Density-to-Taylor Mapping: Techniques & Applications

• Density-to-Taylor Mapping is a set of techniques that convert density expansions into Taylor series, underpinning diverse applications in physics and numerical modeling.
• It employs power-series expansions, discrete transforms, and moment formulas to extract Taylor coefficients, critical for understanding convergence and singular behavior.
• The approach is vital in TDDFT, lattice QCD, and fluid dynamics by linking density data to potential, transport properties, and algorithmic differentiation methods.

The density-to-Taylor mapping refers to the set of mathematical and algorithmic techniques by which information encoded in a density parameter or fugacity expansion is related to, or converted into, a Taylor expansion in some external variable, typically a chemical potential, time, or spatial variable. This mapping is fundamental in many areas of mathematical physics, including time-dependent density functional theory (TDDFT), finite-density lattice field theory, variable-density fluid mechanics, and numerical conformal mapping. The mathematical structure, analytic conditions, and practical efficacy of the mapping depend sensitively on the regularity of the underlying equations, the presence of singularities, and the combinatorial structure of the expansion coefficients.

1. Formal Structure of the Density-to-Taylor Mapping

The essential mechanism of density-to-Taylor mapping is the power-series expansion of a generating function, partition function, or time-evolved observable around some base point (typically $\mu = 0$ or $t = 0$). Suppose the dependence on the relevant variable (e.g., chemical potential $\mu$ or time $t$) is encoded through a "density" expansion: $$f(\mu) = \sum_{k \in \mathbb{Z}} b_k\,e^{k \mu / T}$$ or, in the context of conformal mapping or boundary potentials, via an integral over a density function $\sigma$: $$f(z) = \int_\Gamma K(z, \tau) \sigma(\tau) ds(\tau)$$ The Taylor expansion is then obtained through termwise expansion of the exponential, binomial and combinatorial transforms, or by repeated time or parameter derivatives: $$f(\mu) = \sum_{n=0}^\infty a_n\,\mu^n,\quad a_n = \frac{1}{n!}\,\frac{d^n f}{d\mu^n}\bigg|_{\mu=0}$$ The connection between coefficients is often given by explicit moment formulas, e.g.,

$a_n = \frac{1}{n!}\frac{1}{T^n}\sum_k b_k\,k^n$

and, conversely, via discrete Fourier transforms or combinatorial inversions (e.g., Stirling transforms) which reconstruct the density coefficients from the Taylor data (Wilfling et al., 2013, Grünwald et al., 2013, Grünwald et al., 2014).

2. Density-to-Taylor Mapping in Time-Dependent Density Functional Theory

In TDDFT, the density-to-Taylor mapping is foundational to the Runge–Gross uniqueness theorem, which asserts that, under sufficient regularity, the time-dependent one-particle density $\rho(t,x)$ associated with a many-body wavefunction uniquely determines (up to a time-dependent gauge) the external potential. The mathematical thread is:

• The many-body Schrödinger equation generates $\rho(t,x)$.
• If $\rho(t,x)$ is $C^\infty$ in $t$, its Taylor expansion about $t=0$ is well defined:

$$\rho(t,x) = \sum_{k=0}^\infty \frac{1}{k!} \partial_t^k \rho(0,x) t^k$$

• For systems with smooth or bounded interactions, one can recursively relate the Taylor coefficients of $\rho$ to those of $V(t,x)$, enabling the inversion from the density to the potential (Fournais et al., 2016).

However, in the presence of Coulomb or $\delta$-function singularities, the uniform domain condition on Hamiltonian powers fails at high order. For $N \geq 2$ with Coulomb repulsion, the Taylor expansion for $\rho(t,x)$ exists only to finite order $k_{\max}$ (e.g., $k_{\max}=3$ or 4 depending on symmetry), unless the external $V$ is constant. No infinite Taylor expansion is possible: $$D(H_0^k) = D(H_V^k) \Longleftrightarrow k \leq k_{\max}$$ This obstruction implies the failure of Taylor-based inversion for realistic Coulombic systems, necessitating alternative, non-perturbative approaches (Fournais et al., 2016).

3. Mapping Fugacity/Density to Taylor Expansion in Lattice and Effective Theories

In lattice QCD and effective models, the partition function at finite density or chemical potential is conventionally represented in fugacity (density) form as: $Z(\mu) = \sum_{q} b_q e^{\mu q}$ The mapping to the Taylor expansion proceeds by expanding $e^{\mu q}$: $$e^{q\mu} = \sum_{n=0}^\infty \frac{q^n}{n!} \mu^n, \quad Z(\mu) = \sum_{n=0}^\infty \left[ \frac{1}{n!}\sum_q b_q q^n\right]\mu^n$$ Thus, the $n$th Taylor coefficient is the $n$th moment of the fugacity weights: $a_n = \frac{1}{n!}\sum_q b_q q^n$ Higher-order coefficients directly encode the density fluctuations across canonical sectors. Inversion from Taylor to fugacity coefficients is given by

$$b_k = \frac{1}{2\pi}\int_{-\pi}^\pi d\theta\, e^{-ik\theta} Z(i\theta)$$

ensuring bijective correspondence between the two expansions. Stirling or binomial transforms provide alternate combinatorial mappings (Wilfling et al., 2013, Grünwald et al., 2014, Grünwald et al., 2013).

