# Dissipation + Utilization = Self-Organization

## Abstract

**:**

## 1. Introduction

#### 1.1. The Problem of Evolution

#### 1.2. A Thermocontextual World

## 2. TCI States and Processes

#### 2.1. The Thermocontextual State

_{abs}, into thermocontextual components, which is given by:

_{as}. Ambient state energy is the system’s “ground state” energy, and it is defined by equilibrium with the system’s surroundings. It is the energy of the ambient reference state with respect to the hypothetical zero-energy absolute zero reference state. Since the ambient temperature is always positive (Postulate Two), the ambient state energy is always positive.

_{a}is the ambient temperature.

_{a})dX.

_{V}is the volumetric heat capacity.

_{td}, and the changes in thermal entropy and thermodynamic entropy are identical. They differ only in their zero-entropy reference state. TCI also defines statistical entropy as a transactional property of a transition between states [25] (Appendix A).

_{a}is uniquely defined by E = X = Q = S = 0 and E

_{abs}= E

_{as}.

_{a}at the ambient temperature and as utility υ, which we define as the summed transfers of work and thermal exergy (Figure 1). For an open system, utility can also include the internal exergy of exported components. A system’s exergy, entropy, and entropic energy are empirically defined by measurements of the utility and ambient heat reversibly transferred to the surroundings. A system’s exergy is always non-negative, but its initial entropic energy can be negative or positive.

#### 2.2. Transitions

_{i}; (2) the input of work (represented by −PdV); and (3) the inputs of free energy from added components, dM

_{i}, with chemical potential μ

_{i}(free energy per unit component i). We note that the addition of components’ thermal energy is included in the TdS term.

- Replace TdS with T
_{a}dS=Q (Equation (4)). - Replace PdV (PV work output) with generalized utility output, dυ.
- dE = dX + dQ (Internal energy equals internal exergy plus entropic energy).
- Replace chemical potential μ with specific exergy, $\overline{X}$ (exergy per unit component).

_{diss}is the production of entropic energy by dissipation and dυ is the output of utility to the surroundings associated with the decline in exergy (Figure 1 and Figure 2a). Equation (7) describes the change in a system’s internal exergy due to internal dissipation and the reversible exchanges of components, utility, and entropic energy with the surroundings.

_{a}(exergonic isentropic transition).

_{a}, and of utility, dυ. We can rewrite (8) for an endergonic transition as (9):

_{a}. (endergonic isentropic transition).

#### 2.3. Dissipative Processes

_{ij}are constants describing the linear contribution of the specific exergy gradient in component j to the flow of component i. Onsager expressed potentials by entropy gradients instead of specific exergy gradients, but they differ from each other only by a constant factor (the negative ambient temperature).

_{AB}= 0) at X

_{2}= 2. This is a very special case steady state. For any other fixed value of X

_{2}, the entropy production rate is not minimized at steady state. We also note that the Onsager’s relations (Equation 10) are based on the equality of thermodynamic pressures and negative exergy gradients, and this is yet another special case, not valid for heat flow, diffusion, or chemical reactions. The rates of dissipation and entropy production are generally not minimized at steady state, even very near equilibrium.

_{+}Ax

^{2}− k

_{−}x, where k

_{+}and k

_{−}are the forward and reverse kinetic rate coefficients. This corresponds to a detailed reaction given by A + 2x→3x. The product x partakes in the reaction, making the system’s dynamics autocatalytic and non-linear. Figure 6a illustrates the nodal network representation of the Schlögl reaction.

_{2}is unstable to perturbations. X

_{2}is a bifurcation point separating two basins of attraction around the stable steady state dissipative functions x(t) = x

_{1}and x(t) = x

_{3}.

^{2}Y, making the Brusselator non-linear. Non-linearity allows for the possibility of multiple stationary dissipative solutions for a given set of boundary conditions.

