The hope is to help resolve an apparent disconnect between how we forecast Earth's future during the Anthropocene, by moving away from traditional macroeconomic models and more toward treating civilization as a dissipative physical system like any other on our planet. Section 2. of this paper describes an underlying thermodynamic framework for emergent systems. Section 3. connects this framework to basic economic quantities. Section 4. discusses prognostic solutions for economic innovation and growth. Section 5. identifies formulations for distinct modes of growth in economic systems, and Section 6. summarizes the conclusions of this study.

From the standpoint of forecasting the human role in climate change, a broad brush may be all that is necessary given that carbon dioxide is both well‐mixed and long‐lived in the atmosphere. What the article presents is long‐range prognostic equations for global economic quantities by stepping back and viewing civilization as a whole, as it evolves slowly over “long” timescales and subject to such global externalities as resource availability and increasing natural disasters from a changing global climate.

There have been many prior efforts to link economic models to climate models [e.g., Yohe et al ., 2004 ; Stern , 2007 ; Tol , 2009 ; Nordhaus , 2010 . This paper differs by describing the human system in terms of the same thermodynamic laws that underpin parameterizations of gradients and flows in model representations of the earth's physical processes [e.g., Bitz et al ., 2012 ]. Of course, many might argue that we should not subject human systems to physical laws due to the complexities of human behavior [ Scher and Koomey , 2011 ]. Others might note that even physical systems at their most simple can easily become so sensitive to initial conditions as to become inherently unpredictable. But while we would not dream of predicting local weather beyond a week or so [ Lorenz , 1963 ], forecasting the global mean surface temperature a century out is an accepted challenge. The primary requirement for maintaining predictability is that we degrade temporal and spatial resolution. In this case, little or nothing might be said about the short‐term, finer‐scale details of the system; yet, broader, constrained forecasts can be made for more slowly evolving behaviors [ Bretherton et al ., 2010 ; Temam and Wirosoetisno , 2011 ].

Diagram of the approach taken in this paper where physical first principles are used to derive analytical expressions for the long‐run evolution of the global economy during the Anthropocene. Black arrows indicate a differential process. Red arrows indicate an additive or integral process.

As sketched in Figure 1 , the approach is to develop a general framework for describing the current state of systems and their spontaneous emergence, starting from physical first principles and using a simple theoretical framework outlined previously in Garrett [ 2012c ]. From this point, the paper exploits a fixed link between rates of global primary energy consumption (or power production) and a general measure of global wealth that was described in Garrett [ 2011 ] (see also supporting information). This leads to prognostic formulae for economic innovation and growth that are expressible in units of currency. The equations are presented in a form that can be evaluated against available economic statistics for past behavior. Potentially they may be used to provide physically constrained scenarios for the future, linking human and natural systems where the two are increasingly becoming coupled.

Such cycles are fairly rapid; at least the longest might be the annual periodicities that are tied to agriculture. This paper provides a framework for the slower evolution of civilization, over timescales where rapid cyclical behavior tends to average out, where the material growth and decay of civilization networks is driven by a long‐run imbalance between energy consumption and dissipation.

Burning coal at a power station raises an electrical potential or voltage allowing for a down‐voltage electrical flow. The potential energy is dissipated along the journey from the power station to the appliance. The appliance sustains people, who themselves dissipate heat. And, because what the appliance does is useful in their minds, the cycle is completed with the human desire for more coal to burn. Similarly, energy is dissipated as cars burn gasoline to propel vehicles to and from desirable destinations. Or, people consume food to maintain the circulations of their internal cardiovascular, respiratory, and nervous systems while dissipating heat and renewing their hunger.

As with any other natural system, civilization is composed of matter. Internal circulations are maintained by a dissipation of potential energy. Oil, coal, and other fuels “heat” civilization to raise the potential of its internal components. Dissipative frictional, resistive, radiative, and viscous forces return the potential of civilization to its initial state, ready for the next cycle of energy consumption.

2. Energetic and Material Flows to Systems

Before proceeding to a description of the growth of civilization, the starting point is to define what is meant by “short” and “long” timescales, and then to build from first principles a general thermodynamics for the emergence and evolution of dynamic systems over long timescales.

