Minimum-Tracking Linear Response

• Minimum-tracking linear response is a perturbative framework that selects minimal, optimal perturbations to achieve a desired linear effect under control constraints.
• It is applied across finite-state Markov chains, chaotic systems using shadowing and tangent approximations, and electronic-structure methods for computing Hubbard U and Hund’s J.
• Research demonstrates that leveraging first-order response formulas with explicit tracking or optimization yields computationally efficient and robust solutions with quantifiable error bounds.

Minimum-tracking linear response denotes a class of perturbative constructions in which small changes of a system are evaluated under an explicit tracking, optimization, or minimum-effort criterion. In finite-state Markov chains, the central questions are how to select a perturbation so as to maximise the linear response of the equilibrium distribution, the expectation of a specified observable, or the rate of convergence to equilibrium (Antown et al., 2018). In high-dimensional chaotic dynamics, the reduced formulation keeps the stable regularized tangent equation and omits unstable and neutral terms when the objective function is appropriately aligned with unstable manifolds (Sliwiak et al., 2022). In electronic-structure theory, the minimum-tracking linear-response method computes Hubbard $U$ and Hund’s $J$ from fully relaxed ground-state responses without reference to the Kohn–Sham eigensystem (Chai et al., 2024). A plausible implication is that the phrase identifies a design principle—tracking a desired first-order effect while controlling the perturbation—rather than a single universal formalism.

1. Conceptual scope

Across the literature, the tracked object varies with the application: an invariant density, an observable average, a projected state trajectory, a subspace occupancy, or a thermodynamic work protocol. What remains common is the use of a first-order response formula together with an optimization or admissibility constraint.

Setting	Tracked quantity	Characteristic formulation
Finite-state Markov chains	$h$, $c^\top u_1$, $|\lambda_2|$	$u_1 = Qmh$
High-dimensional chaos	$\frac{d\langle J\rangle}{ds}$	$$\frac{d\langle J\rangle}{ds} \approx \frac{1}{N}\sum_{k=0}^{N-1} DJ_k \cdot v_k$$
DFT+$U$+$J$	$J$0, $J$1	$J$2
Finite-dimensional linear systems	$J$3	$J$4

The topic therefore spans several adjacent research programs. In some of them, “minimum-tracking” refers to minimum-norm or minimum-dissipation tracking; in others, it refers to tracking the fully relaxed ground state or the shadowing component of the tangent dynamics. The technical differences are substantial, but the organizing problem is similar: determine which infinitesimal perturbation or control most efficiently produces, approximates, or suppresses a prescribed linear response.

2. Finite-state Markov chains and transfer-operator optimization

For a column-stochastic Markov matrix $J$5 with invariant vector $J$6, a perturbation $J$7 yields

$J$8

where the linear response $J$9 is characterized by

$h$0

Since $h$1 is singular but invertible on the zero-sum subspace, the explicit solution is

$h$2

The vectorized form

$h$3

turns the problem into a constrained finite-dimensional optimization. The paper studies three objectives: maximizing $h$4, maximizing $h$5 for a specified observable $h$6, and maximizing the decrease in $h$7, where $h$8 is the second eigenvalue of $h$9. Under stochasticity and normalization constraints, the maximizing $c^\top u_1$0 for the invariant-measure problem is the top eigenvector of a reduced matrix derived from $c^\top u_1$1 and projected onto the nullspace of the stochasticity operator $c^\top u_1$2. For observable response, the optimal perturbation has entries

$c^\top u_1$3

and for spectral optimization the first-order eigenvalue variation is

$c^\top u_1$4

The framework extends to inhomogeneous finite sequences $c^\top u_1$5, where the total response at terminal time is

$c^\top u_1$6

For randomly perturbed dynamical systems $c^\top u_1$7, the same machinery is applied after replacing the system by a finite matrix representation of the annealed transfer operator

$c^\top u_1$8

typically via Ulam discretization. The reported computational reduction is to sparse eigenvalue, SVD, or linear-system problems, with uniqueness up to sign under a generic simple-eigenvalue condition and explicit MATLAB implementations (Antown et al., 2018).

3. Shadowing reductions and SRB response in chaotic dynamics

In chaotic systems, linear response is classically formulated through Ruelle’s theory. For an ergodic system $c^\top u_1$9, the derivative of a statistical average satisfies

$|\lambda_2|$0

with $|\lambda_2|$1 and $|\lambda_2|$2 the SRB measure. The S3 method decomposes the perturbation as

$|\lambda_2|$3

leading to stable, neutral, and unstable contributions. The unstable term contains derivatives of the SRB measure along unstable manifolds and is numerically costly.

