Local Chemical Potentials Explained

• Local chemical potentials are spatially dependent measures that quantify the driving force for particle, energy, or spin exchange, bridging global equilibrium with local reactivity.
• They are determined via advanced theoretical and computational methods—including density functional theory, thermodynamic integration, and Hamilton–Jacobi analysis—that overcome challenges in traditional definitions.
• Operational approaches using probe measurements and ensemble averaging in non-equilibrium systems offer practical means to map local variations in phase stability and transport phenomena.

Local chemical potentials quantify the spatially resolved driving force for particle, energy, or spin exchange in diverse physical, chemical, and materials systems. Formally, the chemical potential $\mu$ is defined as the Lagrange multiplier conjugate to a globally conserved quantity—most commonly, particle number or charge—and ensures thermodynamic equilibrium via its uniformity in the absence of external gradients. Extending this notion to "local" chemical potentials allows spatial mapping of reactivity, selective synthesis, transport, and dissipation, but presents formidable challenges both in fundamental definition and in practical computation. Research across density functional theory (DFT), nonequilibrium statistical mechanics, electronic systems, quantum chromodynamics (QCD), and materials synthesis has clarified the mathematical structure, limits, and utility of local chemical potentials in a variety of contexts.

1. Definition and Formal Structure

In a general thermodynamic system, the global chemical potential is given by $$\mu_i = \partial G / \partial n_i |_{T,P,n_{j \neq i}}$$, where $G$ is the Gibbs free energy and $n_i$ is the stoichiometric amount of species $i$. In a spatially extended, multicomponent system, this motivates interpreting $\boldsymbol{\mu} = (\mu_1, ..., \mu_n)$ as a vector in an $n$-dimensional chemical-potential space. A "local chemical potential" is an attempt to assign a spatially dependent value $\mu_i(\mathbf{r})$ or a site-resolved $\mu_i$, reflecting the incremental change in free energy due to the hypothetical addition of a particle at location $\mathbf{r}$ or site $i$ (Todd et al., 2021, Wang, 2011, Du et al., 2016).

However, in DFT and similar frameworks, attempts to define a rigorous local chemical potential via a functional derivative of the energy with respect to the density lead to $$\mu_{\text{loc}}(\mathbf{r}) = \delta E[n]/\delta n(\mathbf{r})|_v$$, with the crucial constraint that the external potential $v(\mathbf{r})$ is held fixed. Chain-rule and variational arguments rigorously demonstrate that for ground-state electron densities, this local chemical potential is strictly constant in space and equals the global $\mu$ (Gal, 2011):

$$\mu_{\text{loc}}(\mathbf{r}) \equiv \mu, \quad \forall\,\mathbf{r}.$$

Therefore, in traditional DFT frameworks, spatial variations of $\mu_{\text{loc}}(\mathbf{r})$ are excluded by construction, and any attempt to relax the external-potential constraint leads to mathematically ill-defined (typically diverging or singular) functionals (Gal, 2011).

2. Local Chemical Potentials in Materials and Phase-Space Design

In multicomponent materials thermodynamics, each phase or compound defines a hyperplane in chemical-potential space via

$$\sum_{i=1}^n x_i (\mu_i-\mu_i^0) = \frac{1}{N} \Delta G_f^0(T),$$

where $x_i$ is the stoichiometric fraction of element $i$ and $\Delta G_f^0$ is the formation free energy per formula unit (Todd et al., 2021). The intersection and convex hull of these hyperplanes partition chemical-potential space into stability polytopes, and local chemical potentials are the coordinates within this space that determine local phase stability.

The distance between two stability polytopes in this space, computed via minimal Euclidean separation between their polytope vertices,

$$\Delta\mu_{\min}(P_a, P_b) = \min_{i,j}\|\boldsymbol{\mu}_{a_i} - \boldsymbol{\mu}_{b_j}\|_2,$$

quantifies the local chemical-potential mismatch at phase boundaries, and thus serves as a metric for selectivity in reaction pathways and materials synthesis. Computational workflows construct phase fields and identify selective synthetic routes by minimizing $\Delta\mu_{\min}$, as demonstrated in selective pyrochlore synthesis in high-dimensional chemical-potential landscapes (Todd et al., 2021).

3. Measurement and Calculation in Non-Equilibrium and Mesoscopic Systems

Local chemical potentials can be probed in systems out of equilibrium, such as mesoscopic transport devices or stochastic mass-transport models. In a tight-binding chain connected to electronic reservoirs at different chemical potentials, the local chemical potential $\mu_i$ at site $i$ is operationally defined by attaching a weakly coupled probe (thermometer), calculating its steady-state occupation $n_d(\varepsilon_d)$, and extracting $\mu_i$ by fitting to a local Fermi distribution:

$$n_d = \frac{1}{\exp[\beta_i(\varepsilon_d - \mu_i)] + 1}$$

(Wang, 2011). In configurations with disorder, large fluctuations in local $\mu_i$ are observed, but ensemble averaging restores a smooth spatial gradient analogous to the classical potential drop along a resistor.