^{2}, the steady state function’s basin of attraction becomes a point and bifurcation boundary. Any perturbation from the steady state dissipative process sends the system on a transient path that converges to a stationary periodic function. The dissipative function for the periodic solution $\left|X\left(t\right),Y\left(t\right)\right|$ is graphically illustrated in Figure 8 as the limit cycle attractor.

_{3}

^{−}); B = 3 × malonic acid (CH

_{2}(CO

_{2}H)

_{2}); D = 9 × CO

_{2}; and E = 4 × bromide (Br

^{−}) + 6 × H

_{2}O. A well-stirred system shows fluctuating colors, reflecting oscillating concentrations (Figure 8). In an unstirred system, in which diffusion occurs, a variety of moving spatial patterns can develop [27]. The models of an unstirred system are represented in a pixelated two- or three-dimensional model space. The generalized state space in this case is many- but finite-dimensional, with separate state-space dimensions for the concentrations of X and Y in each pixel of the model space. The system’s trajectory over time traces out the changes in the concentrations of X and Y across the system’s pixelated volume.

## 3. Evolving Complexity

#### 3.1. An Extremum Principle for Dissipative Systems

_{x}) by the sum of internal dissipation (q

_{diss}) and external dissipation (X

_{exp}):

_{in}= X

_{A}) to a single node and its outputs of utility (υ

_{out}= X

_{B}), dissipated exergy, and work.

_{int}. We then rewrite (15) for an individual node as:

**MaxEff**) constrains how that transition occurs. For an isolated transition, internal dissipation is the only means of reducing exergy, and this means that an isolated transition maximizes dissipation and entropy production. For an open transition, however, work on the surroundings provides another way to reduce the exergy of output to comply with Postulate Four. MaxEff says that given the opportunity, a system selects exergy utilization over exergy dissipation.

_{ext}). Reducing the export and external dissipation of exergy makes additional exergy available for internal utility transfers to other dissipative steps within the overall process of dissipation. From (17) and (18), this reflects an increase in efficiency. The abiogenesis of UV-C pigments under a UV-C flux provides a clear illustration of MaxEff.

^{*}. The high specific energy of A

^{*}creates a barrier that limits the reaction rate from A to B. This relationship is expressed by the Arrhenius equation [28], relating the kinetic rate constant k to temperature:

^{*}, and R is the universal gas constant. The idea is that at a higher temperature, there is a higher probability of a fluctuation reaching the energy necessary to pass over the activation energy’s barrier, resulting in a higher reaction rate.

_{A*}and RT—both defined with respect to absolute zero temperature—with X

_{A*}and Q

_{A}—both defined with respect to the ambient reference state. A higher entropic energy increases the statistical probability of reaching the activated state exergy barrier and increases the reaction rate. Counterintuitively, component A must irreversibly reduce the exergy it had at its input in order to increase its entropic energy and increase its probability of reaching the activation exergy. This results in a higher reaction rate, but lower efficiency. Autocatalysis reduces the activation exergy for the reaction. By reducing the activation exergy, autocatalysis allows a reaction to proceed with a lower entropic energy, a higher utility output, and a higher efficiency.

#### 3.2. The Two Arrows of Evolution

_{F}, which corresponds to external efficiency, and the functional complexity factor, C

_{F}, which corresponds to internal efficiency. A dissipative system can increase its efficiency and stability either by having a positive rate of growth or by increasing its internal utilization and functional complexity.

_{F}. The functional complexity is the ratio of internal utilization to utility input, and it is a measurable and well-defined property of a dissipative process.