In the most abstract sense, the universe is a continuum of matter and potential energy in space. Local gradients drive thermodynamic flows that redistribute matter and energy over time. In the sciences, we invoke the existence of some “system” or “particle” from within this continuum, requiring as a first step that we define some discrete contrast between the system and its surroundings as shown in Figure 2. This discrete contrast can be approximated by an interfacial jump in potential energy Δμ between the system potential μ S and some higher level μ R ; or, Δμ = μ S − μ E with respect to a lower level μ E . Matter that lies along the higher potential μ R has a higher temperature and/or pressure, so it can be viewed as a “reserve” for downhill flows that “pour” into the system potential level μ S . Flows also “drain” from μ S to the lower potential environment lying along the potential surface μ E .

Figure 2 Open in figure viewer PowerPoint Schematic for the thermodynamics of an open system within a fixed volume V. Energy reserves, the system, and the environment lie along distinct constant potential surfaces μ R , μ S , and μ E . Internal material circulations within the system are sustained by heating and dissipation of energy that is coupled to a material flow of diffusion and decay. The level μ S is a time‐averaged potential. Over shorter time‐scales, the legs of a heat‐engine cycle would show the system rising up and down between μ E and μ R in response to heating and dissipation, as shown by the red arrow, allowing for material diffusion to the system and decay from the system. If flows are in balance then the system is at equilibrium and it does not grow.

Viewed from a strictly thermodynamic perspective, any system that is defined by a constant potential must implicitly lie along a surface within which there is no resolved internal contrast, i.e., one where the assumption is that there is a fixed potential energy per unit matter μ S and no internal gradients. This specific potential represents the time‐integrated quantity of work that has been required to displace each unit of matter within the surface through an arbitrary set of force‐fields that point in the opposite direction of the potential vector μ: e.g., the gravitational potential per block in a pyramid is determined by the product of the downward gravitational force on each block and its height.

Although internal gradients and circulations are not resolved within a constant potential surface, the existence of the continuum requires that they exist nonetheless. When a bathtub is filled, internal gradients force the water to slosh from side to side. While the short timescale of these small waves might be of interest to a child, a typical adult cares only about the time‐averaged water level of the bathtub as a whole, and that it gradually rises as the water pours in. The definition of what counts as a “system” is only a matter of perspective. It depends on what timescale is of most interest to the observer looking at the system's variability. As a guiding principle, however, coarse spatial resolution corresponds with coarse time resolution [e.g., Blois et al., 2013].

H S , can be expressed as a product of the amount of matter in the system N S and the specific enthalpy given by (1) The total energy of a system, or its enthalpy, can be expressed as a product of the amount of matter in the systemand the specific enthalpy given by

ν in the system and the oscillatory energy per independent degree of freedom e S (2) The specific enthalpy can be decomposed into the product of the total number of independent degrees of freedomin the system and the oscillatory energy per independent degree of freedom

e S represents the circulatory energy per degree of freedom per unit matter. For example, nitrogen gas has a specific enthalpy that is the product of the specific heat at constant pressure c p and the system temperature T S , or . The specific enthalpy can be decomposed into a total ν = 7 degrees of freedom at atmospheric temperatures and pressures each with a time‐averaged kinetic energy of kT S /2 where k is the Boltzmann constant. Thus, (3) The quantityrepresents the circulatory energy per degree of freedom per unit matter. For example, nitrogen gas has a specific enthalpy that is the product of the specific heat at constant pressureand the system temperature, or. The specific enthalpy can be decomposed into a total= 7 degrees of freedom at atmospheric temperatures and pressures each with a time‐averaged kinetic energy of/2 whereis the Boltzmann constant. Thus,

Zemanksy and Dittman, 1997 (4) a and a dissipation at rate d (5) Conservation of energy considerations dictate that enthalpy is the energetic quantity that rises when there is net heating of the system at a constant pressure [], i.e.and that net heating of the system is a balance between a supply of energy to the system at rateand a dissipation at rate

The Second Law requires that dissipation redistributes enthalpy to some lower potential, draining some higher potential reserve. Not all enthalpy in the reserve H R is necessarily available to the system. For example, unless the temperature of the system is raised to extremely high levels, the nuclear enthalpy of a reserve H R = mc2 might normally be inaccessible. Thus, available enthalpy is distinguished here by the symbol ΔH R .

