A formal framework for the non-equilibrium thermodynamics of civilization growth was laid out in Part 1 . The basis for a model linking economics to physics is a fundamental identity that relates a monetary expression for wealth to the rate at which civilization powers itself with primary energy sources . We all have some sense of the difference between civilization and its uncivilized surroundings since we have farms, buildings, human population, vehicles, and communication networks. As shown in Fig.  and discussed in Appendix A, this difference implies the existence of a gradient between civilization and its environment. Gradients allow for irreversible thermodynamic flows. A consumption and dissipation of potential energy by civilization sustains internal reversible circulations within civilization that characterize all its activities.

The hypothesis that was made in Part 1 is that civilization is effectively a heat engine whose power can be represented in more human terms as economic wealth. Absent any energy consumption, civilization would necessarily decay towards an uncivilized, worthless equilibrium where the gradient ceased to exist and all internal circulations stopped. Wealth is able to grow only when net work is done to grow into the environment. Real economic production occurs when raw materials can be incorporated into civilization's structure faster than civilization decays (Fig. ). Growth at a net positive rate expands civilization's interface with reserves of energy. It enlarges civilization, creating new wealth and a greater overall capacity of civilization to consume and sustain internal circulations. Expressed as an integral, current economic wealth is the net accumulation of past net physical work and real economic production.

In effect, there is no intrinsic wealth in and of itself. Rather, as discussed in Part 1, wealth is built from a network of connections, from the product of a material length density and an energetic potential. Connections are what enable dissipative flows insofar as there exist potential energy gradients to drive the flows. For civilization as a whole, wealth is sustained by primary power consumption through the connections we have to reserves of fossil, nuclear, and renewable energy sources. Within civilization, the interpretation is that wealth is due to the connections between and among ourselves and our “physical capital”, and from the circulations along transportation, telecommunications, and social networks. All aspects of civilization, whether social or material, compete for globally available potential energy. Financial expressions of any element's value reflect the relative extent to which its connections enable civilization to irreversibly consume potential energy in order to sustain the reversible circulations of the global economy.

While energy consumption is required to economically produce and grow, a generally much greater amount is required to sustain circulations within the networks of connections that have accumulated from prior production in the past. An analog is an adult human. People's bodies are also a network of connections that have grown through childhood and adolescence. Far more of current daily food consumption goes towards maintaining life than to any extra weight gain. Similarly, value added to civilization through construction of a house decades ago still contributes to value today, by being part of a larger network that supports the daily rhythms of its inhabitants.

The analytical formulation of this hypothesis is that instantaneous power dissipation, or the rate of primary energy consumption a by all of civilization (expressible in units of energy per time or power), is linked through a constant λ (expressible in units of power per currency) to civilization's inflation-adjusted economic value (or civilization wealth) C (expressible in units of currency). Wealth is defined as an accumulation of the gross world product (GWP) Y, adjusted for inflation at market exchange rates (MER) . MER units are used rather than purchasing power parity units (PPP) since the focus is not on short-term inequalities between people and nations but rather the sum of all activities within the global economy with an eye to variability in the long run. Thus, a=λC=λ∫0tY(t′)dt′. Alternatively, and taking the derivative with respect to time, economic production is a representation of the growth of wealth: dCdt=Y, where, since a = λC, the production function is given by an increase in the capacity to consume energy: Y=1λdadt.

Crucially, Eq. () is a hypothesis that can be tested using available data. As described in greater detail in the Supporting Information of Part 1, GWP estimates from and the United Nations are used for historical estimates of Y. Estimates of the global rate of primary energy consumption a are provided by the US Department of Energy . Expressing a in units of watts, and Y in units of 2005 MER US dollars per second, then wealth has units of 2005 MER US dollars and the constant λ has units of watts per 2005 MER US dollar. What was shown in Table S2 of , and in graphical form in Fig. , is that, for the period 1970 to 2010 for which global statistics for power consumption are available, both a and ∫0tY(t) dt′ have risen nearly in lockstep. The mean value of λ relating the two quantities is 7.1 mW per 2005 US dollar. Even though the GWP more than tripled over this time period, from year to year, the SD in the ratio λ = a/C was just 1 %, implying an uncertainty in the mean at the 95 % confidence level of 0.1 mW per 2005 US dollar.