The reduced minimum-tracking or shadowing-based approximation exploits a specific asymptotic regime. In high-dimensional, statistically homogeneous chaotic systems, if the objective function is appropriately aligned with the most unstable directions, the unstable contribution can be neglected. The resulting reduced algorithm omits the unstable and neutral terms and keeps only the regularized tangent equation, so that

$|\lambda_2|$4

The numerical evidence is sharply regime-dependent. For Lorenz 63, all three terms are comparable and neglecting the unstable term introduces large errors. For Lorenz 96 with $|\lambda_2|$5, the reduced algorithm matches reference finite-difference sensitivity within a few percent except near transitions in the Lyapunov-exponent spectrum. For the Kuramoto–Sivashinsky equation at system size $|\lambda_2|$6, the reduced approach is reported as highly accurate except near transitional parameter values where positive Lyapunov exponents sharply decrease (Sliwiak et al., 2022).

The broader theory of linear response places these reductions in context. In uniformly hyperbolic settings, such as topologically mixing Axiom A diffeomorphisms, linear response holds with Ruelle’s formula

$|\lambda_2|$7

and transfer-operator resolvent formulas are available. By contrast, for piecewise expanding or smooth unimodal families, differentiability may fail unless the perturbation is horizontal, and generic transversal families need not exhibit linear response at all. This makes the shadowing simplification a conditional statement, not a universal replacement for the full SRB-response theory (Baladi, 2014).

4. Ground-state minimum tracking in DFT+$|\lambda_2|$8+$|\lambda_2|$9

In DFT+$u_1 = Qmh$0+$u_1 = Qmh$1, minimum-tracking linear response is a first-principles method for determining Hubbard $u_1 = Qmh$2 and Hund’s $u_1 = Qmh$3 from ground-state properties. The perturbation applied to subspace $u_1 = Qmh$4 and spin channel $u_1 = Qmh$5 is

$u_1 = Qmh$6

implemented by modifying the Kohn–Sham Hamiltonian matrix elements. For each $u_1 = Qmh$7, one performs a self-consistent field calculation and records the subspace occupancy

$u_1 = Qmh$8

and the subspace-averaged Hartree+XC potential

$u_1 = Qmh$9

The spin-resolved “scaled $\frac{d\langle J\rangle}{ds}$0” procedure then extracts

$\frac{d\langle J\rangle}{ds}$1

$\frac{d\langle J\rangle}{ds}$2

with finite differences evaluated over a set of perturbations such as $\frac{d\langle J\rangle}{ds}$3. The defining feature is that all measured quantities are ground-state density-matrix or population-operator elements, with no explicit reference to Kohn–Sham eigenstates.

The implementation in CP2K covers both tensorial and Löwdin subspace representations and includes analytical DFT+$\frac{d\langle J\rangle}{ds}$4+$\frac{d\langle J\rangle}{ds}$5 forces. Benchmarks reported in the paper include band-gap opening in NiO, polaron distributions in TiO$\frac{d\langle J\rangle}{ds}$6, the dependence of the TiO$\frac{d\langle J\rangle}{ds}$7 band gap on $\frac{d\langle J\rangle}{ds}$8 corrections, and a series of hexahydrated transition metals. For bulk rutile TiO$\frac{d\langle J\rangle}{ds}$9, when both $$\frac{d\langle J\rangle}{ds} \approx \frac{1}{N}\sum_{k=0}^{N-1} DJ_k \cdot v_k$$0 and $$\frac{d\langle J\rangle}{ds} \approx \frac{1}{N}\sum_{k=0}^{N-1} DJ_k \cdot v_k$$1 are included using first-principles minimum-tracking parameters, the Kohn–Sham gap is $$\frac{d\langle J\rangle}{ds} \approx \frac{1}{N}\sum_{k=0}^{N-1} DJ_k \cdot v_k$$2 eV in the tensorial representation, compared with an experimental value of $$\frac{d\langle J\rangle}{ds} \approx \frac{1}{N}\sum_{k=0}^{N-1} DJ_k \cdot v_k$$3 eV. For the hexahydrated $$\frac{d\langle J\rangle}{ds} \approx \frac{1}{N}\sum_{k=0}^{N-1} DJ_k \cdot v_k$$4 series in the tensorial representation, metal-$$\frac{d\langle J\rangle}{ds} \approx \frac{1}{N}\sum_{k=0}^{N-1} DJ_k \cdot v_k$$5 $$\frac{d\langle J\rangle}{ds} \approx \frac{1}{N}\sum_{k=0}^{N-1} DJ_k \cdot v_k$$6 values increase from $$\frac{d\langle J\rangle}{ds} \approx \frac{1}{N}\sum_{k=0}^{N-1} DJ_k \cdot v_k$$7 eV for V$$\frac{d\langle J\rangle}{ds} \approx \frac{1}{N}\sum_{k=0}^{N-1} DJ_k \cdot v_k$$8 to $$\frac{d\langle J\rangle}{ds} \approx \frac{1}{N}\sum_{k=0}^{N-1} DJ_k \cdot v_k$$9 eV for Cu$U$0, $U$1 values are about an order of magnitude lower than $U$2, and residuals from the linear fit are negligible for all cases (Chai et al., 2024).

