Finite-Time Scaling in Critical Dynamics

• Finite-Time Scaling (FTS) is a framework that describes how finite observation times or driving rates round off critical singularities, serving as the temporal analogue of finite-size scaling.
• FTS unifies diverse phenomena—from nonequilibrium relaxation and driven critical dynamics to bifurcation and kinetic roughening—by introducing time as an essential scaling field.
• FTS enables precise extraction of dynamic exponents and crossover protocols in systems ranging from quantum critical transitions to fluid criticality and first-order phase transitions.

Finite-Time Scaling (FTS) is a scaling framework for systems whose asymptotic critical or bifurcation behavior is rounded by a finite observation time, a finite driving rate, or a finite iteration number. It is the temporal analogue of finite-size scaling: in equilibrium critical phenomena a finite linear size $L$ cuts off singular behavior, whereas in FTS the relevant cutoff is set by time $t$, iteration number $n$, or a rate $R$. In modern usage the term covers several closely related constructions: nonequilibrium relaxation scaling, finite-time–finite-size scaling (FTFSS), driven critical dynamics in the Kibble–Zurek setting, finite-time scaling of low-dimensional maps, and finite-time analyses of rough interfaces, fluids, quantum systems, and even first-order transitions (Lee et al., 2014, Huang et al., 2014, Corral et al., 2018).

1. Core scaling structure

In the relaxation-based formulation developed for critical many-body dynamics, one introduces a nonequilibrium observable $X(t,L,\epsilon)$, measured at time $t$ in a system of size $L$, with reduced distance to criticality $\epsilon=(K-K_c)/K_c$. The central hypothesis is scale invariance under

$$t\to b^z t,\qquad L\to bL,\qquad \epsilon\to b^{1/\nu}\epsilon,\qquad X\to b^{-\kappa}X,$$

which yields the two-variable scaling form

$$X(t,L,\epsilon)=L^{-\kappa}F\!\bigl(tL^{-z},\,\epsilon L^{1/\nu}\bigr).$$

Here $t$0 is the equilibrium correlation-length exponent, $t$1 the dynamic exponent, and $t$2 the scaling dimension appropriate to $t$3. In globally coupled systems, where the natural variable is the number of degrees of freedom $t$4 rather than a linear size, Lee, Yi and Kim wrote

$t$5

with $t$6 and $t$7; their choice of the sign-averaged real part of the order parameter makes $t$8 (Lee et al., 2014).

Taking the infinite-size limit reduces the two-variable surface to the conventional one-variable FTS form,

$t$9

whose prefactor reproduces the critical power-law decay at $n$0. This relation makes explicit that finite-time decay at criticality and finite-distance relaxation away from criticality are not separate phenomena, but cross-sections of a single scaling structure.

An analogous idea appears in discrete dynamical systems. For maps $n$1 with bifurcation point $n$2, FTS postulates

$n$3

where the iteration number $n$4 plays the role ordinarily played by $n$5. In a related exact formulation for local bifurcations, the distance to the attractor obeys

$n$6

This suggests that FTS is best understood not as a single formula but as a family of scale-covariant representations in which the finite temporal horizon is the singular perturbation (Martin et al., 30 May 2025, Corral et al., 2018).

2. Driven critical dynamics and the Kibble–Zurek connection

A second major branch of FTS concerns systems driven through criticality at a finite rate. For a linear protocol $n$7, the equilibrium relaxation time diverges as $n$8, but the drive itself introduces a competing timescale

$n$9

and a corresponding length scale

$R$0

Huang et al. combined this idea with critical initial slip and wrote the RG scaling form

$R$1

which becomes, upon choosing $R$2,

$R$3

When the initial state is already close to the critical point but far from equilibrium, the standard adiabatic–impulse–adiabatic Kibble–Zurek scenario is replaced by a relaxation–FTS–adiabatic sequence (Huang et al., 2015).

