Dynamical Potential in Complex Systems

• Dynamical potential is a concept that replaces static potentials with time-dependent and trajectory-based control variables, effective interactions, and induced functionals.
• It is applied across diverse fields such as astrophysics and nuclear fusion, where it governs evolving gravitational and interaction potentials.
• Field-theoretic and electronic-structure methods use dynamical potentials to generate energy-dependent embeddings and manage non-equilibrium and chaotic dynamics.

In the works considered here, dynamical potential is not a single universally fixed object. It denotes several distinct constructions that arise when a static potential is replaced by a trajectory-space control variable, an energy-dependent effective interaction, a time-dependent gravitational or galactic potential, an induced geometric object on field space, or a global Lyapunov-type landscape for a flow. In this literature, the phrase can refer to a dynamic field conjugate to event activity (Ye et al., 2022), a cumulant-generating functional for spatiotemporal correlations (Roy et al., 2017), a generalized potential for evolving circular orbits in dynamical spacetimes (Song, 29 Jul 2025), an induced antisymplectic one-form (Batalin et al., 2017), an energy-dependent embedding operator in pseudopotential theory (Quinzi et al., 6 May 2026), or a monotone potential function for a chaotic attractor (Ma et al., 2012). In astrophysics and nuclear theory it also appears in forms such as non-axisymmetric galactic potentials inferred from kinematic substructure (Monari et al., 2011), time-dependent galactic potentials under mass transfer (Illés et al., 2024), and dynamical nucleus–nucleus potentials extracted from transport dynamics (Jiang et al., 2010).

1. Trajectory-space and non-equilibrium statistical meanings

In dynamic phase transition theory, the central intensive quantity is the dynamic field $s$, interpreted as the dynamical potential conjugate to the event number $K$ in a trajectory ensemble. The corresponding dynamic partition function is

$$Z(s)=\sum_{\mathcal{T}(K)} \exp\left\{-\frac{sK+U_0(\mathcal{T})}{k_{\text D}}\right\},$$

where $U_0(\mathcal{T})$ encodes temporal event-event correlations. In this formulation, positive $s$ suppresses events and smaller or negative $s$ enhances them, while zeros of $Z(\gamma)$, with $\gamma=e^{-s/k_{\text D}}$, play the Yang–Lee role for dynamic phase transitions in space-time (Ye et al., 2022).

A closely related but terminologically distinct usage appears in non-equilibrium quantum many-body theory, where dynamical potentials are generating functionals for spatiotemporal correlations. There one introduces an imaginary source field through

$$\mathcal H_s(t)=\mathcal H(t)-i\frac{s(t)}{2}\hat{\mathcal M},$$

and defines

$$\mathcal Z_t[s]=\langle \psi_0|U_t^\dagger[s]U_t[s]|\psi_0\rangle \equiv e^{-\Theta_t[s]}.$$

Here $K$0 is called the dynamical potential, and its derivatives generate time-integrated moments and cumulants of $K$1. This framework is used to diagnose many-body localization, MBL spin-glass order, and discrete time-crystalline order through the scaling and Legendre structure of $K$2 and its associated distribution functions (Roy et al., 2017).

Taken together, these two usages show an important split in the literature. In one case the dynamical potential is the conjugate field $K$3; in the other it is the cumulant-generating functional $K$4. Both constructions move the notion of potential away from static configuration space and into trajectory space.

2. Gravitational, galactic, and cosmological uses

In galactic dynamics, the phrase is often tied to inference of a gravitational potential from orbit structure rather than from equilibrium moments. A proof-of-concept barred-galaxy study constrains the non-axisymmetric Milky Way potential by integrating local stellar phase-space points in trial potentials, extracting principal frequencies $K$5, and identifying the potential that best preserves moving-group substructure in $K$6. The correct barred potential preserves resonant families as coherent structures in frequency space, whereas a wrong potential mixes points that originally lay on resonant lines (Monari et al., 2011).

A different galactic usage makes the potential explicitly time dependent. In a Milky-Way-like axisymmetric model with disk, bulge, and dark-matter halo, secular mass transfer is imposed through

$K$7

so energy is no longer conserved and ordinary autonomous Poincaré sections cease to apply. The analysis then proceeds through a snapshot framework and an ensemble-based stability diagnostic,

$K$8

which quantify breakup of snapshot tori and transport into snapshot chaotic seas during parameter drift (Illés et al., 2024).

In relativistic gravity, a 2025 formulation explicitly introduces a dynamical potential $K$9 for evolving circular orbits in time-dependent spherically symmetric spacetimes. The static conditions $$Z(s)=\sum_{\mathcal{T}(K)} \exp\left\{-\frac{sK+U_0(\mathcal{T})}{k_{\text D}}\right\},$$0 and $$Z(s)=\sum_{\mathcal{T}(K)} \exp\left\{-\frac{sK+U_0(\mathcal{T})}{k_{\text D}}\right\},$$1 are replaced by three requirements: the orbital radius has no explicit affine-parameter dependence, the dynamical potential remains unchanged along the evolving circular orbit, and the angular momentum is conserved. The resulting equations are shown to be equivalent to the quasi-local particle-surface construction (Song, 29 Jul 2025).