The constant λ is not derived from a correlation analysis (something that has been erroneously claimed by others, ), but instead it is obtained from the observation that the ratio of C to a has not changed from year to year even as C and a have. The observation is much like the basic expression of quantum mechanics where it was initially assumed, and then confirmed with measurements, that a photon's energy E and its frequency ν are linked through Planck's constant h. The empirical support for λ or h being effectively a constant stands on its own. But for the purposes of understanding the physics, the quantities they relate are not correlated but instead can be viewed as being interchangeable representations of the same thing.

The challenge might be to comprehend how a psychological construct like money could be tied to a thermodynamic construct like power through a constant. Economic value only goes so far as human judgement. Even with no one home and all the utilities turned off, a house still maintains some worth for as long as it can be perceived as being potentially useful by other active members of the global economy.

The interpretation might be that physical flows tie our brains to the global economy. Brains process a wealth of information from our environment using extraordinarily dense networks of axons and dendrites; patterns of oscillatory neuronal activity lead to the emergence of behavior and cognition; powering this brain activity requires approximately 20 % of the daily caloric input to the body as a whole . Perhaps dissipative neuronal circulations along brain networks reflect our collective perception of real global economic wealth. They march to broader economic circulations along global civilization networks that are sustained by a dissipation of oil, coal, and other primary energy supplies. Eqs. () to () may seem unorthodox by traditional economic standards, but there may be some basis for interpreting λ as a type of psychological constant that links the physics of human perception to the thermodynamic flows that drive the global economy.

As a point of comparison, traditional economic growth models are divorced from expressions of energy dissipation and physical flows, where wealth is expressed in terms of a physical capital, or as a stock that has an intrinsic value. New capital is produced using currently existing labor and capital, and production levels have no explicit dependence on external physical constraints . A more detailed outline and juxtaposition with the model here is described in Appendix B.

The field of macroeconomics is however making steps towards creating links with physics, pointing out that, along with labor and capital, energy must also be a factor of economic production . Quantified links between physical and financial quantities often rely upon a high observed correlation between national or sectoral economic production and energy consumption . A few economic growth models use these data to partially substitute energy for labor and capital as a motive productive force .

The model presented here differs, foremost because civilization is examined only as a whole, as an evolving organism whose growth is a response only to its changing ability to access external resources . Nothing is said about internal trade between countries. Neither is a distinction made between human and physical capital: the capacity to consume external reserves of energy is considered a complete substitute for both these quantities at global scales. This enables the model to be strictly thermodynamic, with no requirement for dimensionally inconsistent fits to prior economic data that are dependent on the time and place that is considered. The model's validity as an economic tool rests only on the observation of a fixed ratio between energy consumption a and the time integral of inflation-adjusted economic production C (Eq. ). The theoretical interpretation is that current energy consumption and dissipation sustains all of civilization's circulations, even human perceptions, insofar as they have accumulated through prior economic production.

The most direct implication of the existence of a constant value for λ is that economic wealth cannot be decoupled from energy consumption. For the past, reconstructions of global rates of energy consumption going back 2000 years are provided in Table S3 of . For the future, CO2 emissions will be inextricably linked to global prosperity for as long as the economy relies on fossil fuels . Increasing energy efficiency may be a commonly supposed mechanism for reducing energy consumption while maintaining wealth. However, as elaborated in Appendix B, this does not appear to be the case.

From Eq. (), the relative growth rate of civilization wealth C and its rate of energy consumption a are equivalent: rateofreturn=η=dln⁡adt=dln⁡Cdt. Effectively, like interest on money in the bank, the parameter η represents the “rate of return” that civilization enjoys on its current wealth C, and that it sustains by consuming ever more power.

Since 1970, rates of return for a and C have varied, but both have averaged ∼ 1.90 % per year (Fig. ). Substituting Eq. () into Eq. () yields a relationship between the rate of return and the inflation-adjusted GWP: Y=ηC=η∫0tY(t′)dt′, or η=Y∫0tY(t′)dt′. So, the current rate of return has inertia since it is tied to the past. It expresses the ratio of current real production to the historical accumulation of past real production.