#### 3.3. Oscillations and Synchronicity

#### 3.4. Whirlpools Disprove the MEPP

## 4. Summary and Discussion

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Thermocontextual Interpretation’s Postulates and Definitions

**Postulate**

**One.**

**Postulate**

**Two.**

**Postulate**

**Three.**

**Postulate**

**Four**

**(Generalized Second Law of Thermodynamics).**

**Postulate**

**Five**

**(Maximum Efficiency Principle—MaxEff).**

**Absolute energy**: A system’s absolute energy, E_{abs}, equals the system’s potential work, as measured on the surroundings in the limit of absolute zero.**Actualization**: Actualization is the reversible work of recording a measurement result during an instantiated microstate’s deterministic transition to its measurement reference state.**Ambient state**: A system is in its ambient state when it is in equilibrium with the ambient surroundings. It is defined by zero exergy and entropic energy.**Ambient state energy**: A system’s ambient state energy, E_{as}, is the ambient reference state’s potential work capacity, as measured at the limit of absolute zero.**Ambient temperature**: A system’s ambient temperature, T_{a}, equals the positive temperature of the system’s actual surroundings with which it interacts or potentially interacts and from which measurements or observations are made.**Dissipative energy function**: A dissipative energy function is a function of a stationary system’s state vector of measurable state properties across its pixelated model space and time. It traces the system’s state properties as energy and mass components pass through it, from input to eventual output. The dissipative energy function is a stationary solution to the dissipative system’s kinetic and boundary constraints.**Efficiency:**Efficiency is the ratio of utilization to the input of work and exergy.**Entropic energy**: A system’s**entropic energy**is defined by system energy minus exergy (Q ≡ E − X).**Entropy (thermal)**: A system’s thermal entropy is defined by entropic energy divided by ambient temperature (S ≡ Q/T_{a}).**Exergy**: A system’s exergy, X, is defined by its potential work as reversibly measured at the ambient surroundings.**Instantiation**: Instantiation [25] randomly selects a measurable zero-entropy microstate from a positive-entropy state comprising multiple microstate potentialities. It is a consequence of derandomization resulting from the reversible export of entropic energy during a transition from a high-entropy state A to a lower entropy state B.**Perfect measurement**: The perfect measurement of state involves a reversible thermodynamically closed process of transition from a system’s initial state to its ambient reference state. Perfect measurement reversibly records the outputs of exergy and entropic energy to the surroundings. The measurement of a positive entropy system is resolved into instantiation and actualization.**Reference time**: Reference time is the time of relativistic causality, as measured by a reference clock in the ambient surroundings.**Refinement**: Refinement (fine graining) is a response to a declining ambient temperature and ambient state energy of the ambient reference state. This leads to increases in the thermocontextually defined exergy and thermal entropy [19].**Statistical Entropy**: Statistical entropy, σ_{AB}, [25] is a transactional property of a transition from state A to state B. It is defined by:$${\sigma}_{AB}\equiv -{\displaystyle \sum}_{i=1}^{N}{P}_{i}\mathrm{ln}\left({P}_{i}\right).$$_{i}is the probability of instantiating measurable microstate i from the positive-entropy system’s N measurable microstate potentialities. Measurable microstate potentialities are defined with respect to the transactional reference state B.**System energy**: System energy is defined by absolute energy minus ambient state energy (E ≡ E_{abs}− E_{as}).**System time**: System time is a complex property of state. The real component of system time indexes the irreversible production of entropy, either by the dissipation of exergy or by refinement. The imaginary component (‘it′ in quantum mechanics) indexes a system’s reversible changes. Both indexes are tracked against the irreversible advance of reference time.**Utilization**: Utilization includes a dissipative system’s work on the surroundings to extend its reach, plus the total transfer of work and exergy between the dissipative system’s dissipative nodes.

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**Figure 1.**Perfect measurement is a reversible closed-system transition from an initial State A to ambient State B. The measurement of a positive-entropy system is resolved into two steps. The first step for a positive entropy state involves derandomization by the reversible transfer of entropic energy to the surroundings. This sets the statistical entropy σ

_{A′B}(Appendix A) [25] to zero and randomly instantiates a zero-entropy microstate for state A′. The second step is the actualization of the measurement results. This involves the reversible and deterministic transfer of state A′, with exergy X

_{A}to ambient state B, and the export of exergy to actualize a measurement result.

**Figure 2.**A component transitions as a thermodynamically closed system, exchanging energy and work but no material components with the surroundings. (

**a**) In an exergonic transition, a component transitions to lower exergy and outputs utility (υ). (

**b**) Utility added to an endergonic transition lifts a component to higher exergy. Dissipation reduces the efficiency of both exergonic and endergonic transitions.