In teleparallel cosmology, the scalar-field potential remains an ordinary $$Z(s)=\sum_{\mathcal{T}(K)} \exp\left\{-\frac{sK+U_0(\mathcal{T})}{k_{\text D}}\right\},$$2, but it is treated as a dynamical-system driver. For

$$Z(s)=\sum_{\mathcal{T}(K)} \exp\left\{-\frac{sK+U_0(\mathcal{T})}{k_{\text D}}\right\},$$3

the slope parameter $$Z(s)=\sum_{\mathcal{T}(K)} \exp\left\{-\frac{sK+U_0(\mathcal{T})}{k_{\text D}}\right\},$$4 is constant, and the potential enters the autonomous system through

$$Z(s)=\sum_{\mathcal{T}(K)} \exp\left\{-\frac{sK+U_0(\mathcal{T})}{k_{\text D}}\right\},$$5

This structure underlies radiation-scaling, matter-scaling, scalar-field-dominated, and de Sitter critical points in $$Z(s)=\sum_{\mathcal{T}(K)} \exp\left\{-\frac{sK+U_0(\mathcal{T})}{k_{\text D}}\right\},$$6 gravity (Kadam et al., 2024).

3. Field-theoretic and geometric constructions

In generalized topological sigma models, the relevant object is the dynamical antisymplectic potential

$$Z(s)=\sum_{\mathcal{T}(K)} \exp\left\{-\frac{sK+U_0(\mathcal{T})}{k_{\text D}}\right\},$$7

Because the local Lagrangian density $$Z(s)=\sum_{\mathcal{T}(K)} \exp\left\{-\frac{sK+U_0(\mathcal{T})}{k_{\text D}}\right\},$$8 is allowed to depend nonlinearly on the de Rham “velocities” $$Z(s)=\sum_{\mathcal{T}(K)} \exp\left\{-\frac{sK+U_0(\mathcal{T})}{k_{\text D}}\right\},$$9, the antisymplectic potential is not fixed a priori on target space. It is induced from the Lagrangian itself and depends on both $U_0(\mathcal{T})$0 and $U_0(\mathcal{T})$1. From it one builds the dynamical antisymplectic metric

$U_0(\mathcal{T})$2

the local and functional antibrackets, and ultimately the master-equation structure of the theory (Batalin et al., 2017).

In graphene under dynamical deformation, a different field-theoretic meaning appears. The longitudinal component of the deformation-induced gauge field can be written as $U_0(\mathcal{T})$3, and its time dependence induces a valley-antisymmetric potential

$U_0(\mathcal{T})$4

This scalar potential enters with opposite signs at the $U_0(\mathcal{T})$5 and $U_0(\mathcal{T})$6 valleys, vanishes in the static limit, and shifts intervalley phonon energies as $U_0(\mathcal{T})$7. The paper argues that this explains why dynamical deformation broadens the Raman $U_0(\mathcal{T})$8 band but does not similarly broaden the $U_0(\mathcal{T})$9 band (Sasaki et al., 2014).

A related usage in high-energy theory concerns potentials that are dynamically generated rather than postulated. In dynamical chaotic inflation, strong supersymmetric gauge dynamics produce an effective inflaton potential through dimensional transmutation,

$s$0

with the power $s$1 fixed by beta-function coefficients. For $s$2 models one obtains $s$3, while adding extra massive flavors yields $s$4. The potential is therefore an IR output of gauge dynamics rather than an elementary monomial input (Harigaya et al., 2014).

4. Nuclear-reaction and lattice-QCD meanings

In heavy-ion fusion, the dynamical nucleus–nucleus potential is extracted from transport evolution rather than from frozen densities. In the ImQMD framework, the large-distance and overlap regimes are defined by

$s$5

and smoothly combined through

$s$6

This construction shows that the fusion pocket is deep for $s$7, becomes shallow for $s$8, and almost disappears for $s$9, where the short-distance dynamical potential remains much higher than the adiabatic limit (Jiang et al., 2010).

A follow-up ImQMD study examined how this potential depends on nuclear-matter incompressibility and on the Gaussian wave-packet width. There the same dynamical extraction

$s$0

is used, and the main conclusion is that the wave-packet width affects both the barrier region and the inner potential, while incompressibility mainly affects the short-distance region. Comparing SKP* and IQ3 at fixed packet width, the short-distance potential increases with increasing $s$1 (Zanganeh et al., 2012).

In deuteron-induced reactions, the relevant object is the dynamical polarization potential generated by coupling the elastic channel to breakup channels. Formally it is non-local and $s$2-dependent, and local equivalents obtained by exact $s$3 inversion exhibit strong undulations and even emissive regions. A central result is that local equivalent DPPs from separate uncoupled channel sectors are not additive, even when the underlying non-local DPPs are additive; this non-additivity is used as direct evidence for dynamical non-locality (Mackintosh et al., 2016).

