Virtual Thermodynamic Potential

• Virtual thermodynamic potential is a constructed free-energy-like functional that extends equilibrium concepts to nontraditional and nonequilibrium settings.
• It organizes non-equilibrium states by mapping them onto equilibrium-like forms, enabling applications such as Jarzynski relations, Landau phenomenology, and off-shell geometric formulations.
• Its use spans thermophoretic trapping, phase transitions, black-hole thermodynamics, moist-air modeling, and quantum passive states, offering new insights into work, heat, and stability.

“Virtual thermodynamic potential” denotes a family of constructions in which a free-energy-like scalar or functional is introduced outside ordinary equilibrium thermodynamics. In one line of work, it is a virtual effective potential $U_{\rm eff}$ that maps a Soret steady state of a thermophoretically trapped colloid onto an equilibrium-like Boltzmann form, thereby enabling Jarzynski-type relations in a nonisothermal, force-free setting (Thalheim et al., 2020). In other contexts, the term refers to a phenomenological Landau potential built from symmetry and fitted coefficients, an off-shell Euclidean free energy on “virtual geometries” in black-hole thermodynamics, or generalized potentials adapted to nonextensive, moist-air, or quantum-passive-state descriptions (Zhang et al., 2017, Dumitru et al., 24 Jul 2025, Gu et al., 2010, Eldred et al., 2022, Sonkar et al., 8 Jan 2026). The unifying feature is not a single formula but a common role: a constructed potential that organizes admissible states, response, work, or criticality when the microscopic or equilibrium interpretation of an ordinary thermodynamic potential is absent, reduced, or deliberately generalized.

1. Conceptual scope and defining features

The literature uses “virtual thermodynamic potential” for several distinct but structurally related objects. One usage is explicitly thermodynamic and nonequilibrium: a nonisothermal stationary distribution $P(\mathbf r)$ is rewritten in Boltzmann form by introducing an effective potential $U_{\rm eff}$, even though the confinement is produced by thermophoretic drift rather than by a conservative force (Thalheim et al., 2020). A second usage is phenomenological: a Landau-type free-energy density $G(\mathbf M;T,\mathbf H)$ is postulated from symmetry, truncated at finite order, and fitted to reproduce observed transitions, without deriving it from a microscopic Hamiltonian (Zhang et al., 2017). A third usage is off shell: a Euclidean action is evaluated on “virtual geometries,” which preserve horizon and asymptotic structure but need not satisfy Einstein equations, thereby producing a “virtual thermodynamic potential” $\mathcal G$ that behaves as a Gibbs free energy with an extra virtual-work term (Dumitru et al., 24 Jul 2025).

Other papers extend the same pattern. In moist-air modeling, thermodynamic consistency is restored by deriving all relations from potentials in their natural variables; within the consistent constant-$\kappa$ system, virtual potential temperature $\theta_v$ becomes an entropic thermodynamic variable (Eldred et al., 2022). In Tsallis thermodynamics, generalized Helmholtz, Gibbs, and enthalpy potentials are constructed from the physical temperature and pressure rather than from the Lagrange multipliers, preserving the Legendre structure in a modified form (Gu et al., 2010). In quantum thermodynamics, virtual temperatures of adjacent levels act as operational quantities that constrain passive-state transformations, heat flow, and Otto efficiency, suggesting a potential-like role played by a set of effective temperatures rather than by a single equilibrium temperature (Sonkar et al., 8 Jan 2026).

This variety suggests a family resemblance rather than a universal definition. A virtual thermodynamic potential is typically introduced when one wants the organizational power of equilibrium thermodynamics—Boltzmann weights, Legendre structure, response, fluctuation relations, or phase-selection criteria—in a setting that is nonequilibrium, coarse-grained, phenomenological, off shell, or otherwise not governed directly by an ordinary microscopic free energy.

2. Virtual effective potential in Soret equilibria

In the most explicit thermodynamic construction, a colloidal particle undergoing thermophoresis in a static temperature field $T(\mathbf r)=T_0+\Delta T(\mathbf r)$ obeys the stationary balance

$$\mathbf J=-D\nabla P-PD_T\nabla T,\qquad \mathbf J=0,$$

which yields the Soret steady-state distribution

$$\frac{P(\mathbf r)}{P_0}=\exp\!\bigl(-S_T\,\Delta T(\mathbf r)\bigr),\qquad S_T=\frac{D_T}{D}.$$

Because this has exactly the form of a Boltzmann distribution, the virtual effective potential is defined by comparison with

$P(\mathbf r)$0

so that

$P(\mathbf r)$1

For a harmonic temperature rise,

$P(\mathbf r)$2

the resulting virtual potential is harmonic,

$P(\mathbf r)$3

The paper stresses that this $P(\mathbf r)$4 is not a mechanical potential exerting a conservative force; it is a virtual thermodynamic potential constructed so that the non-equilibrium Soret steady state can be treated as if it were an equilibrium Boltzmann state at temperature $P(\mathbf r)$5 (Thalheim et al., 2020).

