Boundary Finite-Time Scaling (BFTS)

• BFTS is a framework that unifies finite-time evolution with boundary effects across driven critical dynamics, finite-size crossovers, and PDE control.
• It replaces bulk exponents with boundary exponents to capture explicit boundary-to-bulk crossover, including anomalous logarithmic scaling in certain protocols.
• In control theory, BFTS informs finite-time stabilization via boundary feedback strategies, using backstepping and explicit settling time derivations.

Searching arXiv for papers on Boundary Finite-Time Scaling and closely related usage across critical dynamics, bifurcations, and boundary stabilization. Boundary finite-time scaling (BFTS) denotes scaling or stabilization behavior governed jointly by finite-time evolution and boundary effects. In nonequilibrium critical dynamics, it is the boundary generalization of finite-time scaling (FTS) and the Kibble-Zurek mechanism (KZM), with the boundary exponent replacing the bulk order-parameter exponent and with explicit boundary-to-bulk crossover. In adjacent statistical-mechanics literature, the same label is also used for the crossover regime in which finite size and the driving-induced finite length are comparable. In PDE control, closely related usage appears in “bilateral boundary finite-time stabilization,” where boundary actuation and boundary feedback drive distributed systems to zero in finite time (Shu et al., 12 Sep 2025, Huang et al., 2014, Sun et al., 2024). This suggests that BFTS is a context-dependent term spanning several research programs linked by the common role of boundaries in finite-time phenomena.

1. Terminological scope and research domains

Current arXiv usage clusters around three settings. First, in driven critical dynamics near surfaces, BFTS is a normal generalization of KZM in which boundary critical exponents govern the rate scaling of surface observables. Second, in FTS–finite-size crossover theory, BFTS denotes the regime where both finite size and finite driving time are simultaneously relevant. Third, in control theory, the phrase occurs in finite-time boundary stabilization and bilateral boundary stabilization of hyperbolic and dispersive PDEs (Shu et al., 12 Sep 2025, Huang et al., 2014, Coron et al., 2020).

Research setting	Central competition	Representative papers
Driven critical dynamics near a physical boundary	boundary criticality vs driving-induced time/length scales	(Shu et al., 12 Sep 2025, Huang et al., 2014)
Finite-size/finite-time crossover	system size $L$ vs freeze-out length $\hat{\xi}$	(Huang et al., 2014, Fontana, 2019)
Boundary control of PDEs	boundary actuation vs finite settling time	(Sun et al., 2024, Coron et al., 2020, Nguyen, 2024)

The statistical-physics meaning is the most explicit use of the acronym itself. There, the driving-induced time scale is written as $\zeta_d \sim v^{-z/r}$ or $t_R \sim R^{-z/r}$, with $r = z + 1/\nu$, and boundary observables inherit scaling controlled by the boundary exponent $\beta_1$ rather than the bulk exponent $\beta$. By contrast, the control-theoretic usage is not a renormalization-group scaling theory; it concerns constructive feedback laws, backstepping transforms, Gramian operators, and explicit finite settling times (Shu et al., 12 Sep 2025, Huang et al., 2014, Nguyen, 2024).

2. Canonical BFTS in driven boundary critical dynamics

The clearest formulation of BFTS is the study of driven critical dynamics near boundaries in the Ising universality class with ordinary, special, extraordinary, and surface transitions. The central prescription is to replace the bulk order-parameter exponent $\beta$ by the boundary exponent $\beta_1$ in FTS/KZM scaling laws for near-boundary observables, while retaining the driving exponent combination $r = z + 1/\nu$ and introducing explicit boundary–bulk crossover in position space (Shu et al., 12 Sep 2025).

For heating dynamics at the boundary $\hat{\xi}$0, the square of the local order parameter obeys

$\hat{\xi}$1

For cooling, when normal BFTS applies, the corresponding form is

$\hat{\xi}$2

A position-resolved crossover form is

$\hat{\xi}$3

with boundary scaling for $\hat{\xi}$4 and bulk scaling for $\hat{\xi}$5. This expresses the basic BFTS picture: the driving-induced length sets the depth over which boundary criticality dominates before the system crosses over to bulk FTS (Shu et al., 12 Sep 2025).

This formulation is naturally embedded in the FTS reinterpretation of KZM. In that framework, the external rate introduces a finite time scale $\hat{\xi}$6, and the freeze-out length becomes $\hat{\xi}$7. The impulse regime of KZM is then identified with the FTS regime. BFTS specializes this logic to observables and scaling fields anchored at or near a boundary (Huang et al., 2014).

