Theory of Cross Phenomena (TCP)

• TCP is a thermodynamic theory defined by each independent molar flux being driven by its own conjugate potential gradient, with cross phenomena emerging from variable interdependence.
• The framework redefines transport processes by employing a diagonal kinetic coefficient matrix through an appropriate choice of force basis in both original and revised formulations.
• TCP explains canonical effects like thermoelectricity and thermodiffusion by deriving response coefficients from thermodynamic derivatives, thereby preserving Onsager reciprocity.

Theory of Cross Phenomena (TCP) is a thermodynamic framework in which each independent molar flux is governed by the gradient of its own conjugate potential, while observable cross effects are attributed to the thermodynamic interdependence of those potentials rather than to independent off-diagonal kinetic coefficients. In Liu’s 2022 formulation, the coefficient matrix of linear response is intrinsically diagonal when the driving forces are chosen as the gradients of the conjugate potentials appearing in the combined law of thermodynamics with internal processes; in the 2025 revision, the flux equations are rederived from the first law of thermodynamics and the definitions of the total entropy change and the total work change of an internal process, but the conclusions with respect to Onsager’s theorem remain similar (Liu, 2022, Liu, 14 Jul 2025).

1. Definition and conceptual basis

In TCP, a cross phenomenon is defined as a situation in which the flux of one molar quantity responds to a non-conjugate thermodynamic force. Canonical examples given in the literature include thermoelectricity, where electrical current is driven by a temperature gradient, and thermodiffusion, where mass flux is driven by $\nabla T$ (Liu, 2022). The classical Onsager formulation represents such behavior through linear phenomenological laws of the form

$J_i=-\sum_j L_{ij}\,\nabla Y_j,$

with reciprocal relations $L_{ij}=L_{ji}$ near equilibrium. TCP retains the linear-response setting but argues that the matrix becomes diagonal when the force basis is chosen correctly.

The original formulation defines the conjugate potentials from the internal energy $U(S,V,N_i)$ as

$$T=\frac{\partial U}{\partial S}\Big|_{V,N},\qquad -P=\frac{\partial U}{\partial V}\Big|_{S,N},\qquad U_i=\frac{\partial U}{\partial N_i}\Big|_{S,V,N_{j\neq i}}.$$

Here $T$ is temperature, $P$ hydrostatic pressure, and $U_i$ the partial internal energy of species $i$ (Liu, 14 Jul 2025). In that setting, the independent molar flux is postulated as

$J_i=-\,L_i\,\nabla\phi_i,$

with $J_i=-\sum_j L_{ij}\,\nabla Y_j,$0, and no cross-terms between different species or between heat and mass flux are allowed at the level of independent processes. The foundational claim is that, in the basis of truly independent potentials, the kinetic coefficient matrix is diagonal, and that cross phenomena arise because the potentials themselves depend on other state variables.

2. Original thermodynamic construction

The 2022 formulation derives TCP from the combined first and second law of thermodynamics including internal processes. The entropy balance is written as

$J_i=-\sum_j L_{ij}\,\nabla Y_j,$1

and the internal-energy balance as

$J_i=-\sum_j L_{ij}\,\nabla Y_j,$2

The irreversible contribution is expressed through internal variables $J_i=-\sum_j L_{ij}\,\nabla Y_j,$3: $J_i=-\sum_j L_{ij}\,\nabla Y_j,$4 This introduces a process-by-process pairing of generalized displacements and conjugate forces (Liu, 2022).

TCP then considers a small slab of area $J_i=-\sum_j L_{ij}\,\nabla Y_j,$5 and thickness $J_i=-\sum_j L_{ij}\,\nabla Y_j,$6 and defines the thermodynamic driving force and flux for process $J_i=-\sum_j L_{ij}\,\nabla Y_j,$7 as

$J_i=-\sum_j L_{ij}\,\nabla Y_j,$8

Under the linear-proportionality approximation, the fundamental kinetic law becomes

$J_i=-\sum_j L_{ij}\,\nabla Y_j,$9

with $L_{ij}=L_{ji}$0 the kinetic coefficient of the $L_{ij}=L_{ji}$1-th process. If one writes the most general linear response as

$L_{ij}=L_{ji}$2

then choosing the driving forces to be the gradients of the conjugate potentials of the corresponding processes gives

$L_{ij}=L_{ji}$3

The matrix $L_{ij}=L_{ji}$4 is therefore intrinsically diagonal in this basis. The 2022 presentation also gives an orthogonality argument: since a real symmetric Onsager matrix can be diagonalized by an orthogonal transformation, TCP identifies the chosen force basis with the principal axes of that matrix.