The rate of change of civilization's rate of return can be referred to as an “innovation rate”: innovationrate=dln⁡η/dt. Referring to an acceleration term dln⁡η/dt as an innovation might seem a bit arbitrary. However, in Appendix B it is shown that it corresponds directly to more traditional economic descriptions of innovation such as increases in the “total factor productivity” or the “production efficiency”. For example, it is easy to show from Eqs. () and () that η = λY/a. Since λ is a constant, it follows that increases in the production efficiency (or inverse energy intensity) Y/a are equivalent to the expression for innovation dln⁡η/dt. Innovation is a driving force for economic growth and energy consumption since it follows directly from Eqs. () and () that the real GWP relative growth rate is governed by the relationship dln⁡Ydt=η+dln⁡ηdtGWPgrowth=rateofreturn+innovationrate.

The rate of return η is equivalent to the time integral of past innovations through η = ∫0t dη/dt′ dt′, and so Eq. () can be expressed as dln⁡Ydt=∫0t(dη/dt′)dt′+dη/dt∫0t(dη/dt′)dt′. The implication here is that current rates of GWP growth can be considered to be a consequence of past innovations (the first term) and current innovations insofar as they are not diluted by past innovations (the second term). The first term implies that current GWP growth rates will tend to persist because past innovation is carried to the present; the second term implies that new technological advances will always struggle to replace older advances that are already in place . Placing an internal combustion engine on a carriage was revolutionary for its time, but only a series of more incremental changes have been made to the concept since. Any new dramatic change has to compete with the large vehicular infrastructure that has already been put in place.

Figure  shows how rates of return, innovation rates, and GWP growth have been changing in recent decades based on the data sets provided in the Supporting Information of Part 1. The rate of return η generally has had an upward trend. In 2008, the rate of return on global wealth reached an all time historical high of 2.24 % per year, up from 1.93 % per year in 1990 and 0.71 % per year in 1950.

Meanwhile, innovation rates have declined. The rate of growth of the rate of return, or dln⁡η/dt, has dropped from around 4 % per year in 1950 to near stagnation today. Unprecedented gains in production efficiency that were obtained in the two decades after World War II appear to have since given way to much more incremental innovation.

From Eq. (), GWP growth is the sum of these two pressures. On the one hand, positive innovation has had a lasting positive impact on the GWP since it has led to an ever increasing rate of return η. On the other hand, innovation rates have declined. Figure  shows that, between 1950 and 1970, GWP growth rates were between 4 and 5 % per year. Since 1980, they have been closer to 3 % per year. Increasingly, the long-term increase in civilization's rate of return η has been offset by the long-term decrease in innovation dln⁡η/dt. The turning point was in the late 1970s, when, as shown in Fig. , innovation rates dipped below rates of return. Between 1950 and 1975, current innovation was the largest contributor to current GWP growth rates. Since then, continued GWP growth has relied increasingly on innovations made in the first two decades since the end of World War II (Eq. ).

For the special case that there is positive net convergence of matter in a system, the system grows (see Fig.  and Appendix A). It extends its interface with accessible reserves of energy and matter. An enlarged interface allows for faster rates of consumption. The result is a positive feedback that allows growth to accelerate. This is a basic recipe for emergent or exponential growth. One important aspect of this feedback, however, is that rates of exponential growth are never constant. Rather, they increase when new reserves of energy or matter are discovered and they decrease when there is accelerated decay.

The thermodynamics of this recipe were applied in Part 1 to the emergent growth of civilization and its rates of return on wealth . It was shown that the rate of return η can be broken down into the proportionality η∝(1-δ)ΔHRNS2/3eStot. Here, NS represents the amount of matter or mass within civilization. NS grows from a positive imbalance between civilization's incorporation of raw materials from the environment and civilization's material decay. δ is the decay parameter that accounts for how rapidly NS falls apart due to natural causes. ΔHR represents the size of the energy reserves that are available to be consumed by civilization. The term eStot represents how much of this energy must be consumed by civilization in order that raw materials can be added to civilization's fabric, thereby adding to NS. The exponent 2/3 arises from how flows are down a gradient and across an interface.

Building on the identity a = λC, it was argued in that Eq. () implies that rates of economic innovation can be represented by dln⁡ηdt=-2η+ηtechinnovationrate=diminishingreturns+technologicalchange. The first term represents a drag on innovation due to a law of diminishing returns -2η. The second term expresses a rate of technological change ηtech due to changes in δ, ΔHR, and eStot. A distinction is made here between a technological advance and an innovation. Technological change only counts as an innovation if it overcomes diminishing returns to lead to a real increase in the rate of return η.