Heating is coupled to material flows through an idealized four step cycle or “heat engine”, whose circulation is shown by the red arrow in Figure 2. A system that is initially in equilibrium with the environment at level μ E is heated, which raises the potential level of the system μ S an amount 2Δμ to level μ R with a timescale of τ heat ∼ 2Δμ/a. It is at this point that, according to the Gibbs‐Duhem equation, the surface μ S comes into diffusive equilibrium with respect to external sources of raw materials, allowing for a material flow to the system [Kittel and Kroemer, 1980]. There is then cooling through dissipation of heat to the environment with timescale τ diss ∼ 2Δμ/d, which brings the system back into diffusive equilibrium with surface μ E , allowing for material decay.

How the thermodynamics should be treated depends on the question at hand, and whether the timescale of interest is short or long compared to τ heat .

2.1. Systems in Material Equilibrium Over Short Timescales τ heat , the legs of the heat engine are resolved, so that the amount of matter in a system N S would appear to change sufficiently slowly that it could be considered to be fixed. In this case, the response to net heating would be that the specific enthalpy per unit matter rises at rate (6) Over timescales much shorter than, the legs of the heat engine are resolved, so that the amount of matter in a systemwould appear to change sufficiently slowly that it could be considered to be fixed. In this case, the response to net heating would be that the specific enthalpy per unit matter rises at rate (7) c p is the specific heat of the substance at constant pressure and ∂Qnet/∂t is the radiative heating. In a materially closed system, the response to net heating is for the temperature to rise. For the example that heating is a response to radiative flux convergence, then it may be that the temperature rises according towhereis the specific heat of the substance at constant pressure and ∂/∂is the radiative heating. In a materially closed system, the response to net heating is for the temperature to rise. In the atmospheric sciences, equation 7 expresses the short‐term temperature response to radiative heating [Liou, 2002]. At timescales longer than τ heat , however, the establishment of a temperature gradient ultimately leads to a diffusive, material flow that restores equilibrium and that we call the wind.

2.2. Systems in Material Disequilibrium Over Long Timescales Over timescales much longer than τ heat , the legs of the heat engine are not resolved. Instead, because the heat engine cycles are much faster than the timescales that are of interest to the observer, what is seen is only some average level of μ S that lies in between the points of maximum and minimum potential energy, μ R and μ E (Figure 2). μ S corresponds with growth of the system enthalpy at rate (8) (9) In this case, energetic and material flows have the appearance of being instantaneously coupled. An illustration of this coupling is shown in Figure 3 , which recasts Figure 2 in terms of a single co‐ordinate. Where there is a disequilibrium, material convergence along a surface of constant potentialcorresponds with growth of the system enthalpy at rateso that from equation 5 , the bulk grows at rate Figure 3 Open in figure viewer PowerPoint V. Energy reserves, the system, and the environment lie along distinct constant potential surfaces μ R , μ S , and μ E . The size of an interface between surfaces determines the rate of heating a and the speed of downhill material flow j a . The system grows or shrinks according to a net material flux convergence j a − j d along μ S . System growth is related to expansion work w that is done to grow the interface, extending the system's access to previously inaccessible energy reserves. The efficiency of work is determined by ε = w/a. Schematic for the thermodynamic evolution of a system within a constant volume. Energy reserves, the system, and the environment lie along distinct constant potential surfaces, and. The size of an interfacebetween surfaces determines the rate of heatingand the speed of downhill material flow. The system grows or shrinks according to a net material flux convergencealong. System growth is related to expansion workthat is done to grow the interface, extending the system's access to previously inaccessible energy reserves. The efficiency of work is determined by If there is zero time‐averaged net heating, then because ⟨a⟩ = ⟨d⟩, in which case the size of the system N S does not change. Like water pouring into and draining from a bathtub at equal rates, circulations within the system maintain a steady‐state. Although local entropy production (∂Qnet/∂t) μ /μ is zero, global entropy ∑ μ (∂Qnet/∂t) μ /μ grows from a continuous redistribution of matter through a flow from high to low values of μ. . There is a net convergence of matter along the potential surface μ S at rate jnet. Material flows into civilization at rate j a , and out of civilization at the decay rate j d , to form a balance defined by (10) τ growth ∼ N S /jnet. Combined with equation (11) (12) (13) Material growth occurs when there is the nonequilibrium condition that energy consumption exceeds dissipation, in which case. There is a net convergence of matter along the potential surfaceat rate. Material flows into civilization at rate, and out of civilization at the decay rate, to form a balance defined byso that the timescale for growth of the system is. Combined with equation 9 , this implies that A straightforward and familiar example of this physics is what happens when we boil a pot of water. Once the water reaches the boiling point, the temperature of the water is maintained at a constant 100°C. Any energy input from the stove goes into turning liquid water into bubbles. Setting aside the energetics of forming the bubble surface, and assuming the pot is well insulated, the energy input that is required to vaporize a single liquid water molecule is where l v is the latent heat of evaporation at boiling. Thus, vapor molecules contained in the bubbles are created at a rate that is proportional to the rate of energetic input: . Heating creates an internal circulation of bubbles that we call a boil. When bubbles rise to the surface, molecules escape the fluid at rate j d , and there is an associated evaporative cooling of the water at rate . With a steady simmer, a constant vapor concentration N S is maintained within the pot because heating equals cooling. In this case, from equation 13, j a ≃ j d and jnet = 0. If the output from the heating element is suddenly raised to high, then there is a nonequilibrium adjustment period of τ growth ∼ N S /(j a − j d ) during which heating temporarily exceeds dissipation and bubble production at the bottom of the pot j a exceeds bubble popping at its top j d . The size and number of vapor bubbles in the water increases, and a new stasis is attained only when evaporative cooling d rises to come into equilibrium with the element heating a. At this point, the pot has gone from a simmer to a rolling boil.

