Dynamic Scaling in Physics and Computation

• Dynamic scaling is a framework that uses time and space invariance to simplify the description of complex, evolving systems, with applications ranging from critical phenomena to computational resource allocation.
• Foundational theories, such as the Kibble–Zurek mechanism and self-similar evolution, provide key insights into how critical exponents and dynamic scaling laws govern the behavior near phase transitions.
• Practical implementations in computational systems, including cloud resource management and adaptive deep learning, leverage dynamic scaling principles to optimize performance and minimize costs.

Dynamic scaling is the principle that the time-dependent evolution of physical observables or statistics in a given system admits a reduced description via specific combinations of temporal, spatial, or control parameters—invariant under reparameterization by relevant length, time, or energy scales. Originating in nonequilibrium statistical physics, the concept is crucial for understanding critical phenomena, glassy dynamics, phase ordering, and scaling strategies in modern computational systems. Dynamic scaling emerges whenever macroscopic dynamics are governed by few relevant variables, and can also inform algorithmic and resource allocation problems in computational practice.

1. Fundamental Theory and Universal Hypotheses

In equilibrium and nonequilibrium statistical physics, dynamic scaling characterizes how observables behave near critical points or during self-similar evolution. At criticality, correlation length $\xi$ diverges and the characteristic time scale diverges as $\tau \sim \xi^{z}$, where $z$ is the dynamic critical exponent. The key dynamic scaling hypotheses state that, for disturbance or time evolution (e.g., after a quench), observables $O(t, L, x)$ admit the scaling form

$$O(t, L, x) \simeq L^{-y_o} F\left( t\,L^{-z}, X \right),$$

with $L$ the system size, $t$ the time, $x$ a control parameter (temperature, field, etc.), exponents $z$ and $y_o$ determined by universality class, and $X$ an appropriate scaling variable (e.g., $xL^{y_{x}}$) (Liu et al., 2014, Pelissetto et al., 2018, Pelissetto et al., 2017).

In nonequilibrium scenarios, the dynamic scaling form may reduce to functions combining time, system size, and rate of external change (such as a “quench” velocity) (Liu et al., 2014). The Kibble–Zurek mechanism predicts how such scaling emerges when crossing a phase transition at finite speed, leading to a relation between quench rate and frozen-in correlation scales.

Dynamic scaling also underpins spatiotemporal self-similarity in systems such as phase-ordering kinetics, phase-separating quantum gases, and the post-quench evolution of superfluids or magnets (Ito et al., 7 May 2025, Cui et al., 2022).

2. Canonical Systems and Exponents

Spin Glasses

In 3D Ising spin glasses, dynamic scaling is identified via a non-equilibrium Monte Carlo “quench” protocol. The order parameter $\langle q^2 \rangle(v, L)$ obeys

$$\langle q^2 \rangle(v, L) \simeq L^{-2\beta/\nu} F_2\big(v\,L^{z+1/\nu}\big),$$

where $v$ is quench velocity, $L$ system size, $\beta$ and $\nu$ static critical exponents, $z$ the dynamic exponent. Data collapse is attained with

$$z_{\pm J} = 5.85(9), \qquad z_{\rm Gaussian} = 6.00(10)$$

demonstrating universality across disorder distributions (Liu et al., 2014).

Disordered and Anisotropic $N$-Vector Models

In the presence of isotropic quenched disorder, the dynamic exponent is $z = 2 + O(\epsilon)$, with disorder generically slowing dynamics near second order transitions (Mukherjee et al., 2020). However, symmetry-breaking disorder causes breakdown of dynamic scaling, with runaway flows and loss of finite-$z$ scaling forms.

Thin-Film Roughening

The Clarke–Vvedensky (CV) model of surface growth produces dynamic scaling for the roughness $W(t; R, \epsilon)$:

$$W(t; R, \epsilon) \sim \left(\frac{t}{R^{3/2}(\epsilon + a)}\right)^{\beta},$$

with $R$ the diffusion-to-deposition ratio, $\epsilon$ the step-edge detachment probability, $a$ a nonuniversal constant, $\beta\simeq 0.2$ (VLDS universality class), and dynamic exponent $z\approx 3.3$ (Assis et al., 2015). The scaling variable $R^{3/2}(\epsilon + a)$ governs both roughness and correlation-length evolution.

Swarm Dynamics

Natural insect swarms display dynamic scaling for spatiotemporal correlations of the form

$$\hat C(k, t) = F\left( k^z t \right), \qquad \tau_k \sim k^{-z}$$

with $z\simeq 1.2$ experimentally, as opposed to $z\simeq 2$ for Vicsek-model simulations. The presence of inertial (“spin-wave”) relaxation modes in real swarms implies a novel, non-dissipative universality class distinct from standard flocking models (Cavagna et al., 2016).

Quantum and First-Order Transitions

Dynamic finite-size scaling describes nonequilibrium quench dynamics at both continuous and first-order transitions, with scaling governed by the gap, exponents, and perturbation dimensions. For FOQT, observables depend on $t/\Delta(L)$, and two-state Poisson dynamics describe universal scaling functions (Pelissetto et al., 2017, Pelissetto et al., 2018).

Quantum Gases and Phase Separation

Universal dynamic scaling in quantum gases after a quench is controlled by the scaling symmetry of the Hamiltonian and initial density matrix. For the momentum distribution $n(\mathbf{k}, t)$,

$$n(\mathbf{k}, t) = t^{\alpha} f(\mathbf{k} t^{\beta}), \quad \text{with} \; \alpha=3,\,\beta=1/2 \;\; (\text{unitarity, 3D})$$

(Cui et al., 2022). In phase-separating superfluid mixtures, the vorticity structure factor scales as $S_n(k, t) = l^{-1}(t) F(k l(t))$, where $l(t)\propto t^{2/3}$ is the coarsening scale (Ito et al., 7 May 2025).