A law of diminishing returns is a characteristic feature of emergent systems. As indicated by Eq. (), the exponential growth rates of larger, older objects with high values of NS tend to be lower than for smaller, younger ones. In our case, our bodies are a complex network of nerves, neurons, veins, gastrointestinal tracts, and pulmonary tubes. We use this network so that we can interact with a network of electrical circuits, communication lines, plumbing, roads, shipping lanes, and aviation routes . Such networks have been built from a net accumulation of matter. So, as civilization grows, any given addition becomes increasingly incremental.

The implication of Eq. () is that, absent sufficiently rapid technological change, relative growth rates η will tend to decline, and innovation will turn negative. For example, from Eq. (), innovation requires that ηtech > 2η. Or, by substituting Eq. () into Eq. (), an expression for GWP growth is dln⁡Y/dt = -η + ηtech, in which case maintenance of positive GWP growth requires that ηtech > η. That economic growth has been sustained over the past 150 years is a testament to the importance of technological change for overcoming diminishing returns.

The rate of technological change follows from the first derivate of Eq. (): ηtech=dln⁡(1-δ)dt+dln⁡ΔHRdt-dln⁡eStotdtηtech=ηδ+ηRnet+ηe technologicalchange=improvedlongevity+netreservediscovery+extractionefficiencygains. The first of these three forces is improved longevity. Suppose that civilization decays by losing matter at rate jd. At the same time, it incorporates new matter at rate ja, as discussed in Appendix A. A dimensionless decay parameter δ = jd/ja can be introduced that expresses the relative importance of material decay to material growth: if δ is zero, new material growth is not offset by decay. If δ declines then it is because new or existing matter lasts longer, representing an increase in civilization's longevity. For example, δ might decrease as civilization shifts from wood to steel as a construction material. Alternatively, it might increase due to more frequent natural disasters from climate change.

In Part 1, it was shown how the nominal GWP can be tied to the incorporation of new matter into civilization ja; the real inflation-adjusted GWP can be tied to the net incorporation of new matter ja - jd. This yields the interesting result that physical decay is related to economic inflation. At domestic scales, the so-called “GDP deflator” is often used as an analog for the annual inflation rate 〈i〉 since it represents the fractional downward adjustment that is imposed on the nominal GDP to obtain the real GDP. For civilization as a whole, the implication is that declining decay, or increased longevity, corresponds to a smaller GDP deflator, declining inflation, and faster real GWP growth, i.e., ηδ≃-d〈δ〉dt≃-d〈i〉dt.

The second force for technological change in Eq. () is discovery of new energy reserves. Where discovery exceeds reserve depletion, it accelerates economic innovation through an increase in the size of available energy reserves ΔHR . Energy reserves decline as they are consumed at rate a. Meanwhile, civilization discovers new reserves at rate D. The rate of net discovery is ηRnet=D-aΔHR. Provided that reserves expand faster than they are depleted, the rate ηRnet is positive. It represents a technological advance because there is reduced competition for available resources. From Eq. (), larger reserves enable higher rates of return for the relative growth of wealth and energy consumption.

The specific enthalpy of civilization eStot in Eq. () is an expression of the amount of power a that is required for civilization to extract raw materials and to incorporate them into civilization's fabric at rate ja. If the ratio a/ja declines, then civilization becomes more energy-efficient.

For example, mining and forestry is currently powered by large diesel engines rather than human and animal labor. Civilization is able to extract raw materials with comparative efficiency and lengthen civilization networks at a correspondingly greater rate. Using less energy, we are able to build more roads, lengthen communications networks, and even increase population, as we too are made of matter and are part of civilization's fabric. Where the extraction efficiency of raw materials improves, it is an effective force for technological change defined by ηe=dln⁡jadt-dln⁡(a)dt.

Equation () for innovation is logistic in form. That is, it could be expressed as dη/dt = ηtechη - 2η2 with a rate of exponential growth ηtech and a drag rate on growth of -2η. An initial exponential growth phase yields to diminishing returns where rates of return stabilize (Fig. ). If ηtech is constant, then the solution for the rate of return η is η(t)=ηtech/21+(G-1)exp⁡-ηtecht, where G=12ηtechη0 represents a “growth number” and the subscript 0 indicates the initial observed value for η0. The solution for η in Eq. () is sigmoidal. Provided G is greater than 1, rates of return initially increase exponentially and saturate at a rate of ηtech/2. So, for example, if ηtech is sustained at 5 % per year, then one would expect rates of return to grow sigmoidally towards 2.5 % per year. The characteristic time for the exponential growth phase would be 1/ηtech, or 20 years.

