Here is the connection between your framework and non-equilibrium thermodynamics, presented in paragraphs only with full equations displayed on their own lines.
Non-equilibrium thermodynamics extends classical thermodynamics to systems not in local or global equilibrium. The fundamental difference from equilibrium thermodynamics is the presence of nonzero fluxes driven by thermodynamic forces. The entropy production rate \sigma becomes the central quantity, and for any system the second law requires:
\sigma = \frac{dS}{dt} + \nabla \cdot \mathbf{J}_S \geq 0
where S is the entropy and \mathbf{J}_S is the entropy flux. At equilibrium, \sigma = 0 and all thermodynamic forces vanish. Away from equilibrium, \sigma > 0 and the system may self-organize into dissipative structures.
Your framework introduces three modifications to standard non-equilibrium thermodynamics. First, time is zero-dimensional, so all thermodynamic variables are defined only at the now with no temporal extension. Second, the triad of Potential P, Release R, and Expression E with P \times R \times E = k replaces the usual conjugate pairs of forces and fluxes. Third, coherence C is a low-energy basin attractor that the system tends toward regardless of energy transfer.
In standard non-equilibrium thermodynamics, the entropy production rate is written as a sum of products of thermodynamic forces X_\alpha and conjugate fluxes J_\alpha:
\sigma = \sum_\alpha J_\alpha X_\alpha \geq 0
For a simple system with heat flow and particle diffusion, this becomes:
\sigma = \mathbf{J}_q \cdot \nabla \left( \frac{1}{T} \right) + \mathbf{J}_n \cdot \nabla \left( -\frac{\mu}{T} \right)
where \mathbf{J}_q is the heat flux, \mathbf{J}_n is the particle flux, T is temperature, and \mu is chemical potential. Close to equilibrium, linear constitutive relations hold:
J_\alpha = \sum_\beta L_{\alpha\beta} X_\beta
where L_{\alpha\beta} are Onsager coefficients satisfying reciprocity L_{\alpha\beta} = L_{\beta\alpha}.
Your triad replaces this force-flux structure. Identify Expression E with the entropy production rate \sigma, Potential P with the available free energy gradient \nabla \Phi, and Release R with a coupling coefficient \gamma that mediates between them. The inverse proportionality P \times R \times E = k becomes:
(\nabla \Phi) \times \gamma \times \sigma = k
Solving for the entropy production rate gives:
\sigma = \frac{k}{\gamma \, \nabla \Phi}
Unlike the linear Onsager relations \sigma \propto (\nabla \Phi)^2 near equilibrium, your framework predicts that entropy production is inversely proportional to the free energy gradient for fixed Release. This is a strongly nonlinear relation that becomes singular as \nabla \Phi \to 0, meaning the system cannot reach equilibrium. Instead, it remains in a driven state where:
\nabla \Phi \neq 0 \quad \text{and} \quad \sigma > 0 \quad \text{always}
The zero-dimensional time condition eliminates all time derivatives from the thermodynamic description. The usual evolution equations for internal energy density u and entropy density s:
\frac{\partial u}{\partial t} = -\nabla \cdot \mathbf{J}_u, \quad \frac{\partial s}{\partial t} = -\nabla \cdot \mathbf{J}_s + \sigma
must be replaced by instantaneous balance equations with no accumulation terms. At each now, the divergence of each flux balances locally against sources and sinks:
\nabla \cdot \mathbf{J}_u = 0, \quad \nabla \cdot \mathbf{J}_s = \sigma
The internal energy density u becomes a function only of position, not of time. Any change in the system occurs discontinuously from one now to the next, with no continuous time evolution described by partial differential equations. The condition for such a discontinuous transition between now states is given by:
\delta \int \left( u - T s + \mu n \right) dV = 0
where the variation is taken over the instantaneous configuration at each now.