**Figure 3.**(

**a**): Continuous heat flow in the classical limit. Input energy q

_{T}is resolved into thermal exergy X

_{q}and entropic energy Q. The input of entropic energy is output as ambient heat, q

_{a}. Thermal exergy is dissipated and is also output as ambient heat, but as q

_{diss}. Produced entropy is exported, and the transition is, therefore, isentropic. (

**b**): Discontinuous heat flow through an elementary transition node.

**Figure 4.**Dissipative system model. The system’s nonequilibrium surroundings include utility source(s)—either directly (e.g., sunlight) or indirectly—by high-exergy components. Other components in the surroundings (e.g., water, air, detritus) are also freely available to the system for processing and discharge. A stationary system is defined by a function describing its stationary dissipative process, with equal time-averaged inputs and outputs of materials and total energy.

**Figure 5.**Coupled transitions. The flows of components 1 and 2 are based on Equation (11) with L

_{11}= L

_{22}= 1 and L

_{12}= L

_{21}= 0.2. (

**a**) For X

_{2}< 2, exergonic node A outputs power to endergonic node B ( $\dot{\upsilon}$

_{AB}> 0), which lifts component 2 to higher exergy. For X

_{2}> 2, the flow of component 2 is reversed, and node B becomes exergonic, outputting power to node A. (

**b**) Graphs of power input for component 1 (black line) and total power input (blue line). Total power input equals total dissipation $\dot{Q}$. The figure also shows the rate of utility transfers between nodes ($\dot{\mathsf{\upsilon}}$

_{AB}) (red line).

**Figure 6.**(

**a**) Nodal network diagram for the Schlögl Reaction. Transition 1 is an autocatalytic reaction A + 2x→3x. Transition 2 is a simple transition x→B. (

**b**) The black curve shows the rate of change in the concentration of x versus concentration. The Schlögl reaction’s three steady states, at x

_{1}, x

_{2}, and x

_{3}, are defined by dx/dt = 0. Perturbation analysis shows that x

_{1}and x

_{3}are stable to perturbations. The red curve shows the system’s rate of dissipation as a function of the concentration of x.

**Figure 7.**The Brusselator network model. Letters refer to the components’ states. The overall reaction is A + B→E + D. States higher on the diagram have higher specific exergy. The Brusselator comprises four reaction steps: R1: External source A → X; R2: Y + 2X → 3X; R3: External source B + X → Y + D (discharge); and R4: X → E (discharge). The arrowhead expressions describe reaction rates.

**Figure 8.**The Brusselator’s stationary dissipative functions in state space, spanned by its state variables. The arrows show the direction of change in the stationary oscillating state over time within a single well-stirred volume.

**Figure 9.**Internal and external utility. (

**a**) Thermodynamics describes an isolated transition by the conservation of energy and production of entropy. (

**b**) Postulate Four extends the Second Law to open transitions, in which exergy decline can be by dissipation to q

_{X}and by external work w. (

**c**) For a network of dissipators, an elementary node’s utility output is resolved into external utility (work) and internal utility, which is input to other nodes within the network.

**Figure 10.**Simple feedback loop. A component flows through a system (straight vectors) at a rate (J) of one mass unit/s. The component’s specific exergy ($\overline{X}$) equals two units at input and zero at output. The rates of net exergy input (J × $\overline{X}$) and total dissipation rate ($\dot{Q}$) equal two energy units/s. Mass and energy inputs and outputs are balanced. Exergonic node 2 takes the component with 12 units of specific exergy, dissipates one unit, and performs work on endergonic node 1 (wavy vector) at a rate of 11 units/s. Endergonic node 1 applies ten units of work to lift the component’s specific exergy from 2 units to 12 units, and in the process, it dissipates one unit of work. The system’s rate of internal work on the component equals 10 units. With 2 units of exergy input, its functional complexity C

_{F}and efficiency equals five.