In the Kibble–Zurek interpretation, the nonadiabatic regime begins when the externally imposed timescale becomes shorter than the instantaneous equilibrium relaxation time. Huang et al. argued that the “impulse” regime is precisely the FTS regime generated by the finite external timescale $R$4, and that the Kibble–Zurek defect-density law

$R$5

arises because the true correlation length is cut off by the effective scale $R$6, not because correlations literally stop growing. The same work showed that finite-time–finite-size scaling depends sensitively on protocol and observable: the Liu–Polkovnikov–Sandvik power law applies only to some observables in cooling, and fails for heating or when an external field is present (Huang et al., 2014).

This driven formulation has become the standard language for nonequilibrium critical ramps because it connects static exponents, dynamic exponents, drive protocols, and finite-size effects within a single scaling ansatz.

3. Collapse procedures and exponent extraction

In FTFSS the two most useful collapse protocols follow immediately from the two-variable ansatz. At criticality, $R$7,

$R$8

so plotting $R$9 versus $X(t,L,\epsilon)$0 yields a dynamic collapse from which $X(t,L,\epsilon)$1 is adjusted and $X(t,L,\epsilon)$2 is read from the small-argument slope. At fixed time-scale ratio $X(t,L,\epsilon)$3,

$X(t,L,\epsilon)$4

and plotting $X(t,L,\epsilon)$5 against $X(t,L,\epsilon)$6 determines $X(t,L,\epsilon)$7 and $X(t,L,\epsilon)$8. Lee, Yi and Kim emphasized practical rules that have remained standard: start from a fully ordered state, evolve only up to times satisfying $X(t,L,\epsilon)$9, average over many samples, and exclude both microscopic transients and late-time saturation (Lee et al., 2014).

In driven systems the principal crossover variable is $t$0. At $t$1, $t$2 gives the thermodynamic-limit FTS regime, $t$3 gives equilibrium finite-size scaling, and the interval between them is a genuine finite-time–finite-size regime. For susceptibilities and moments, the appearance or disappearance of plateaus in rescaled plots is used to refine $t$4 and to test the rate exponent $t$5 (Huang et al., 2014).

For kinetic rough interfaces, FTS is formulated in Fourier space through the dynamical structure factor

$t$6

With $t$7, the scaling form

$t$8

permits direct extraction of exponents from simple power laws. If $t$9 at small $L$0, $L$1 at large $L$2, and $L$3, then

$L$4

Rescaling $L$5 or $L$6 then produces the universal scaling functions $L$7 and $L$8 (Chhimpa et al., 2023).

4. Canonical models and benchmark results

The Kuramoto model provided one of the earliest systematic demonstrations of FTFSS. For globally coupled oscillators with pure quenched disorder, Lee, Yi and Kim found $L$9, $\epsilon=(K-K_c)/K_c$0, and $\epsilon=(K-K_c)/K_c$1, corresponding to $\epsilon=(K-K_c)/K_c$2, $\epsilon=(K-K_c)/K_c$3, and $\epsilon=(K-K_c)/K_c$4 for the chosen observable $\epsilon=(K-K_c)/K_c$5. For pure thermal noise they found $\epsilon=(K-K_c)/K_c$6, $\epsilon=(K-K_c)/K_c$7, and $\epsilon=(K-K_c)/K_c$8, again giving $\epsilon=(K-K_c)/K_c$9 and $$t\to b^z t,\qquad L\to bL,\qquad \epsilon\to b^{1/\nu}\epsilon,\qquad X\to b^{-\kappa}X,$$0. In both cases the full two-variable surface $$t\to b^z t,\qquad L\to bL,\qquad \epsilon\to b^{1/\nu}\epsilon,\qquad X\to b^{-\kappa}X,$$1 as a function of $$t\to b^z t,\qquad L\to bL,\qquad \epsilon\to b^{1/\nu}\epsilon,\qquad X\to b^{-\kappa}X,$$2 was reported to be extremely smooth over $$t\to b^z t,\qquad L\to bL,\qquad \epsilon\to b^{1/\nu}\epsilon,\qquad X\to b^{-\kappa}X,$$3, with conventional dynamic and finite-size scaling curves appearing as cross-sections. The same scheme was also applied to small-world Kuramoto networks and to the globally coupled $$t\to b^z t,\qquad L\to bL,\qquad \epsilon\to b^{1/\nu}\epsilon,\qquad X\to b^{-\kappa}X,$$4-state clock model under Monte Carlo dynamics, where standard exponents were extracted without equilibration (Lee et al., 2014).