In stochastic mass transport, flux-balance considerations are central. On homogeneous rings (or networks without branching), a unique global chemical potential can be probed locally and is spatially uniform. In contrast, in branching geometries, locally measured chemical potentials $\tilde{\lambda}_\nu$ (inferred by probe or fluctuation methods) can differ across subsystems, reflecting nontrivial flux partitioning despite an underlying unique global $\mu$ defined by the partition function

$$Z(M) = \int\prod_{i=1}^N [dm_i f_i(m_i)]\, \delta\bigl(\sum_i m_i - M\bigr).$$

Branching points induce nonperturbative, nonlocal contributions to subsystem partition functions, so that flux-balance, not just density, determines local chemical potentials (Martens et al., 2011).

4. Approaches to Genuine Spatially Resolved Chemical Potentials

To circumvent the limitations of traditional functional-derivative definitions, alternative frameworks construct local chemical potentials as property densities emerging from local energy densities. For electronic systems, this is accomplished via a Hamilton–Jacobi analysis of the many-body Schrödinger equation, yielding a unique, physically motivated local energy density

$$e(\mathbf{r}) = t(\mathbf{r}) + v_{ee}(\mathbf{r}) + v_{ne}(\mathbf{r}) = \frac{E}{N} n(\mathbf{r}),$$

where $t$, $v_{ee}$, and $v_{ne}$ are the kinetic, electron–electron, and electron–nuclear energy contributions, respectively (Gal, 2011). The local chemical potential is then defined as

$$\mu(\mathbf{r}) = \left(\frac{\partial e(\mathbf{r})}{\partial N}\right)_{v},$$

which yields, after differentiation,

$$\mu(\mathbf{r}) = f(\mathbf{r}) + \frac{n(\mathbf{r})}{N} \left(\mu - \frac{E}{N}\right),$$

where $f(\mathbf{r})$ is the Fukui function. The negative of this local chemical potential provides a spatially resolved electronegativity, and this approach naturally extends to define local hardness as the second derivative with respect to $N$, thereby generating nontrivial, spatially varying maps of reactivity that integrate to their global analogues (Gal, 2011). These property densities avoid the pathologies associated with ill-defined functional derivatives in traditional DFT.

5. Local Chemical Potentials in Quantum Fluids and QCD

In relativistic quantum fluids, such as the QCD matter produced in heavy-ion collisions, local chemical potentials of conserved charges—baryon number ($\mu_B$), electric charge ($\mu_Q$), and strangeness ($\mu_S$)—are introduced as Lagrange multipliers in the grand-canonical partition function and enter the hydrodynamic equations as conjugate variables to locally conserved currents. The NEOS-4D framework blends lattice QCD and hadron resonance gas equations of state to produce lookup tables in pseudo-variables that enable efficient inversion from local energy and density to $(T, \mu_B, \mu_Q, \mu_S)$ in each space-time cell (Monnai et al., 17 Jun 2024). Local chemical potentials determined in this way serve as essential inputs for multi-conserved-charge evolution, charge diffusion, and flavor-specific observables in hydrodynamic simulations of nuclear matter.

6. Applications: Fluids, Spin Systems, and Selective Synthesis

Spatially resolved chemical potentials enable critical advancements in simulation, control, and design. In atomistic simulations of dense fluids, position-dependent excess chemical potentials $\mu_{\mathrm{ex}}(\mathbf{r})$ are computed using spatially resolved thermodynamic integration schemes (SPARTIAN), which interpolate between atomistic and ideal gas representations across simulation domains (Heidari et al., 2018). The compensation potential $\Delta H(\lambda(R))$ required to maintain uniform density directly yields the local excess chemical potential, circumventing the insertion difficulties of traditional methods at high density.

In spintronics, the magnon (spin) chemical potential $\mu_m(\mathbf{r})$ quantifies the nonequilibrium occupancy of magnon modes and serves as the hydrodynamic variable for spin transport in magnetic insulators. Nanoscale measurement is achieved via NV center relaxometry, translating local magnon occupation $n(\omega_k, \mu_m)$ into changes in NV spin relaxation rates. Local control and mapping of $\mu_m$ are enabled by driving ferromagnetic resonance or injecting spin current, enabling the experimental determination of local thermomagnonic torques and the spatial imaging of spin chemical potential landscapes (Du et al., 2016).

For phase-selective synthesis, the precise evaluation of local chemical-potential mismatches at all possible reaction interfaces in hyperdimensional compositional spaces (e.g., Y–Mn–O–A–Cl) allows for the identification of "thermodynamic wormholes" through which selective transformations to metastable phases can proceed, as demonstrated for Na-based metathesis to Y$_2$Mn$_2$O$_7$ (Todd et al., 2021).

7. Implications, Limitations, and Outlook

The definition and measurement of local chemical potentials are foundational for understanding transport, reactivity, and selectivity in extended, inhomogeneous, or nonequilibrium systems. However, in frameworks where the chemical potential is inherently a global, spatially uniform quantity (e.g., ground-state DFT with fixed external potential), attempts to extract spatially resolved values via functional differentiation lead either to trivial (constant) results or intractable singularities associated with asymptotic nonlocality of the energy functional kernel (Gal, 2011). Methodologies that focus on local energy densities or property densities, operational probe-based measurement, or spatially resolved thermodynamic cycles provide legitimate routes to constructing nontrivial, meaningful local chemical potential landscapes.

Ongoing developments seek to integrate explicit defect energetics, interface models, and kinetic effects, further bridging the gap between idealized thermodynamics and the complex, spatially heterogeneous realities of experimental systems. These advances are critical across fields ranging from catalysis and solid-state synthesis to spintronics and the hydrodynamics of high-energy nuclear matter.