3. Applications in Computational and Machine Learning Systems

Dynamic scaling principles inform resource allocation, tuning, and inference in computational systems.

Resource and Inference Scaling

Adaptive allocation of computational resources, termed “dynamic scaling,” is central in distributed systems, cloud computing, and edge environments. Systems such as PerfEnforce dynamically scale analytical clusters to meet SLA constraints using feedback control, reinforcement learning, or online perceptron regression (Ortiz et al., 2016). DynScaling for LLM inference casts compute-budget allocation as a bandit problem, dynamically prioritizing uncertain or “difficult” queries for additional sampling; this achieves higher task performance per unit cost under fixed compute budgets (Wang et al., 19 Jun 2025). Similarly, dynamic scaling of unit tests in code reward modeling reallocates tests where marginal gains in reward accuracy are largest, leading to improved Pass@1 with fixed or lower test-generation budgets (Ma et al., 2 Jan 2025).

Distributed Graph and Stream Processing

Graph dynamic scaling methods perform rapid repartitioning as computational resources fluctuate, with optimized methods (graph edge ordering + chunk-based partitioning) achieving near-constant-time repartitioning and communication cost near the theoretical minimum for billion-edge graphs (Hanai et al., 2021). In distributed dataflow systems, context-aware dynamic scaling is achieved via attributed-graph propagation models that predict runtime under varying scale-out decisions and tune executor allocations for deadline compliance, even under failure (Scheinert et al., 2021).

Deep Learning Model Scaling

In on-device deep learning, dynamic scaling methods such as AdaScale compose multi-branch elastic DNNs with operator-ensemble compression, real-time context (resource) awareness, and an automated adaptation loop to attain device-specific latency, energy, and accuracy targets (Wang et al., 1 Dec 2024). Learned dynamic scaling policies (ELASTIC) within CNNs enable instance-specific, computationally neutral adjustment of spatial resolutions, consistently outperforming fixed-scale policies in classification and segmentation tasks (Wang et al., 2018).

Approximate Computing and Adaptive Precision

Dynamic Precision Scaling (DPS) exploits temporal variations in noise tolerance by adaptively lowering arithmetic precision on a phase-by-phase basis, constrained by a global accuracy bound. By statistical profiling and runtime bit-width control, DPS provides up to $64\%$ energy savings in scientific kernels under controlled output error (Yesil et al., 2017).

4. Engineering and Systems Design Considerations

Dynamic scaling strategies appear in the vertical scaling of edge resources (DYVERSE), buffer-based hybrid scaling in geo-distributed network function virtualization (ScalIMS), and runtime process scaling in HPC (MPI). Typical implementation involves:

• Periodic monitoring of workload, latency, resource usage, and application priorities.
• Priority- or feedback-based decision logic for triggering resource scale-up or scale-down actions.
• Rapid, low-overhead mechanisms for provisioning and deprovisioning workers, partitioning state, and balancing migration/routing costs.
• Hybrid architectures combining proactive (predictive) and reactive (threshold/event-driven) scaling (Duan et al., 2017, Wang et al., 2018, Hanai et al., 2019).

Strategically, dynamic scaling must balance cost, QoS/SLA constraints, scaling latency, and the risk/reward of prediction error or short-term traffic bursts.

5. Experimental Signatures and Universality

Direct confirmation of dynamic scaling hypotheses relies on “data collapse” across system size, time, or velocity, and on the robustness of critical exponents to microscopic details. Selected results include:

System	Scaling Observable	Exponent(s)	Universality Confirmation
3D Ising spin glass	Non-equil. order parameter $\langle q^2 \rangle$	$z_{\pm J}=5.85(9)$, $6.00(10)$	Common $z$ for bimodal/Gaussian disorder
Insect swarms	$\hat C(k, t)$ at $k=1/\xi$	$z\simeq 1.2$	Swarms vs. Vicsek model ($z=2$)
CV thin films	Global roughness $W(t)$	$\beta\simeq 0.2$, $z\approx 3.3$	VLDS class, RG-derived scaling variable
Quantum gases	$n(\mathbf{k}, t)$ after quench	$\alpha=3$, $\beta=1/2$	Robust for multi-body, approximate cases

These patterns indicate both the predictive strength of dynamic scaling theory and the degree to which universality persists or is broken under system-specific perturbations.

6. Open Problems and Extensions

• Robustness to Disorder and Symmetry Breaking: Dynamic scaling can break down under certain symmetry-breaking disorders (e.g., component-diagonal random coupling in $N$-vector models), driving crossover to new behaviors—such as fluctuation-driven first-order transitions or diverging dynamic exponents (Mukherjee et al., 2020).
• Finite-Time and Preasymptotic Corrections: Strong corrections to scaling (e.g., anomalous roughness in low-$T$ surface growth, or non-universal exponents under approximate symmetry) remain important in interpreting experimental data and benchmarking practical systems (Assis et al., 2015, Cui et al., 2022).
• Algorithmic Scalability: Scaling methods for massive distributed or streaming systems present challenges both for theoretical analysis and for maintaining scaling quality under structural or workload change (Hanai et al., 2021, Scheinert et al., 2021).
• Multi-dimensional Trade-offs: In resource-constrained environments, dynamic scaling must optimize across multi-objective fronts (latency, energy, accuracy, cost), often by real-time adaptation and policy learning (Wang et al., 1 Dec 2024, Ortiz et al., 2016).

Dynamic scaling thus remains a central unifying paradigm, bridging statistical physics, quantum dynamics, and the design of responsive, efficient computational infrastructures.