Low-dimensional maps furnish the rare case in which finite-time scaling functions can be derived analytically. For transcritical bifurcations, the universal scaling function is

$$t\to b^z t,\qquad L\to bL,\qquad \epsilon\to b^{1/\nu}\epsilon,\qquad X\to b^{-\kappa}X,$$5

with exponents $$t\to b^z t,\qquad L\to bL,\qquad \epsilon\to b^{1/\nu}\epsilon,\qquad X\to b^{-\kappa}X,$$6, $$t\to b^z t,\qquad L\to bL,\qquad \epsilon\to b^{1/\nu}\epsilon,\qquad X\to b^{-\kappa}X,$$7, and $$t\to b^z t,\qquad L\to bL,\qquad \epsilon\to b^{1/\nu}\epsilon,\qquad X\to b^{-\kappa}X,$$8. For saddle-node bifurcations, the corresponding exponents are $$t\to b^z t,\qquad L\to bL,\qquad \epsilon\to b^{1/\nu}\epsilon,\qquad X\to b^{-\kappa}X,$$9, $$X(t,L,\epsilon)=L^{-\kappa}F\!\bigl(tL^{-z},\,\epsilon L^{1/\nu}\bigr).$$0, and $$X(t,L,\epsilon)=L^{-\kappa}F\!\bigl(tL^{-z},\,\epsilon L^{1/\nu}\bigr).$$1, with

$$X(t,L,\epsilon)=L^{-\kappa}F\!\bigl(tL^{-z},\,\epsilon L^{1/\nu}\bigr).$$2

These results make precise the claim that the distance-to-attractor law is universal to leading order within a local bifurcation class (Corral et al., 2018).

A later extension studied period-doubling and discontinuous transitions in one-dimensional maps, introduced the finite-time susceptibility

$$X(t,L,\epsilon)=L^{-\kappa}F\!\bigl(tL^{-z},\,\epsilon L^{1/\nu}\bigr).$$3

and the finite-time Lyapunov exponent

$$X(t,L,\epsilon)=L^{-\kappa}F\!\bigl(tL^{-z},\,\epsilon L^{1/\nu}\bigr).$$4

For the logistic map at its first period-doubling, $$X(t,L,\epsilon)=L^{-\kappa}F\!\bigl(tL^{-z},\,\epsilon L^{1/\nu}\bigr).$$5, the reported laws are $$X(t,L,\epsilon)=L^{-\kappa}F\!\bigl(tL^{-z},\,\epsilon L^{1/\nu}\bigr).$$6, $$X(t,L,\epsilon)=L^{-\kappa}F\!\bigl(tL^{-z},\,\epsilon L^{1/\nu}\bigr).$$7, and $$X(t,L,\epsilon)=L^{-\kappa}F\!\bigl(tL^{-z},\,\epsilon L^{1/\nu}\bigr).$$8. The same paper generalized FTS to the 2D Chialvo map by using $$X(t,L,\epsilon)=L^{-\kappa}F\!\bigl(tL^{-z},\,\epsilon L^{1/\nu}\bigr).$$9 as the analogue of the one-dimensional slope, obtaining collapses of $t$00 and $t$01 onto universal curves (Martin et al., 30 May 2025).

5. Quantum, interface, and fluid realizations

In dynamic quantum criticality, Yin, Qin, Lee and Zhong identified three competing time scales: the intrinsic reaction time $t$02, the thermal imaginary-time scale $t$03, and the externally imposed scale $t$04. Their FTS ansatz includes temperature, finite system size, and, for open systems, a dissipation rate $t$05 that must itself be treated as an independent scaling field. For a ramp $t$06, one obtains

$t$07

while a field ramp $t$08 requires the alternative rate exponent $t$09. In the one-dimensional transverse-field Ising chain, this framework recovered $t$10, $t$11, $t$12, $t$13, and $t$14 with high accuracy (Yin et al., 2012).