**Figure 11.**Coupled oscillators. The figure shows sixteen oscillators linked in a circle. All oscillators have identical unit rates of utility input (wavy arrows) into nodes, which pump an ambient component (not shown) into the pods. When the concentration in a pod reaches a critical value, the component is released, and the cycle resumes. Coupling between adjacent pods allows components to leak from one pod to another. The leak rate equals the difference in concentrations, multiplied by a coupling coefficient.

**Figure 12.**Synchronization of coupled oscillators. Top—Pod Concentrations: Oscillators are randomly assigned periods between 49.5 and 50.5 time units. They start with randomly assigned concentrations. After about 1500 time steps, the oscillators synchronize. Bottom—Average Rate of Exergy Accumulation: Each oscillator has a unit rate of work input, which is used to pump component into its pod. When oscillators are not synchronized, some exergy is lost to diffusive leakage between adjacent oscillators. When all oscillators synchronize after 1500 time steps, concentrations are equal, there is no diffusive loss, and the pods’ rate of exergy accumulation is maximized and equal to the work input (except at pod discharge). The rate of internal work is equal to the time-averaged accumulation of exergy by the pods, and this is maximized by synchronization. Synchronization, therefore, increases the efficiency of the system’s dissipative process.

**Figure 13.**Models for radial flow and whirlpool. Left: radial flow. Right: whirlpool. Each concentric shell is a single zone with uniform pressure (water elevation) and fluid speed. Arrows illustrate fluid flow directions only. Speed is constant within each zone but the radial speed increases inwards in both cases due to the incompressibility of water. In addition, the conservation of angular momentum requires that the rotational velocity for the whirlpool is inversely proportional to the radial distance and increases inwards.

**Figure 14.**Paired nodes at whirlpool zonal interface. As water flows across a zonal interface, it undergoes both a decline in pressure and an increase in velocity and kinetic energy. An elementary node represents only a single transition, so each interface has two nodes. The first node is exergonic. It transfers exergy at a rate of ${\dot{\mathrm{X}}}_{in}=\mathrm{J}\Delta \mathrm{P}$ to endergonic node 2. Node 2 uses this exergy for the internal work of accelerating the water, which is given by ${\dot{\mathsf{\upsilon}}}_{int}$ = ${\dot{\mathrm{X}}}_{in}-{\dot{\mathrm{Q}}}_{diss}=\text{\xbd}\rho {\Delta \mathrm{V}}^{2}\mathrm{J}$, where ${\dot{\mathrm{Q}}}_{diss}$ is the rate of dissipation by node 2. The internal work for the system is summed over all interfaces.

**Figure 15.**Pressure and kinetic energy profiles for whirlpool and radial flow. Fluid velocity and kinetic energy increase towards the drain for both the whirlpool and radial flow. The solid lines show an imperceptible drop in the pressure and a similarly imperceptible acceleration of water for radial flow. The dashed lines show an 8.1 cm (40%) pressure drop and a sharp acceleration of water for the whirlpool near its vortex. The lower pressure at the drain for the whirlpool corresponds to lower rates of water discharge, dissipation, and entropy production compared to radial flow.

Process | Component | Thermodynamic Pressure (Generalized Concentration) | Phenomenological Rate Law | Specific Exergy (dX) |
---|---|---|---|---|

Conductive heat flow, J | Unit heat | Temperature, T | J = −k∇T | X = q(T − T_{a})/T; dX=qT _{a}dT/T^{2} |

Chemical diffusion, J | Unit mass | Chemical activity, ɑ. ɑ ∝ C for dilute concentration, C. | J = −D∇C | X = RT_{a}ln(ɑ)dX=RT _{a}dɑ/ɑ |

Electrical flow, I | Unit charge | Voltage, V | I = −σ∇V (σ electrical conductance) | dX = dV |