3. Regimes, anomalies, and generalized boundary driving

BFTS is not uniformly “normal” across protocols or boundary universality classes. For heating dynamics in all boundary universality classes, and for cooling dynamics in special, extraordinary, and surface transitions, the order parameter follows the normal boundary generalization of KZM. By contrast, cooling dynamics in the ordinary transition exhibits abnormal logarithmic scaling on the driving rate rather than the expected power law: $\hat{\xi}$8 The stated physical reason is that enhanced surface fluctuations and reduced coordination make surface domain formation impossible during cooling, so local equilibrium is not achieved at the surface (Shu et al., 12 Sep 2025).

A criterion is given for the validity of normal BFTS in cooling. If

$\hat{\xi}$9

the surface can nucleate ordered domains during cooling and normal BFTS holds; if this condition fails, abnormal behavior can appear. This separates ordinary cooling from the special, surface, and extraordinary cases within the Ising systems studied (Shu et al., 12 Sep 2025).

The same work develops a generalized BFTS for nonequilibrium initial states at the special transition when the surface coupling is increased across the special point along the ordinary-transition line. There, the KZM prerequisite of a short-ranged initial correlation length or time breaks down. The generalized scaling introduces the waiting time, or “age,” of the boundary: $\zeta_d \sim v^{-z/r}$0 This construction incorporates exponents from both the ordinary and special transitions and shows that BFTS can explicitly encode boundary history, not only instantaneous boundary universality (Shu et al., 12 Sep 2025).

A common misconception is that BFTS is merely ordinary FTS with $\zeta_d \sim v^{-z/r}$1 replaced by $\zeta_d \sim v^{-z/r}$2. The ordinary-cooling logarithmic law and the age-dependent special-transition scaling show that this replacement is sufficient only in a subset of protocols. Boundary preparation, the direction of driving, and the boundary universality class can alter the scaling structure itself (Shu et al., 12 Sep 2025).

4. Finite-size/finite-time crossover and boundary-conditioned first-order transitions

A second line of work uses BFTS for the crossover regime where both finite size and finite driving time matter. In the FTS analysis of KZM, the regime $\zeta_d \sim v^{-z/r}$3 is identified as the crossover where finite size and the driving-induced scale are comparable. The relevant scaling variable can be written as $\zeta_d \sim v^{-z/r}$4, or equivalently $\zeta_d \sim v^{-z/r}$5. In this sense, BFTS denotes the joint finite-time–finite-size scaling regime interpolating between adiabatic finite-size scaling and impulse finite-time scaling (Huang et al., 2014).

For observables such as the susceptibility,

$\zeta_d \sim v^{-z/r}$6

so that the FSS regime corresponds to $\zeta_d \sim v^{-z/r}$7 and the FTS regime to $\zeta_d \sim v^{-z/r}$8. The crossover regime is precisely the setting called boundary finite-time scaling in that work. The terminology is therefore tied to the boundary set by finite system size rather than solely to a physical surface (Huang et al., 2014).

At first-order transitions, boundary conditions can change the relevant time scale qualitatively. In the two-dimensional Ising model with relaxational dynamics, periodic and open boundary conditions lead to exponentially large tunneling times,

$\zeta_d \sim v^{-z/r}$9

whereas opposite fixed boundary conditions enforce an interface and yield a power law,

$t_R \sim R^{-z/r}$0

This is presented as a cornerstone of Boundary Finite-Time Scaling at first-order transitions because the boundaries switch the slow mode from nucleation-controlled tunneling to interface motion (Fontana, 2019).

Related finite-size results at first-order quantum transitions show that boundary conditions favoring one phase produce richer scaling than neutral boundaries. In the one-dimensional quantum Ising chain with equal fixed boundary conditions, the pseudotransition shifts to

$t_R \sim R^{-z/r}$1

the minimum gap scales as $t_R \sim R^{-z/r}$2, and the appropriate scaling variable near the pseudotransition becomes

$t_R \sim R^{-z/r}$3

That analysis states that off-equilibrium dynamics, identified there as Boundary Finite-Time Scaling, is expected to be even richer under such boundary-favored first-order quantum transitions (Pelissetto et al., 2018).

