-
\begin{bmatrix}
\dot z_1\\
\partial_{z_2}H\\
z_3
\end{bmatrix}^{\!T}
R
\begin{bmatrix}
\dot z_1\\
\partial_{z_2}H\\
z_3
\end{bmatrix}
+
\langle y,u\rangle
\le
\langle y,u\rangle.
\]
Potential energy contributes through gradients of \(H\), but dissipation is generated by the symmetric positive semidefinite operator \(R\), not by potential energy alone [2406.12391].

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 \(\mathcal F_p\) and \(\mathcal F_s\). The power balance becomes
\[
\dot H
=
-
\nabla_{\mathbf p}H_{qp}^{\top}\nabla_{\mathbf p}\mathcal F_p
-
\nabla_{\mathbf s}H_s^{\top}\nabla_{\mathbf s}\mathcal F_s
+
\mathbf y^\top \mathbf u
\le
\mathbf y^\top \mathbf u.
\]
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 [2605.12971].

A classical precursor is Rayleigh’s dissipation function, where conservative and dissipative forces are generated by different scalar functions:
\[
Q_i^{(\mathrm{cons})}=-\frac{\partial V}{\partial q_i},
\qquad
Q_i^{(\mathrm{diss})}=-\frac{\partial \mathcal R}{\partial \dot q_i}.
\]
The paper “Rayleigh's dissipation function at work” emphasizes that \(\mathcal R\) 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 [1409.4041].

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
\[
\frac12 E\left\|(-\Delta)^{\frac{1+\alpha}{4}}u\right\|_{L^2}^2
\quad\text{or}\quad
\frac12 E\|h_\alpha\ast_x \partial_x u\|_{L^2}^2.
\]
Here the constitutive law changes the correct notion of potential energy rather than creating a dissipative term [1912.00510].

## 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
\[
-\left(\mathbf P_\alpha\cdot\nabla\right)\cdot \mathbf u_\alpha.
\]
The exact species internal-energy equation is
\[
\partial_t E^{th}_\alpha + \nabla\cdot\left(E^{th}_\alpha \mathbf u_\alpha+\mathbf h_\alpha\right)
=
-\left(\mathbf P_\alpha\cdot\nabla\right)\cdot \mathbf u_\alpha,
\]
and the paper argues that this directly tracks internal-energy increase and temperature enhancement, whereas \( \mathbf j\cdot\mathbf E \) tracks electromagnetic-energy loss [2202.02409].

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 \(\varepsilon_p\) is not a reversible conversion
\[
APE \rightarrow {\rm GPE}_r,
\]
but a heating term that enters irreversible entropy production in the same way as viscous dissipation. The paper’s revised pathway is
\[
APE \xrightarrow{\varepsilon_p} {\cal B}_0,
\qquad
{\cal B}_{ex}\rightarrow {\rm GPE}_r,
\]
with \({\cal B}_0\) the dead internal energy and \({\cal B}_{ex}\) the exergy of stratification [1806.11303]. 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
\[
G=\Gamma_{\mathrm{tot}},
\]
while residual frictional heating, prolonged slip-tail processes, and distributed off-fault damage belong to the broader event-scale budget. Accordingly,
\[
\Gamma_{\mathrm{tot}}=W_b
\]
holds only under restrictive localization assumptions, and otherwise
\[
\Gamma_{\mathrm{tot}}\le W_b
\]
in the absence of additional dissipative channels [2403.06916].

Even in breaking-wave DNS, the strongest potential-energy reading remains partly interpretive. The paper defines and tracks
\[
E_p=\int_V \rho gy\,dV-E_{p0},
\qquad
E_m=E_k+E_p,
\qquad
\epsilon_l=\frac{\Delta E_m}{\Delta t},
\]
and finds \(\epsilon_l\propto H_b\) through inertial scaling. However, it does not derive an explicit law of the form \(\epsilon_l\propto E_p/\Delta t\); the interpretation of \(H_b\) as a proxy for available gravitational potential energy is therefore an informed reading rather than an author-stated identity [2302.01699].

## 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 \(C(r)\) and the inferred pair \((\epsilon,\beta)\) are proposed as quantitative metrics for comparing powder-specific interactions such as humidity, particle size, and particle shape [2604.10709]. In convection, the relation
\[
\langle \varepsilon_u\rangle\sim Nu^\eta Ra^{-1/2}Pr^{-1/2}
\]
provides the missing kinetic-dissipation backbone for extending Grossmann–Lohse theory to RBC, HC, VC, and NOB cases within a unified framework [2508.12289].

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
\[
e=e_{com}+e_{int},
\qquad
e_{com}=\frac12 m|\mathbf v|^2+\frac12 m\omega_i^2 r_i^2-\mathbf f\cdot \mathbf r,
\]
and the working 1D extraction formula
\[
\Delta e_{int}(t)
=
-
\int_0^t du\,(\dot f\,y)(u)
-\frac12 mv^2
-\frac12 m\omega^2 y^2
+fy.
\]
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 [2508.15626].

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
\[
W\equiv -\ln \Gamma,
\]
shows that pattern onset requires
\[
W>W_c(d),
\]
and derives an accuracy–dissipation tradeoff in which positional error decreases with excess dissipation and robustness grows asymptotically as
\[
N_{\max}\propto e^{W/2}.
\]
This broadens the topic from mechanical potential energy to nonequilibrium thermodynamic driving [2206.01769].

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.

Source: https://www.emergentmind.com/topics/potential-energy-based-dissipation-relationship