The extension to strongly interacting Dirac systems showed that a gapped initial state is not required for FTS. In two-dimensional Dirac semimetal-to-Mott transitions, the criterion for observing the universal driven regime becomes

$t$15

with the same $t$16 as in conventional Kibble–Zurek theory. The resulting scaling forms for the squared order parameter $t$17 differ between disordered and ordered initial states, but the overall FTS structure survives even though the initial phase is gapless (Zeng et al., 2024).

A further generalization, proposed for quantum critical and tricritical points, incorporates the initial distance from criticality $t$18 explicitly: $t$19 This form is designed to remain valid for arbitrary driving rates within the critical region, interpolating between the slow-driving Kibble–Zurek limit and the sudden-quench De Grandi–Gritsev–Polkovnikov limit (Yin, 29 May 2026).

In kinetic roughening, FTS has been used to identify universality classes from transient structure-factor data. For the isotropic Sneppen model A and the Maslov–Zhang B-1 variant, the reported values are $t$20, $t$21, $t$22, giving $t$23, $t$24, $t$25, and $t$26; this corresponds to a super-rough, faceted class. For the anisotropic Sneppen model A and the Maslov–Zhang B-2 variant, $t$27, $t$28, and $t$29 imply $t$30, $t$31, and $t$32, consistent with the tensionless one-dimensional KPZ universality class (Chhimpa et al., 2023).

Fluid criticality provides a distinct variant. Along the critical isobar of a Lennard-Jones fluid, complete scaling modifies the effective rate exponent from the thermal form $t$33 to

$t$34

because the mixed ordering field $t$35 becomes the leading relevant field. Molecular dynamics in the $t$36 ensemble yielded $t$37, $t$38, and $t$39, together with static and dynamic exponents consistent with the three-dimensional Ising universality class (Wang et al., 2015).

6. Anomalous, multiscale, and first-order forms

The simplest FTS picture assumes that the relevant critical fluctuations remain self-similar under the finite-rate drive. Zhong et al. argued that this can fail in the phase-fluctuation sector of driven Ising systems, producing anomalous nonequilibrium scaling. They introduced a “bressy” exponent $t$40 associated with singular dependence on

$t$41

On the disordered side, conventional FTS survives, but on the ordered side observables acquire extra factors of $t$42, for example

$t$43

A central consequence is that ordered and disordered phases can exhibit different leading exponents under nonequilibrium driving, rather than the same exponents with different amplitudes (Yuan et al., 2020).

A different departure from the single-scale theory occurs at critical points with emergent symmetry and a dangerously irrelevant scaling variable. In the three-dimensional $t$44-state clock model, driven heating from the ordered phase reveals two driving-induced time scales,

$t$45

where $t$46 and $t$47. The amplitude observable $t$48 follows the usual FTS form, but the angular order parameter $t$49 has two regimes: for small $t$50 it is controlled by $t$51, whereas for large $t$52 it is controlled by $t$53. Numerical work on isotropic and anisotropic $t$54 models reported, for example, $t$55, $t$56, $t$57, $t$58, $t$59, and $t$60 in the isotropic case (Shu et al., 21 Mar 2025).

FTS is not confined to continuous transitions. For finite-time first-order transitions in the Curie–Weiss model, the excess work under a field quench at rate $t$61 obeys

$t$62

for sufficiently large systems, with crossover to $t$63 for smaller systems or faster quenches. Near the spinodal, the dynamics reduces to the universal saddle-node normal form

$t$64

whose solution is expressed in terms of Airy functions. The delay time and transition time scale as $t$65, and the excess-work exponent $t$66 follows from the same saddle-node structure (Wu et al., 2024).

Taken together, these developments show that FTS is a unifying but not monolithic framework. Its most basic claim is that finite temporal cutoffs enter critical dynamics as scaling fields, but the concrete implementation depends on whether the dominant cutoff comes from observation time, finite rate, finite size, initial slip, dissipation, spectral structure, dangerously irrelevant variables, or metastability. The most important technical caution is therefore that apparent one-parameter collapse may conceal missing scaling variables; this is precisely why later work introduced explicit dependence on $t$67, $t$68, $t$69, bressy exponents, or second rate exponents $t$70 when the simpler theory is insufficient (Huang et al., 2014, Yin et al., 2012, Yin, 29 May 2026).