Chemical reaction | Σν_{i}R_{i}⇌ Σν_{j}P_{j} ^{(1)} | $\mathsf{\Theta}\equiv \frac{\prod {a}_{{\mathrm{P}}_{\mathrm{i}}}^{{\mathsf{\nu}}_{\mathrm{i}}}}{\prod {a}_{{\mathrm{R}}_{\mathrm{j}}}^{{\mathsf{\nu}}_{\mathrm{j}}}}\equiv \frac{{\mathsf{\Theta}}_{P}}{{\mathsf{\Theta}}_{R}}$ | $\frac{d\zeta}{dt}={k}_{+}{\mathsf{\Theta}}_{R}-{k}_{-}{\mathsf{\Theta}}_{P}$^{(2)} | $\begin{array}{c}d\overline{\mathrm{X}}\equiv {\mathrm{d}\mathsf{\Theta}}_{\overline{X}}\equiv \\ \frac{\prod {\overline{X}}_{{\mathrm{P}}_{\mathrm{i}}}^{{\mathsf{\nu}}_{\mathrm{i}}\zeta}}{\prod {\overline{X}}_{{\mathrm{R}}_{\mathrm{j}}}^{{\mathsf{\nu}}_{\mathrm{j}}\left(1-\zeta \right)}}d\zeta \end{array}$^{(3)} |

Laminar flow, J | Fluid | Pressure, P | J = −K∇P ^{(4)} | dX = dP |

^{(1)}Reactants R

_{i}transition to products P

_{j}as a closed system. ν

_{i}are stoichiometric coefficients.

^{(2)}zeta ζ is a reaction progress variable (0→1) and k

_{+}and k

_{−}are forward and reverse kinetic rate constants.

^{(3)}Specific exergy $\overline{X}$ = RT

_{a}ln(ɑ), where activity for the ambient reference state is set to unity.

^{(4)}Hydraulic conductivity K is constant for Newtonian laminar flow, but at higher flow rates, turbulent flow becomes non-linear with pressure gradient.

Steady State Flow Rate J (m ^{3}/s) | ${\dot{\mathbf{X}}}_{in}$ ρgh _{o}J (J/s) | ${\dot{\mathbf{X}}}_{out}$ ρJ ^{3}/A_{drain}^{2} (J/s) | Net Power ${\dot{\mathbf{X}}}_{in}-{\dot{\mathbf{X}}}_{out}$ (J/s) | ${\dot{\mathsf{\upsilon}}}_{\mathrm{int}}$ Σ${\dot{\mathsf{\upsilon}}}_{\mathrm{int},i}$ (J/s) | C_{F}${\dot{\mathsf{\upsilon}}}_{\mathrm{int}}$${\dot{\mathbf{X}}}_{\mathrm{in}}$ |
---|---|---|---|---|---|

Whirlpool3.00 × 10 ^{−5} | 0.029 | 0.0022 | 0.027 | 2.9 × 10^{−3} | 0.1 |

Radial flow3.88 ×10 ^{−5} | 0.038 | 0.0047 | 0.033 | 7.4 × 10^{−7} | 1.9 × 10^{−5} |

_{out}= Kinetic exergy of water exiting a 1 cm diameter drain with area A

_{drain}. ρ = Fluid density (1000 kg/m

^{3}). h

_{o}= Water depth at perimeter (20 cm). ${\dot{\mathrm{W}}}_{i}=\frac{1}{2}\rho \Delta {V}_{i}^{2}$ is the increase in kinetic energy per volume of water at interface i (Figure 14). $\dot{\mathrm{W}}$

_{int}= Internal work of accelerating water from the perimeter zone to the core zone.

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Crecraft, H.
Dissipation + Utilization = Self-Organization. *Entropy* **2023**, *25*, 229.
https://doi.org/10.3390/e25020229

**AMA Style**

Crecraft H.
Dissipation + Utilization = Self-Organization. *Entropy*. 2023; 25(2):229.
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**Chicago/Turabian Style**

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2023. "Dissipation + Utilization = Self-Organization" *Entropy* 25, no. 2: 229.
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