Physics & Dynamics Randomization

• Physics and dynamics randomization is a multidisciplinary approach that integrates stochastic methods into classical and quantum frameworks to model inherent uncertainties.
• It utilizes techniques like stochastic perturbations, Monte Carlo sampling, and domain randomization to improve simulation fidelity and control robustness.
• These methodologies enable efficient statistical inference and sim-to-real transfer, significantly impacting applications from robotics to high-energy physics.

Physics and dynamics randomization encompasses a diverse set of theoretical, computational, and algorithmic methodologies by which randomness, stochasticity, or parameter variability are introduced into physical systems, models of their dynamics, or the mechanistic frameworks describing their evolution. The motivations span from capturing intrinsic fluctuations (as in thermal or quantum contexts), reproducing observed macroscopic irreversibility, enabling robust control or statistical inference, to bridging the gap between simulation and experimental or real-world behavior. Techniques range from formal modifications of foundational variational principles, stochastic perturbation of classical systems, and probabilistic modeling of long-term dynamics to modern Monte Carlo sampling and domain randomization strategies in computational physics and robotics.

1. Stochastic Extensions of Classical Mechanics

A fundamental approach to physics and dynamics randomization is the extension of core variational principles to ensembles of random trajectories. In the probabilistic mechanics formalism (Wang, 2010), the classical virtual work and least action principles—originally expressed as $dW = 0$ and $dA = 0$ for every trajectory—are generalized to statistical statements

$$\overline{\delta W} = \sum_j p_j \delta W_j = 0, \qquad \overline{\delta A} = \sum_k p_k \delta A_k = 0,$$

where the averaging is over the probability distribution of random dynamical paths $j, k$. This ensemble-based extension enforces that the variational conditions hold on average rather than for each realization.

This probabilistic mechanics framework softens deterministic Newtonian laws to allow for stochastic evolutions, making it structurally compatible with systems where fluctuations—whether due to thermal noise, external perturbations, or measurement imprecision—cannot be neglected. The theory recovers classical results such as equiprobability in the microcanonical ensemble in the zero-randomness limit, but also yields exponential path-probability distributions for random processes (e.g., $p_k \propto \exp(-\eta A_k)$ for action $A_k$ along a random path).

A notable implication is the violation of the Liouville and Poincaré recurrence theorems: under random forces, the phase–space density evolves non-conservatively ($p(t) = p(0)\exp[-\Lambda(t)]$), and the system loses its recurrence property as the phase–volume grows or shrinks with entropy production determined by the cumulative random work. This provides a dynamical origin for macroscopic irreversibility.

2. Randomization in Statistical and Quantum Physics

In quantum theory, the emergence of probability is formalized through the ergodic randomization of deterministic unitary dynamics (Hofmann, 2016). The Hilbert space framework, with its unitary evolution operators

$U(t) = \sum_n e^{-i S_n/\hbar} \ket{U_n}\bra{U_n},$

maps "states" to closed orbits under deterministic motion; probabilities such as

$$P(b|a) = \operatorname{Tr}(\ket{b}\bra{b}\ket{a}\bra{a})$$

arise from randomized sampling along these orbits during state preparation and measurement. Here, observable outcomes are statistical manifestations of entire deterministic trajectories.

Stochastic Bäcklund transformations (O'Connell, 2014) advance this concept for integrable Hamiltonian systems by augmenting deterministic evolution with stochastic differential equations—introducing noise to constants of motion. This produces diffusive dynamics whose generators coincide (after a ground-state transformation) with quantum Hamiltonians, thereby probabilistically bridging classical integrability and quantum mechanics. The and noise parameter create a direct link between stochastic dynamics and properties of quantum systems, including the statistics of measurement outcomes.

Spatial symmetries further constrain the possibilities for randomization. As shown in (Jones et al., 2022), imposing rotational invariance (SO(2) or higher) tightly restricts the correlations and potential for randomness in measurement outcomes, quantifying the extractable randomness as a function of geometric properties, such as maximal spin and transformation overlap.

3. Sampling, Algorithmic, and Monte Carlo Randomization

Practical computation and simulation of physical systems often require efficient stochastic sampling of high-dimensional distributions. Algorithms for random thermal momenta generation (Molnar, 2012) are essential for simulations in heavy-ion physics and statistical mechanics. State-of-the-art methods utilize automated rejection sampling with staircase comparison functions and tabulated interpolation of the probability density

$$g(x) \propto \frac{x^2}{\exp(\sqrt{x^2 + z^2}) + \alpha}$$

for efficient momentum sampling. For massless Boltzmann particles, direct sampling can be performed via

$x = -\ln[(1 - \xi_1)(1 - \xi_2)(1 - \xi_3)]$

with uniform deviates $\xi_i$, maximizing both fidelity and computational speed.