This construction is then used to extend Jarzynski’s equality to a nonisothermal setting. Work is defined trajectory-wise in the standard stochastic-thermodynamic form, but with the virtual potential,

$P(\mathbf r)$6

and heat is likewise defined as

$P(\mathbf r)$7

For translation of a harmonic virtual trap, the free-energy difference vanishes because the partition function is translation invariant, so the Jarzynski equality reduces to

$P(\mathbf r)$8

Using $P(\mathbf r)$9, the same paper rewrites the relation in terms of an applied temperature change,

$U_{\rm eff}$0

which leads to the thermophoretic Jarzynski equality

$U_{\rm eff}$1

In the reported experiment, $U_{\rm eff}$2, so $U_{\rm eff}$3 (Thalheim et al., 2020).

The same virtual potential also emerges dynamically. In overdamped thermophoretic trapping,

$U_{\rm eff}$4

and with harmonic $U_{\rm eff}$5,

$U_{\rm eff}$6

Comparing this with mechanical trapping drift $U_{\rm eff}$7 and using $U_{\rm eff}$8 gives

$U_{\rm eff}$9

Thus the same $G(\mathbf M;T,\mathbf H)$0 reproduces both the steady-state distribution and the drift term in the Langevin equation. The paper distinguishes “virtual temperature minimum,” “virtual effective potential,” and “equivalent potential” from a real mechanical potential (Thalheim et al., 2020).

3. Phenomenological and effective potentials

A second major usage treats a virtual thermodynamic potential as a phenomenological or reduced functional defined in an order-parameter space. For DyCo$G(\mathbf M;T,\mathbf H)$1, the thermodynamic potential $G(\mathbf M;T,\mathbf H)$2 is constructed as an eighth-order Landau expansion in the spontaneous magnetization components $G(\mathbf M;T,\mathbf H)$3, constrained by the cubic symmetry of the paramagnetic phase and coupled to an external field by $G(\mathbf M;T,\mathbf H)$4. In the tetragonal ferromagnetic phase, the symmetry reduction $G(\mathbf M;T,\mathbf H)$5, $G(\mathbf M;T,\mathbf H)$6 gives

$G(\mathbf M;T,\mathbf H)$7

This potential is explicitly described as phenomenological rather than microscopic: it is guided by symmetry, truncated at eighth order, and fitted to reproduce the ferromagnetic transition temperature, magnetization curve, temperature dependence of magnetization, and metamagnetic behavior (Zhang et al., 2017).

A related many-body usage appears in the strong-coupling Hubbard model. There the grand potential is written as

$G(\mathbf M;T,\mathbf H)$8

and an auxiliary coupling constant $G(\mathbf M;T,\mathbf H)$9 is introduced so that $\mathcal G$0 interpolates between the atomic limit and the full model. The paper constructs a functional

$\mathcal G$1

where $\mathcal G$2 is a sum of vacuum skeleton diagrams built from irreducible Green’s functions and renormalized tunneling lines, and proves the stationarity relations

$\mathcal G$3

together with

$\mathcal G$4

In this setting, the functional behaves as a strong-coupling analogue of a Luttinger–Ward functional: it is a thermodynamic-potential functional of virtual or trial propagators whose stationary point reproduces the physical grand potential (Moskalenko et al., 2010).

NJL and PNJL models provide another effective-potential construction. Integrating the gap equations yields

$\mathcal G$5

with the integration constant fixed by Stefan–Boltzmann asymptotics. In the PNJL extension,

$\mathcal G$6

The paper emphasizes that the potential is defined only up to an integration constant invisible to the gap equations, and that fixing $\mathcal G$7 is essential for correct asymptotics of pressure, constituent masses, and Polyakov-loop behavior (Moreira et al., 2010).

A further effective-potential construction appears in superfluid $\mathcal G$8He in aerogel. Starting from the Serene–Rainer reduction of the Luttinger–Ward functional, the paper derives an impurity-averaged thermodynamic-potential functional for the superfluid order parameter,

$\mathcal G$9

valid at all temperatures and reducing to the Ginzburg–Landau functional near $\kappa$0. Its stationary condition reproduces the impurity-renormalized gap equation, and the same functional is used to compute the thermodynamic potential, entropy, heat capacity, and density of states (Ali et al., 2010).