A useful terminological contrast is provided by lattice QCD with dynamical fermions. There the object of interest is still the static quark–antiquark potential $s$4, extracted from Wilson loops, while “dynamical” refers to the inclusion of the fermion determinant rather than to a time-dependent potential. The paper’s main technical point is that HYP and HYP2 smearing permit precise extraction of the continuum static potential up to $s$5 effects on dynamical gauge configurations (Donnellan et al., 2010).

5. Electronic-structure meanings and the limits of effective-potential dynamics

In electronic-structure theory, dynamical pseudopotentials are introduced as energy-dependent embedding operators obtained by tracing out the core sector. The valence-only problem is

$s$6

with

$s$7

This turns pseudopotential theory into a Dyson-like problem and leads to a generalized norm-conservation condition involving $s$8. In a sum-over-poles representation, the number of reference energies is disentangled from the number of projectors, allowing all-electron scattering to be reproduced over extended energy ranges; the same formalism is then inserted into a Green’s-function total-energy functional for a unified treatment of atom, pseudo-atom, and solid (Quinzi et al., 6 May 2026).

In correlated-electron theory, “potential energy” may itself be a dynamical quantity reconstructed from frequency-dependent one- and two-particle objects. In ladder D$s$9A for the half-filled three-dimensional Hubbard model, the ambiguity is between

$Z(\gamma)$0

and

$Z(\gamma)$1

The paper restores a unique value by introducing Moriya-like mass renormalizations $Z(\gamma)$2 and $Z(\gamma)$3 in the charge and spin susceptibilities, thereby enforcing both a Pauli-principle sum rule and one-/two-particle consistency for the interaction energy (Stobbe et al., 2022).

A major cautionary result concerns the use of effective potentials in real-time dynamics. For a time-dependent mean field $Z(\gamma)$4, the paper argues that inserting the ordinary static effective potential into the equation of motion generally fails because it omits the exact mode functions $Z(\gamma)$5 and their backreaction, thereby violating energy conservation. Even an adiabatic effective potential has only a narrow regime of validity and breaks down under parametric amplification or spinodal instability, precisely when particle production becomes important. The consistent replacement is a coupled, renormalized, energy-conserving system for $Z(\gamma)$6 and the fluctuation modes rather than a closed equation driven by a single effective potential (Herring et al., 2024).

6. Deterministic dynamical systems and induction-based potential theory

For continuous dissipative chaotic flows, a potential function can be defined globally by the conditions

$Z(\gamma)$7

where $Z(\gamma)$8 is the attractor. In a Lorenz-like piecewise system, the potential is built explicitly from a self-similar seed function $Z(\gamma)$9 whose zero set is the middle-third Cantor set, so the resulting $\gamma=e^{-s/k_{\text D}}$0 reveals the fractal structure of the strange attractor. The vector field is then decomposed as

$\gamma=e^{-s/k_{\text D}}$1

with $\gamma=e^{-s/k_{\text D}}$2 symmetric semipositive and $\gamma=e^{-s/k_{\text D}}$3 skew-symmetric. Within this framework, the gradient part determines attraction and the rotational part sustains motion on the attractor, allowing the paper to separate the origins of “strangeness” and “chaos” (Ma et al., 2012).

A different potential-theoretic tradition appears in measure-preserving dynamics through the Poisson equation

$\gamma=e^{-s/k_{\text D}}$4

For a recurrent system $\gamma=e^{-s/k_{\text D}}$5 and an induced subsystem on $\gamma=e^{-s/k_{\text D}}$6, the paper proves a balayage formula: if $\gamma=e^{-s/k_{\text D}}$7 on $\gamma=e^{-s/k_{\text D}}$8 and $\gamma=e^{-s/k_{\text D}}$9, then

$$\mathcal H_s(t)=\mathcal H(t)-i\frac{s(t)}{2}\hat{\mathcal M},$$0

This generalizes the standard fact that induced maps preserve the restricted invariant measure. The induced observable is the excursion sum

$$\mathcal H_s(t)=\mathcal H(t)-i\frac{s(t)}{2}\hat{\mathcal M},$$1

and it is used to prove induction invariance of the Green–Kubo form and, up to an explicit correction term, of a degree-3 invariant $$\mathcal H_s(t)=\mathcal H(t)-i\frac{s(t)}{2}\hat{\mathcal M},$$2 (Pène et al., 2019).

Across these usages, a common pattern emerges. A dynamical potential is rarely just a static scalar function of configuration. It is instead an object that encodes how reduced descriptions inherit dynamics: through trajectory weighting, time-dependent embedding, induced return maps, geometric structures on field space, or conserved backreaction frameworks. The term therefore marks a shift from equilibrium or kinematic description to operators, functionals, or landscapes that are defined by the dynamics itself.