5. Tracking controllability, minimum norm, and lower bounds

A distinct but closely related strand treats tracking as an exact controllability problem. For the finite-dimensional linear system

$U$3

with output projector $U$4, the tracking objective is

$U$5

The associated control-to-output map is

$U$6

Using the Hilbert Uniqueness Method and duality, tracking controllability is characterized by surjectivity of $U$7 onto $U$8, equivalently by a non-standard observability inequality for the adjoint system,

$U$9

The minimum-norm tracking control solves

$J$0

subject to the tracking constraint, and is synthesized as $J$1. A central structural point is the regularity loss: for a target $J$2, an $J$3 control may not exist, and in the scalar Brunovský case the required regularity can be as high as $J$4, depending on $J$5, $J$6, and $J$7 (Zamorano et al., 2024).

For monotonic tracking of a constant step reference in MIMO systems, a separate geometric-control formulation shows that monotonicity from all initial conditions is achieved when each output error component is governed by a single real stable closed-loop mode and the remaining closed-loop modes are invisible at the tracking error. The constructive condition is expressed through the output-nulling subspace $J$8 and per-output subspaces $J$9, with dimension inequalities such as

$J$00

When the condition is satisfied, rapid settling and monotonicity are not necessarily competing objectives, even for non-minimum phase MIMO systems (Ntogramatzidis et al., 2014).

In stochastic tracking of an Itô semi-martingale target, the objective is to minimize both deviation and intervention costs. Asymptotic lower bounds are derived as intervention costs become small, and the limiting problem is related to time-average control of Brownian motion via occupation measures and an infinite-dimensional linear program. In the scalar quadratic regular-control case, the explicit lower bound is

$J$01

attained by Ornstein–Uhlenbeck feedback $J$02 with $J$03 (Cai et al., 2015).

A thermodynamic analogue appears in the finite-time Otto cycle with a classical harmonic oscillator working substance. There, shortcut protocols valid in the linear response regime can make the excess work exactly zero for switching times satisfying

$J$04

so the cycle duration is reduced to the sum of the thermalization relaxation times. In that setting, the “minimum-tracking” behavior refers to minimizing total cycle time while retaining quasistatic work (Bonança, 2018).

6. Rigorous computation, optimal perturbations, and limits of validity

A rigorous computational approach to linear response formulates the response of invariant densities through

$J$05

for a family of transfer operators $J$06 acting on nested Banach spaces. The computational scheme discretizes $J$07 by a finite-rank operator $J$08, approximates $J$09 by $J$10, and computes

$J$11

with explicit control of tail, discretization, and approximation errors. Reported examples include expanding circle maps under stochastic and deterministic perturbations with $J$12 error bounds, and the doubling map at the boundary of the intermittent family with an $J$13 error bound (Bahsoun et al., 2015).

Recent work on Anosov diffeomorphisms shifts the emphasis from computing response to optimizing it. For a $J$14 Anosov diffeomorphism $J$15 of $J$16 and observation $J$17, the linear-response operator is

$J$18

and the optimization problem is to maximize

$J$19

over unit-norm perturbations in $J$20. For non-constant $J$21, the optimizing perturbation exists and is unique, and its Fourier coefficients are given explicitly. The numerical method uses Fourier discretization, Fejér-kernel smoothing, matrix representations of the transfer operator, FFTs, and automatic differentiation, with convergence to the analytical optimum as the number of modes tends to infinity (Froyland et al., 23 Apr 2025).

These results clarify both the reach and the limits of minimum-tracking linear response. In some regimes, first-order response is robust, uniquely optimizable, and rigorously computable. In others, exact tracking requires high regularity, unstable SRB contributions cannot be omitted, or differentiability of the invariant measure fails altogether. The subject is therefore best understood as a family of tightly related perturbative and control-theoretic problems whose shared concern is not merely response, but the principled selection of perturbations that track, minimize, or optimize that response.