These examples share a common pattern: the potential is not treated as the exact microscopic free energy of the full system, but as a reduced, phenomenological, or stationary functional that still generates equilibrium conditions, response, and phase structure.

4. Off-shell and geometric formulations

In black-hole thermodynamics, the term “virtual thermodynamic potential” is used explicitly for an off-shell Gibbs-like free energy defined on a family of geometries that need not satisfy Einstein equations. The Euclidean formalism is generalized by considering “virtual geometries,” preserving horizon existence and asymptotic structure while leaving one metric function arbitrary. The resulting potential is

$\kappa$1

where $\kappa$2 is the regularized Euclidean action, $\kappa$3 is the horizon radius, and $\kappa$4 are control parameters (Dumitru et al., 24 Jul 2025).

The decisive feature is that $\kappa$5 is evaluated off shell. Its differential takes the form

$\kappa$6

so that

$\kappa$7

The extra term is interpreted as virtual work. The same paper shows that

$\kappa$8

so extremizing $\kappa$9 with respect to the order parameter $\theta_v$0 enforces the horizon Einstein equation. The conditions

$\theta_v$1

are then used to study criticality and swallow-tail structure, making $\theta_v$2 function as a Landau–Ginzburg potential in the horizon radius (Dumitru et al., 24 Jul 2025).

A formally related idea appears in compactified bosonic strings, where the “thermodynamic potential” is the one-loop free energy, or with chemical potentials the grand-canonical thermodynamic potential, derived from the Euclidean torus path integral,

$\theta_v$3

Chemical potentials are introduced by constant background gauge fields,

$\theta_v$4

and the resulting one-loop potential depends on compactification radius and chemical potentials, displays T-duality at zero density, and serves as a generating functional for entropy, charge densities, and energy (Shiraishi, 2013). This usage is not labeled “virtual” in the title, but it illustrates the same formal move: a free-energy-like functional is constructed on an analytically continued or background-dependent configuration space rather than on ordinary equilibrium macrostates.

The off-shell black-hole and path-integral string usages therefore extend the concept from effective thermodynamic landscapes in state space to functionals defined on families of geometries or backgrounds. In both cases, the potential organizes admissibility, equilibrium conditions, and critical structure even before the full equations of motion or microscopic consistency conditions are imposed.

5. Generalized variables, virtual temperatures, and nonstandard thermodynamic coordinates

Some papers do not introduce a single new scalar potential but instead redefine the thermodynamic coordinates so that a potential-based formulation remains possible. In moist-air thermodynamics, the guiding principle is that thermodynamic consistency requires the use of thermodynamic potentials in their natural variables, from which all thermodynamic quantities and relationships are derived (Eldred et al., 2022). In the consistent constant-$\theta_v$5 approximation, the virtual temperature

$\theta_v$6

and the virtual potential temperature

$\theta_v$7

become especially important. In that approximation, $\theta_v$8 is shown to be a genuine entropic variable. The corresponding internal energy and enthalpy take the forms

$\theta_v$9

$T(\mathbf r)=T_0+\Delta T(\mathbf r)$0

and the derived equations of state,

$T(\mathbf r)=T_0+\Delta T(\mathbf r)$1

are independent of the partition of water among vapor, liquid, and ice for reversible adiabatic dynamics (Eldred et al., 2022). In this sense, virtual thermodynamic potential can refer to a consistent replacement of ordinary temperature-like variables by “virtual” ones that absorb composition dependence into a single thermodynamic coordinate.

In nonextensive thermodynamics, generalized potentials are constructed from the physical temperature

$T(\mathbf r)=T_0+\Delta T(\mathbf r)$2

and physical pressure rather than from the Lagrange multipliers. The generalized Helmholtz and Gibbs potentials are

$T(\mathbf r)=T_0+\Delta T(\mathbf r)$3

$T(\mathbf r)=T_0+\Delta T(\mathbf r)$4

Their Legendre structure is preserved, but the conjugate of $T(\mathbf r)=T_0+\Delta T(\mathbf r)$5 is no longer $T(\mathbf r)=T_0+\Delta T(\mathbf r)$6 itself: $T(\mathbf r)=T_0+\Delta T(\mathbf r)$7 Here the “virtual” aspect lies in replacing the usual entropy term by a nonlinear function that restores thermodynamic consistency under a nonadditive entropy functional (Gu et al., 2010).