Such mappings underlie the formulation and truncation analysis of QCD observables at finite density, including the baryon number, pressure, and susceptibilities (Nagata et al., 2012, Huovinen et al., 2014, Karsch et al., 2011). The convergence radius of the Taylor expansion, estimated by ratios of successive coefficients,

$$r(T) = \lim_{n \to \infty} \sqrt{|c_{2n}/c_{2n+2}|}$$

sets the domain of validity for calculations at finite chemical potential (Karsch et al., 2011).

4. Modified Taylor Expansions, Convergence, and Algorithmic Considerations

Modified Taylor expansions (MTEs), notably expansions in $e^{\mu}-1$ or $e^{\pm\mu/T}-1$, reorganize the Taylor series to partially resum large-fugacity behavior, moving singularities farther from the origin and generally increasing convergence radius over standard Taylor expansions. This is particularly effective in models where the underlying physical structure is encoded in fugacity-like factors. The coefficients are obtained by collecting moments and binomial coefficients: $$P(\mu) = \sum_{m=0}^\infty \tilde{a}_m \rho^m,\quad \tilde{a}_m = \sum_{k \geq m} b_k \binom{k}{m}$$ This technique extends the usable range in $\mu$ by $\approx 50\%$ or higher relative to regular Taylor expansions for the same computational effort (Wilfling et al., 2013, Grünwald et al., 2014, Grünwald et al., 2013). Fugacity expansions themselves are the most robust at high density but carry a substantially greater computational burden because all $b_q$ must be computed or Fourier-projected in the canonical ensemble.

Algorithmic differentiation enables the direct calculation of high-order Taylor coefficients (up to order 24 or higher) by automatically propagating derivatives through implicit density minimization and field dependencies, as in mean-field QCD models (Karsch et al., 2011). For stochastic lattice applications, spectral moment methods and the Cauchy residue theorem can be used to extract Taylor coefficients as contour integrals over random-matrix traces of shifted Dirac operators, thus overcoming explicit differentiation or explicit moment calculation bottlenecks (Forcrand et al., 2018).

5. Density-to-Taylor Mapping in Variable-Density and Flow Problems

In computational fluid dynamics, especially for variable-density flows, Taylor time-stepping methods rely fundamentally on the mapping of density time-derivatives onto Taylor coefficients. The density and velocity fields are recursively advanced by computing time derivatives through the continuity and momentum equations: $$\rho^{n+1} = \rho^n + \tau \rho_1^n + \frac{\tau^2}{2} \rho_2^n + \cdots$$ with, for example,

$$\rho_1^n = -u_0^n \cdot \nabla \rho_0^n, \quad \rho_2^n = - (u_0^n \cdot \nabla \rho_1^n + u_1^n \cdot \nabla \rho_0^n)$$

The recursive structure allows for high-order accurate time advancement provided that sufficient regularity is present. When solution regularity is lost, as in Rayleigh–Taylor flow, higher time-derivatives become unbounded and the Taylor mapping ceases to be valid (Lundgren et al., 2022).

In the analysis of Taylor dispersion in pulsatile variable-density flows, the density-to-Taylor mapping emerges via multiple-scale asymptotics, where the cross-sectional mean scalar evolves according to a Taylor–Aris effective one-dimensional diffusion equation, with the density modulating the Taylor dispersion correction: $$D_{\mathrm{eff}} = D_0 + \frac{Pe^2\,\rho_0^2}{48\,D_0} + \text{pulsatile correction}$$ This demonstrates a direct mapping from the spatial and temporal evolution of local density to the Taylor-corrected transport coefficient (Rajamanickam et al., 25 Mar 2025).

6. Density-to-Taylor Mapping in Potential Theory and Numerical Conformal Mapping

In potential theory, the problem of extracting Taylor (or Laurent) coefficients of analytic maps, such as conformal maps, from boundary layer densities provides a geometrically rich example. The interior Riemann map $f(z) = \sum_n a_n z^n$ can be reconstructed from a density $\sigma$ solving a second-kind integral equation associated with the double-layer operator: $$f(z) = -\frac{1}{2\pi i} \int_\Gamma \frac{\sigma(\tau)}{\tau - z} d\tau$$ yielding Taylor coefficients directly: $$a_n = -\frac{1}{2\pi i} \int_\Gamma \frac{\sigma(\tau)}{\tau^{n+1}} d\tau$$ Given a numerically computed density on $\Gamma$, this mapping allows for accurate spectral recovery of Taylor coefficients, extending to cornered domains and exterior maps with appropriate changes of variables and integral operators (Wala et al., 2016).

---

The structure and limitations of the density-to-Taylor mapping are context dependent. In systems with adequate analyticity and regularity, the mapping provides a direct route to extract Taylor-series data (response, cumulants, expansion coefficients) from density-encoded or canonical-ensemble information, enabling efficient numerical and analytic computations. When singularities or loss of regularity are present, as in the Coulomb-interacting many-body problem, the mapping breaks down at finite order, and alternative techniques must be used (Fournais et al., 2016). Modified expansions and algorithmic advances continue to enlarge the practical and theoretical reach of these mappings across physics and applied mathematics.