2.3. Gradients and Flows in Material Disequilibrium j can be regarded as the diffusion of matter downhill, across a material interface toward the system. The interface between the system and its higher potential reservoirs can be defined by a potential step with a rise Δμ = μ R − μ S and an orthogonal quantity of material that lies along the interface . The total energy required to grow the interface is the product of these two quantities: i.e., . Because the presence of the gradient is required to enable flows, there is a proportional dissipation of available potential energy ΔH R at rate (14) α is a rate coefficient with units of inverse time. The quantity in equation ΔH R = N R Δμ. The available enthalpy is a reserve of energy that is eventually available to be consumed. In contrast, ΔG represents the gradient that is instantly available to drive flows due to a material interface between the system and ΔH. As shown in Figure 3 , a material flow at ratecan be regarded as the diffusion of matter downhill, across a material interface toward the system. The interface between the system and its higher potential reservoirs can be defined by a potential step with a riseand an orthogonal quantity of material that lies along the interface. The total energy required to grow the interface is the product of these two quantities: i.e.,. Because the presence of the gradient is required to enable flows, there is a proportional dissipation of available potential energy Δat ratewhereis a rate coefficient with units of inverse time. The quantityin equation 14 is a related but different quantity from the available enthalpy. The available enthalpy is a reserve of energy that is eventually available to be consumed. In contrast, Δrepresents the gradient that is instantly available to drive flows due to a material interface between the system and Δ . Thus, from equation (15) From equations 10 and 11 , when a system is considered over long timescales, then energy consumption is coupled to a material flux. Thus, from equation 14 reflects the respective sizes of the two components it separates. In general, when there is a diffusive flow to a system, is proportional to a product of the available enthalpy within a high potential energy “reservoir” ΔH R = N R Δμ and the size of the system N S taken to a one third power [Garrett, 2012c (16) k is related to the object shape. For a sphere, k = (48π2)1/3 [Garrett, 2012c The magnitude of the interfacereflects the respective sizes of the two components it separates. In general, when there is a diffusive flow to a system,is proportional to a product of the available enthalpy within a high potential energy “reservoir”and the size of the systemtaken to a one third power [], or thatwhere the dimensionless coefficientis related to the object shape. For a sphere,= (48]. At first glance, one might guess that the system interface should be proportional to N S N R instead of : both the size of the system and the size of the reserve are what drive flows between the two. In general, a system's size is proportional to its volume V S = N S /n S , where N S is the number of elements in the system and n S is the internal density; V S and N S are proportional to a dimension of length cubed, or volume. What is important to recognize is that flows to a system are not determined by a volume. Rather, flows are down a linear gradient that lies normal to a surface. The surface area has dimensions of length squared or , and the linear gradient has dimensions of inverse length or . Both factors control the flow rate across the interface, and their product yields a one third power, or a length dimension: . The significance is that, if it were assumed that is proportional to the product N S N R , then the implication would be that wholes interact with wholes, implying a perfect mixture of the system and its reserve. The objection to this formulation is that any supposed existence of a perfect mixture would mean that it would be impossible to resolve flows between N S and N R : the two components of the mixture would be indistinguishable. A second consequence is that assuming a unity exponent for N S removes any element of persistence or memory from rates of system growth, as will be shown below. Unphysically it would isolate what happens in the present from and what has happened in the past. As a general principle, the Second Law allows for neither perfection nor isolation in our universe. ΔH R = N R Δμ, equations (17) (18) Since, equations 14 and 15 for energy dissipation and material flows can now be expressed as In Garrett [2012c] it was shown that the quantity can be expressed in an equivalent fashion in terms of a length density times a diffusivity , where the length density is analogous to the electrostatic capacitance within a volume and the diffusivity has dimensions of area per time. For the diffusional growth of a spherical cloud droplet of radius r, vapor condenses at rate , where N R /V is equivalent to the excess vapor density relative to saturation. In this case . For more dendritic structures like snowflakes, there is no clearly definable “radius”, yet it is still a length dimension within a volume Λ or “capacitance density” that drives diffusive growth [Pruppacher and Klett, 1997]. (19) (20) Thus, the flow and dissipation equations can be alternatively expressed as The rate of material flows is proportional to a rate of energy dissipation a, which in turn is proportional to some measure of the length density within the system Λ or its accumulated size N S to a one third power, and the number of potential energy units in the reserve N R = ΔH R /Δμ. The final component is , which expresses the amount of energy that must be dissipated to enable each unit of material flow toward the system.