A quantum analogue appears in passive-state thermodynamics. For a passive state

$T(\mathbf r)=T_0+\Delta T(\mathbf r)$8

adjacent virtual temperatures are defined by

$T(\mathbf r)=T_0+\Delta T(\mathbf r)$9

The paper proves that the extremal virtual temperatures $$\mathbf J=-D\nabla P-PD_T\nabla T,\qquad \mathbf J=0,$$0 and $$\mathbf J=-D\nabla P-PD_T\nabla T,\qquad \mathbf J=0,$$1 are always realized by adjacent pairs, and defines the mean virtual temperature

$$\mathbf J=-D\nabla P-PD_T\nabla T,\qquad \mathbf J=0,$$2

These quantities control majorization relations between passive states, bound the direction of heat flow with a bath, and provide Otto-efficiency bounds such as

$$\mathbf J=-D\nabla P-PD_T\nabla T,\qquad \mathbf J=0,$$3

Virtual temperatures therefore act as potential-like state variables that constrain work extraction and heat flow even when the state is nonthermal (Sonkar et al., 8 Jan 2026).

A related quantum-thermodynamic development concerns coherent thermal machines with a virtual qubit. There the steady-state currents and entropy production are reproduced by an effective classical Markov process, but current fluctuations acquire an additional purely quantum correction originating from coherence localized in a virtual-qubit subspace (Li et al., 15 Jan 2026). This suggests a further extension: in quantum nonequilibrium settings, virtual temperatures or virtual qubits may supply the intensive, effective variables from which a potential-like description of transport is reconstructed.

6. Terminology, limitations, and common structure

A recurrent source of confusion is the distinction between a real mechanical potential and a virtual thermodynamic one. In thermophoretic trapping, the motion is force-free in the sense that no conservative body force acts on the particle; the virtual potential is introduced only because the stationary distribution and drift can be mapped onto those of an isothermal harmonic trap (Thalheim et al., 2020). In Landau theory, the potential is real as a coarse-grained free-energy density but phenomenological in origin, because its coefficients are fitted and fluctuations are neglected (Zhang et al., 2017). In black-hole thermodynamics, the potential is explicitly off shell: it is defined on configurations that do not satisfy the equations of motion until its first derivative is set to zero (Dumitru et al., 24 Jul 2025).

These constructions also have explicit domains of validity. The virtual effective potential for Soret equilibria assumes overdamped Langevin dynamics, constant $$\mathbf J=-D\nabla P-PD_T\nabla T,\qquad \mathbf J=0,$$4 and $$\mathbf J=-D\nabla P-PD_T\nabla T,\qquad \mathbf J=0,$$5, small temperature variations, and an exponential stationary distribution of Boltzmann type; large gradients or multiplicative noise can invalidate the mapping (Thalheim et al., 2020). Landau-type phenomenological potentials may fail near critical regions where fluctuations or additional order parameters become important (Zhang et al., 2017). The moist-air constant-$$\mathbf J=-D\nabla P-PD_T\nabla T,\qquad \mathbf J=0,$$6 system achieves consistency only if the approximation is applied at the level of the potentials themselves; treating $$\mathbf J=-D\nabla P-PD_T\nabla T,\qquad \mathbf J=0,$$7 as an entropic variable without consistently modifying the potentials breaks energy and entropy conservation (Eldred et al., 2022). In the black-hole setting, the construction depends on an integrability condition that allows one metric function to remain arbitrary during the Euclidean off-shell evaluation (Dumitru et al., 24 Jul 2025).

Despite these differences, the usages share a stable formal core. A virtual thermodynamic potential is introduced when one needs a scalar function or functional that preserves some part of thermodynamic reasoning under generalized circumstances. It may map a nonequilibrium steady state to an equilibrium-like Boltzmann form, provide a phenomenological Gibbs landscape in an order-parameter space, represent an off-shell Euclidean free energy, or encode admissibility through virtual temperatures. In each case, the potential is useful not because it is a universal object, but because it supplies a thermodynamic ordering principle—through minimization, stationarity, monotonicity, or fluctuation relations—in a setting where the ordinary microscopic free energy is unavailable, incomplete, or intentionally replaced.

That common role clarifies the broader significance of the topic. Virtual thermodynamic potentials unify disparate problems by allowing work, heat, free-energy difference, stability, or criticality to be formulated in equilibrium-like terms while retaining the actual nonmechanical, off-shell, or coarse-grained origin of the dynamics. A plausible implication is that the concept is best understood not as a single formal definition, but as a recurrent methodological device for extending thermodynamic structure beyond standard equilibrium domains.