3. Revised formulation from the first law

The 2025 revision begins explicitly from the first law for a closed system exchanging heat, work, and moles,

$L_{ij}=L_{ji}$5

where

$L_{ij}=L_{ji}$6

is the partial internal energy of component $L_{ij}=L_{ji}$7 (Liu, 14 Jul 2025). The entropy balance is written as

$L_{ij}=L_{ji}$8

with

$L_{ij}=L_{ji}$9

the partial entropy of species $U(S,V,N_i)$0.

Combining the first and second laws and including hydrostatic work gives

$U(S,V,N_i)$1

This identifies the chemical potential as

$U(S,V,N_i)$2

The revised flux law is then constructed by considering unidirectional transport of species $U(S,V,N_i)$3 over area $U(S,V,N_i)$4 and distance $U(S,V,N_i)$5 in time $U(S,V,N_i)$6: $U(S,V,N_i)$7 TCP postulates

$U(S,V,N_i)$8

in exact analogy with Fourier’s law $U(S,V,N_i)$9.

Using Eq. (2.4), the gradient of partial internal energy is re-expressed as

$$T=\frac{\partial U}{\partial S}\Big|_{V,N},\qquad -P=\frac{\partial U}{\partial V}\Big|_{S,N},\qquad U_i=\frac{\partial U}{\partial N_i}\Big|_{S,V,N_{j\neq i}}.$$0

Substitution yields the revised flux equation

$$T=\frac{\partial U}{\partial S}\Big|_{V,N},\qquad -P=\frac{\partial U}{\partial V}\Big|_{S,N},\qquad U_i=\frac{\partial U}{\partial N_i}\Big|_{S,V,N_{j\neq i}}.$$1

Under isothermal and isobaric conditions this reduces to

$$T=\frac{\partial U}{\partial S}\Big|_{V,N},\qquad -P=\frac{\partial U}{\partial V}\Big|_{S,N},\qquad U_i=\frac{\partial U}{\partial N_i}\Big|_{S,V,N_{j\neq i}}.$$2

which the paper states is identical to Ågren’s diagonal formulation for multicomponent diffusion. The same revision gives the entropy flux in the presence of heat and mass transport as

$$T=\frac{\partial U}{\partial S}\Big|_{V,N},\qquad -P=\frac{\partial U}{\partial V}\Big|_{S,N},\qquad U_i=\frac{\partial U}{\partial N_i}\Big|_{S,V,N_{j\neq i}}.$$3

and states that this automatically satisfies the second law $$T=\frac{\partial U}{\partial S}\Big|_{V,N},\qquad -P=\frac{\partial U}{\partial V}\Big|_{S,N},\qquad U_i=\frac{\partial U}{\partial N_i}\Big|_{S,V,N_{j\neq i}}.$$4.

4. Relation to Onsager’s theorem and reciprocity

The central controversy surrounding TCP concerns the status of Onsager’s theorem. In Onsager’s general linear formalism for $$T=\frac{\partial U}{\partial S}\Big|_{V,N},\qquad -P=\frac{\partial U}{\partial V}\Big|_{S,N},\qquad U_i=\frac{\partial U}{\partial N_i}\Big|_{S,V,N_{j\neq i}}.$$5 fluxes and forces,

$$T=\frac{\partial U}{\partial S}\Big|_{V,N},\qquad -P=\frac{\partial U}{\partial V}\Big|_{S,N},\qquad U_i=\frac{\partial U}{\partial N_i}\Big|_{S,V,N_{j\neq i}}.$$6

the coefficient matrix is full and symmetric, with $$T=\frac{\partial U}{\partial S}\Big|_{V,N},\qquad -P=\frac{\partial U}{\partial V}\Big|_{S,N},\qquad U_i=\frac{\partial U}{\partial N_i}\Big|_{S,V,N_{j\neq i}}.$$7 independent coefficients. TCP disputes the necessity of treating those off-diagonal entries as fundamental, while not discarding reciprocity itself (Liu, 14 Jul 2025).

The comparison among Onsager, original TCP, and revised TCP can be summarized as follows.