2.4. Efficiency and Growth As described above, a system grows if there is an imbalance so that net heating drives an accumulation of matter in the system through diffusive material flows (equations 10 and 13). Growth of the size of the system N S increases an interface with the energy reserves that enable diffusive flows. Δμ is fixed, then future flows evolve because the magnitude of the “step” grows laterally in response to a convergence (or divergence) of current flows (Figure and the potential difference ΔG is termed “work” w, where (21) Taking the approach that the resolved “rise” of the interfaceis fixed, then future flows evolve because the magnitude of the “step”grows laterally in response to a convergence (or divergence) of current flows (Figure 3 ). Here, this material expansion or “stretching” of the interfaceand the potential difference Δis termed “work”, where (22) The efficiency of converting heating to a rate of doing work is normally defined by the ratio Here efficiency can be either positive or negative depending on whether the interface is shrinking or growing in response to heating, and therefore on the sign of w (equation 21). (23) η has units of inverse time. In other words, 1/η is the characteristic time for exponential growth of ΔG and . From equation 21 , the relative growth rate of the interface can be defined bywherehas units of inverse time. In other words, 1/is the characteristic time for exponential growth of Δand w = dΔG/dt = εa and from equation a = αΔG, it follows that the relationship between the growth rate η and the efficiency ε and the heating rate a is given by (24) (25) Since from equations 21 and 22 = d/dand from equation 14 , it follows that the relationship between the growth rateand the efficiencyand the heating rateis given by This formulation has the advantage of expressing η in terms of a measurable flux a. The implication is that systems that are efficient are able to incorporate matter more quickly; such efficient incorporation causes the system to accelerate growth of an interface with respect to energy reserves. Ultimately, higher efficiency allows the system to consume energy more rather than less. For the special case of pure exponential growth where η is a constant, then a = a 0 exp(ηt), but, more generally, nothing is ever fixed in time: η constantly changes as the interface evolves, and it can even change sign if it shrinks. The growth rate η is positive if the efficiency ε is greater than zero meaning that the system is able to do net work on its surroundings in response to heating (i.e., ). Otherwise, the growth rate is negative and the system collapses (i.e., ε < 0 and .