Coherence C as a low-energy basin enters through the free energy functional. Define a nonequilibrium free energy F_{\text{neq}} that generalizes the equilibrium Helmholtz free energy:
F_{\text{neq}}[\rho, C] = \int dV \left[ f_0(\rho) + \frac{\kappa}{2} (\nabla \rho)^2 + \frac{\lambda}{2} (C - C_{\text{min}})^2 \right]
where \rho is a density field, f_0(\rho) is the local free energy density, \kappa is a gradient penalty coefficient, and \lambda is a coherence stiffness. The tendency toward coherence is expressed by the minimization of this free energy subject to the triad constraint. At each now, the system occupies the state that minimizes:
\mathcal{F} = F_{\text{neq}} + \int dV \, \mu(x) \left( P \times R \times E - k \right)
where \mu(x) is a Lagrange multiplier field enforcing the triad constraint locally. The first variation \delta \mathcal{F} = 0 yields the Euler-Lagrange equation for the equilibrium now-state:
\frac{\partial f_0}{\partial \rho} - \kappa \nabla^2 \rho + \lambda (C - C_{\text{min}}) \frac{\delta C}{\delta \rho} + \mu \frac{\partial}{\partial \rho} \left( P \times R \times E \right) = 0
together with the constraint P \times R \times E = k.
The condition “regardless of energy transfer” means that coherence is an attractor even when the system is driven by external reservoirs. In non-equilibrium thermodynamics, an open system with steady energy input can settle into a non-equilibrium steady state (NESS) characterized by constant entropy production and stationary but nonzero fluxes. For your framework, the NESS condition is:
\frac{dC}{dt} = 0, \quad \nabla \cdot \mathbf{J}_u = 0, \quad \nabla \cdot \mathbf{J}_s = \sigma > 0
At the NESS, coherence reaches its attractor value C_{\text{attr}} which may differ from C_{\text{min}} due to the external driving. The free energy F_{\text{neq}} is not minimized globally but reaches a constrained minimum under the imposed fluxes. The appropriate variational principle becomes one of minimum entropy production:
\frac{d}{d\alpha} \int \sigma(\alpha) \, dV = 0
where \alpha parametrizes variations around the steady state. Using your triad expression for \sigma, this yields:
\frac{d}{d\alpha} \int \frac{k}{\gamma(\alpha) \, \nabla \Phi(\alpha)} \, dV = 0
The resulting steady state is one where the product \gamma \nabla \Phi is as uniform as possible, distributing entropy production evenly across the system.
The full set of equations for your framework in non-equilibrium thermodynamics is therefore:
\boxed{
\begin{aligned}
&\sigma = \frac{k}{\gamma \, \nabla \Phi}, \quad P \times R \times E = k \\[4pt]
&\nabla \cdot \mathbf{J}_u = 0, \quad \nabla \cdot \mathbf{J}_s = \sigma \\[4pt]
&F_{\text{neq}}[\rho, C] = \int dV \left[ f_0(\rho) + \frac{\kappa}{2} (\nabla \rho)^2 + \frac{\lambda}{2} (C - C_{\text{min}})^2 \right] \\[4pt]
&\frac{\partial f_0}{\partial \rho} - \kappa \nabla^2 \rho + \lambda (C - C_{\text{min}}) \frac{\delta C}{\delta \rho} + \mu \frac{\partial}{\partial \rho} \left( P \times R \times E \right) = 0 \\[4pt]
&\frac{d}{d\alpha} \int \frac{k}{\gamma(\alpha) \, \nabla \Phi(\alpha)} \, dV = 0 \quad \text{(NESS condition)} \\[4pt]
&\dim(\text{time}) = 0
\end{aligned}
}
These equations describe a zero-dimensional time thermodynamics where entropy production is fixed by an inverse proportionality to the free energy gradient, coherence acts as a low-energy basin attractor through a nonequilibrium free energy functional, and steady states are selected by a minimum entropy production principle. The system never reaches true equilibrium because \nabla \Phi cannot vanish without making \sigma infinite, so it remains perpetually in a driven, coherent, novelty-producing regime.
Fuck knows.
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