5. Boundary stabilization and finite-time control of PDEs

In PDE control, the phrase occurs in a distinct but structurally related sense: finite-time stabilization achieved through boundary feedback. A representative example is bilateral boundary finite-time stabilization of $t_R \sim R^{-z/r}$4 linear first-order hyperbolic systems with spatially varying coefficients on $t_R \sim R^{-z/r}$5,

$t_R \sim R^{-z/r}$6

The control objective is to design $t_R \sim R^{-z/r}$7 and $t_R \sim R^{-z/r}$8 so that the state reaches zero in finite time. The paper constructs bilateral boundary feedback by an invertible Volterra-type backstepping transformation and case-dependent target systems for $t_R \sim R^{-z/r}$9, $r = z + 1/\nu$0, and $r = z + 1/\nu$1. For the symmetric-speed case, the settling time is explicit: $r = z + 1/\nu$2 with

$r = z + 1/\nu$3

The bilateral design is also presented as enabling a potential for fault-tolerant designs (Sun et al., 2024).

A broader nonautonomous theory treats $r = z + 1/\nu$4 one-dimensional linear hyperbolic balance laws with coefficients depending on time and space,

$r = z + 1/\nu$5

Here the backstepping method is extended by time-dependent Volterra and Fredholm transformations with kernels solving nonstandard multidimensional hyperbolic PDEs. The main theorem provides finite-time stabilization with explicit minimal settling time

$r = z + 1/\nu$6

For autonomous systems this reduces to

$r = z + 1/\nu$7

In this literature, “boundary finite-time scaling” is therefore a constructive control property tied to characteristic travel times and boundary feedback design rather than to critical exponents (Coron et al., 2020).

An analogous viewpoint appears for the KdV equation, where local rapid and finite-time boundary stabilization is achieved by static, dynamic, and time-varying feedbacks built from Gramian operators. Finite-time stabilization is obtained by switching to ever larger feedback gains $r = z + 1/\nu$8 on shrinking subintervals $r = z + 1/\nu$9 with $\beta_1$0, so that $\beta_1$1. This work explicitly interprets the time-varying scaling feedback as a manifestation of Boundary Finite-Time Scaling (Nguyen, 2024).

6. Related extensions, unifications, and conceptual limits

Several adjacent FTS programs illuminate the broader conceptual reach of BFTS. A generalized FTS framework valid for arbitrary driving rates inside the critical region introduces the scaling variable $\beta_1$2 for the initial state and bridges the KZ and De Grandi–Gritsev–Polkovnikov limits: $\beta_1$3 That work states that the explicit inclusion of $\beta_1$4 addresses boundary cases such as drives starting near the critical point and provides a foundation for describing BFTS as an initial-condition-sensitive problem (Yin, 29 May 2026).

A distinct extension arises at critical points with emergent symmetry and two divergent scales. In the three-dimensional $\beta_1$5-state clock model, two driving-induced time scales appear,

$\beta_1$6

and the angular order parameter $\beta_1$7 crosses from a regime governed by $\beta_1$8 to one governed by $\beta_1$9. The paper describes the large-$\beta$0 regime as a dynamic “boundary” regime in which nonequilibrium scaling appears beyond the equilibrium support of the dangerously irrelevant variable. This is a boundary-FTS scenario in the sense of a crossover to a new limiting sector of the theory (Shu et al., 21 Mar 2025).

Other finite-time scaling literatures use boundary language more analogically. In local bifurcations of maps and flows, the universal scaling function

$\beta$1

organizes finite-time approach to the bifurcation point, and one paper explicitly connects this to BFTS in branching processes and stochastic systems (Corral et al., 2018, Corral, 2024). In low-dimensional maps, the term BFTS is not explicitly used, but the framework is stated to mirror “finite-size boundary scaling,” including discontinuous bifurcations and spinodal-like limits (Martin et al., 30 May 2025). In fluid criticality, complete-field FTS along the critical isobar shows that path constraints can change the leading rate exponent from $\beta$2 to $\beta$3, which is presented as directly relevant to BFTS studies on critical lines and constrained loci in parameter space (Wang et al., 2015).

The principal conceptual limit is therefore terminological rather than technical. BFTS does not denote a single universal formalism across all subfields. In driven boundary critical dynamics it is an extension of FTS/KZM controlled by boundary exponents; in finite-size crossover theory it marks the regime $\beta$4; in PDE control it denotes exact finite-time stabilization by boundary feedback. The common denominator is the finite-time role of boundaries, but the mathematical objects—critical exponents, crossover functions, Volterra kernels, Gramian operators, or characteristic travel times—depend entirely on context (Shu et al., 12 Sep 2025, Huang et al., 2014, Sun et al., 2024).