In reinforcement learning for robotics and control, physics and dynamics randomization is implemented via explicit parameter-level and dynamics-level domain randomization during policy training. Crucial parameters such as mass, friction, joint damping, and control delays are randomized per training episode to produce robust controllers capable of withstanding sim-to-real discrepancies (Peng et al., 2017, Exarchos et al., 2020, Campanaro et al., 2022, Lei et al., 2 Oct 2025). Enhanced schemes include:

• Adaptive randomization via active domain randomization (ADR) (Mehta et al., 2019), where the sampling distribution is dynamically focused on physically "difficult" or informative parts of the parameter space, guided by discriminative feedback from policy rollouts.
• Minimal randomization strategies such as random force injection (RFI) and its extension (ERFI) (Campanaro et al., 2022), which perturb the system dynamics with per-step or per-episode additive forces/torques, modeling both local and global variations without the need to hand-tune myriad physical parameters.
• Kinematic domain randomization (Exarchos et al., 2020), wherein geometric aspects (e.g., link lengths) are varied, often yielding greater transferability across unmodeled environments than dynamics-only randomization.
• Multi-simulator dynamics randomization (Lei et al., 2 Oct 2025), where heterogeneous simulators each contribute different dynamic inductive biases, and the policy is optimized over their convex combination, demonstrably reducing the sim-to-real gap by enveloping a broader swath of conceivable real-world dynamics.

4. Randomized Models of Conservative and Stochastic Dynamics

For deterministic conservative systems, such as Newtonian motion in single-well potentials, physical randomization can be effected by operations like hard velocity reversals at randomly chosen times, conserving energy but altering the phase-space trajectory (Mandrysz et al., 2019). Analytical results indicate that the long-time stationary distributions for position and velocity coincide with those from purely deterministic motion, demonstrating insensitivity to the fine details of randomization protocol provided the energy is held fixed. This highlights the universality of ergodic results under energy-conserving randomizations.

In higher-dimensional stochastic settings (Cinque et al., 2023), random motions with a finite, minimal set of velocities are characterized via joint distributions for both position and count of displacements in each direction. The general framework encompasses cyclic and complete motions, which, respectively, impose deterministic switching orders or allow transitions among all possible velocities with prescribed probabilities. The time evolution of the probability density is governed by a system of first-order PDEs, reducible in symmetric cases to higher-order hyperbolic equations; explicit closed-form solutions are provided for several key configurations.

For iterative and map-based dynamics, new probabilistic methods (Cirier, 6 Jan 2025) analyze the asymptotic distribution of states under repeated action of deterministic or stochastic maps. By associating uniquely invariant probability measures and leveraging tools such as the Fourier–Laplace transform and Plancherel–Rotach function (with critical saddles found by $d y(a)/d a = s\, (df/da) - 1/a = 0$), one uncovers the formation of random periodic cycles and "self-averaged" densities, even when the underlying ODE possesses a unique solution.

5. Emergence, Symmetry, and Foundational Perspectives

At the foundational level, the random dynamics program (Tarzi, 2014) frames the very emergence of physical laws, including symmetries like Lorentz and gauge invariance, as arising not from postulated fundamental invariants but from maximally random combinatorial or relational structures (e.g., multicoloured random graphs). Under this perspective, randomness at the substrate level is "whittled away" via selection or confusion mechanisms to yield emergent, effective symmetries and the familiar regularities of low-energy physics, including features specific to the Standard Model's group structure.

Relatedly, in quantum-correlated protocols, theory-independent randomness is shown to be enforced by the structure of spatial symmetries alone (Jones et al., 2022), decoupling the origin of unpredictability from the details of quantum theory and rooting it in deeper geometrical constraints.

6. Applications and Impact Across Disciplines

Physics and dynamics randomization is central to a variety of applications:

• Statistical mechanics and thermodynamics: The probabilistic mechanics formalism directly yields entropy production and irreversibility, maximal entropy inference as a derived consequence, and reproduces transport laws such as Fourier's law via averaged virtual work (Wang, 2010).
• Numerical simulation and Monte Carlo studies: Efficient generation of randomized thermal momenta and sampling from complex distributions underpin current approaches to transport and event generation in high-energy and condensed matter physics (Molnar, 2012).
• High-dimensional optimization for PINNs: Randomized linear algebra techniques such as Nyström approximations dramatically accelerate natural gradient optimization without sacrificing final accuracy in training PDE-solving neural networks (Guzmán-Cordero et al., 17 May 2025).
• Robotic control and sim-to-real transfer: Domain and dynamics-level randomization, extended by multi-simulator approaches and active adaptation, have achieved substantial success in producing policies that generalize to real hardware with minimal or no fine-tuning (Peng et al., 2017, Chebotar et al., 2018, Campanaro et al., 2022, Lei et al., 2 Oct 2025).
• Turbulence, distributed chaos, and complex fluids: Particle-laden flow studies attribute qualitative shifts in energy spectra and randomization to two-way coupling, symmetry breaking, and moment invariants dictated by the parameters controlling the degree and form of stochasticity (Bershadskii, 2023).

These examples illustrate a unifying theme: randomness, when properly structured—via stochasticization of dynamics, randomized parameterization, or probabilistic reformulation of classical equations—allows both the reproduction of observed physical phenomena and the robust solution of complex control, inference, and simulation problems across classical, quantum, and computational physics.

---

In summary, physics and dynamics randomization unifies disparate methodologies that inject, analyze, or exploit stochasticity at the heart of physical modeling. This not only reconciles experiment and theory in systems where determinism is empirically inadequate, but also equips modern computational physics, control, and learning theory with indispensable tools for robust performance, efficient simulation, and a deeper foundational understanding of the origins and consequences of randomness in the physical world.