Framework	Force basis	Independent coefficients
Onsager	$$T=\frac{\partial U}{\partial S}\Big|_{V,N},\qquad -P=\frac{\partial U}{\partial V}\Big|_{S,N},\qquad U_i=\frac{\partial U}{\partial N_i}\Big|_{S,V,N_{j\neq i}}.$$8	$$T=\frac{\partial U}{\partial S}\Big|_{V,N},\qquad -P=\frac{\partial U}{\partial V}\Big|_{S,N},\qquad U_i=\frac{\partial U}{\partial N_i}\Big|_{S,V,N_{j\neq i}}.$$9
Original TCP	$T$0	$T$1
Revised TCP	diagonal in $T$2; effective form in $T$3	$T$4

In the original TCP, the kinetic matrix is strictly diagonal in the conjugate-potential basis: $T$5 In the revised TCP, coupling reappears when the variables are changed from $T$6 to $T$7, but the number of independent coefficients does not increase. Writing

$T$8

the effective matrix becomes

$T$9

with

$P$0

All off-diagonal entries in the first row and column are fixed by the partial entropies $P$1, not by new kinetic coefficients. The matrix is symmetric, $P$2, and still has exactly $P$3 independent parameters $P$4 (Liu, 14 Jul 2025). Accordingly, the revised TCP presents reciprocity as recovered in an effective basis, but with cross-coefficients computed from thermodynamic quantities rather than introduced as independent phenomenological constants.

5. Canonical cross phenomena and materials examples

TCP explains cross phenomena by expanding the conjugate potential gradients in terms of other independent gradients. In the 2022 presentation, thermoelectric transport is written as

$P$5

The Seebeck coefficient is then defined by

$P$6

so that under open-circuit conditions $P$7 adjusts until $P$8, giving $P$9. The Peltier coefficient is said to follow by Onsager reciprocity or by direct calculation of the heat carried by $U_i$0 (Liu, 2022).

For thermodiffusion, the atomic flux of species $U_i$1 is written as

$U_i$2

and under steady state $U_i$3 one defines the Soret coefficient through

$U_i$4

In the revised 2025 presentation, this same logic is summarized by stating that under open-circuit conditions the Seebeck or Soret coefficient

$U_i$5

emerges directly from thermodynamics, not as an independent kinetic parameter (Liu, 14 Jul 2025).

Several materials examples are used to illustrate the framework. Uphill diffusion is described as a situation in which a species can migrate up its concentration gradient if its chemical potential is strongly coupled to another species. The revised paper cites uphill diffusion of carbon in steels, where a high Si concentration raises the carbon chemical potential so that $U_i$6 drives $U_i$7 up its concentration gradient, and it also cites the Kirkendall effect, where marker motion and pore formation are captured by atomic mobility databases built on $U_i$8 and Gibbs energies. The same paper further states that, although the full Maxwell–Stefan formalism uses many diffusion coefficients, TCP shows that only $U_i$9 mobilities are truly independent and that the rest follow from thermodynamic factors (Liu, 14 Jul 2025).

The 2022 paper extends the same interpretive scheme to electromigration and electrocaloric effects,

$i$0

and

$i$1

as well as to converse and conventional piezoelectric, piezomagnetic, magnetoelectric, and flexoelectric effects, for which the coefficient expressions are said to reduce to appropriate second derivatives of a free-energy function (Liu, 2022).

6. Scope, interpretation, and acronym disambiguation

Within Liu’s usage, TCP denotes a general thermodynamic theory of cross-coupled transport and response. Its characteristic claim is not that cross effects are absent, but that they are carried by the dependence of each conjugate potential on the other independent variables, mathematically encoded in derivatives such as

$i$2

rather than by independent off-diagonal kinetic coefficients (Liu, 2022). A plausible implication is that TCP should be read as a reorganization of nonequilibrium thermodynamics around a particular choice of force basis, rather than as a denial of reciprocal transport.

The acronym is not unique across arXiv. In condensed-matter transport, closely related language appears in the theory of “transport phenomena cross-correlated among the heat and electric currents of magnons and Dirac electrons on the surface of ferromagnetic topological insulators,” where Onsager reciprocity, drag Nernst and Seebeck effects, Ettingshausen and Peltier responses, and thermal Hall conductivities are analyzed for coupled electron-magnon systems (Imai et al., 2018). In a different part of the literature, TCP is used to denote “topological charge pumping” in the non-Hermitian Rice–Mele model, where finite-size generalized Brillouin zone methods, non-Bloch topological invariants, skin effects, exceptional points, and anomalous pumping phases are studied (Kumar et al., 2021). These usages are terminologically adjacent but conceptually distinct from Liu’s thermodynamic Theory of Cross Phenomena.

Taken in its own intended sense, TCP proposes a unified framework in which the fundamental transport parameters are the diagonal kinetic coefficients and the measurable cross responses are derived from thermodynamic interdependence. The 2025 revision preserves that structure while grounding the flux equations more directly in the first law of thermodynamics and showing that the resulting effective response matrix remains symmetric without introducing extra, unconstrained cross-coefficients (Liu, 14 Jul 2025).