From Eq. (), the corresponding time-dependent solution for GWP growth assuming a fixed rate of technological change is dln⁡Ydt(t)=ηtech21+2(G-1)exp⁡-ηtecht1+(G-1)exp⁡-ηtecht. Here, GWP growth rates also saturate at a value of ηtech/2, but if G > 1 then this is by way of decline rather than growth. Thus, rates of return on wealth (Eq. ) and rates of GWP growth (Eq. ) should have a tendency to converge with time. This is in fact precisely the behavior that has been observed in the past few decades. Figure  shows values of η and dln⁡Y/dt that differ by about a factor of 4 in 1950 but that are approaching from opposite directions towards a common value of about 2.5 % per year.

Figure  shows the relationship between innovation rates and rates of return over the past three centuries derived from GWP estimates from and the United Nations , using Eqs. () and () (see Sect. 2 and the Supporting Information of Part 1 for methods and associated statistics). Rapid innovation and accelerating rates of return characterized the industrial revolution and the late 1940s. Periods of subsiding innovation followed 1910 and 1950.

Equation () implies that, if rates of technological change ηtech are roughly a constant, innovation rates dln⁡η/dt should be related to rates of return η by a slope of about -2; the intercept should be equivalent to the rate of technological change ηtech given by Eq. (). Focussing on the period since 1950, where statistical reconstructions of GWP are yearly and presumably most reliable , Fig.  shows that the past 60 years have been characterized by a least-squares fit relationship between innovation rates dln⁡η/dt and rates of return η (with 95 % uncertainty bounds) given by dln⁡ηdt=-(2.54±0.54)η+(0.06±0.01). Within the stated uncertainty, the observed slope relating innovation to rates of return is consistent with the theoretically expected value of -2 that comes from a law of diminishing returns. The implied rate of technological discovery for this time period ηtech is the intercept of the fit, or about 6 % per year. The magnitude of the difference of the fit from the anticipated slope might be an indication that ηtech has not in fact been a strict constant but rather has declined with slowly with time, as discussed below.

The second test is to approach the problem as a hindcast. A hypothetical economic forecaster in 1960 might have noted that the average values of η and dln⁡η/dt between 1950 and 1960 were 0.9 % and 3.3 % per year, respectively. From Eq. (), this implies that ηtech was 5.1 % per year during this period. Applying Eqs. () and (), and using an initial value for η0 of 1.0 % per year in 1960, the forecaster could then have obtained the trajectories for economic innovation and growth that are shown in Fig. .

Fifty-year hindcasts are summarized in Table  along with skill scores defined relative to a reference model of persistence in trends : Skillscore=1-|error(hindcast)||error(persistence)|. Skill scores are positive when the hindcast beats persistence in trends, and zero when they do not. For example, average rates of energy consumption growth in the past decade would have been forecast to be 2.3 % per year relative to an observed average of 2.4 % per year. Relative to a persistence prediction of 1.0 % per year, the skill score is 96 %. Alternatively, a forecast of the GWP growth rate for the first decade of this century would have been 2.8 % per year compared to the actual observed rate of 2.6 % per year. The persistence forecast based on the 1950 to 1960 period is 4.0 % per year, so the skill score is 91 %.

High skill scores suggest that it is possible to provide physically constrained scenarios for civilization evolution over the coming century using a simple logistic model given by Eq. (). There do not appear to be other macroeconomic forecast models that are equally successful, and in any case, macroeconomic models are not normally evaluated through comparisons to multi-decadal historical data. If a comparison with data is made, it is not in the form of a true hindcast. The model is judged by the extent to which a sufficiently complex production function can be tuned to provide an accurate fit to prior observations e.g.,.

Using only a fixed value for ηtech as input to the model presented here appears to work very well, at least for global scales. Still, a more fully deterministic model would not rely on an assumed value for ηtech, even if it is a fit to data prior to the date of model initialization. It is reasonable to anticipate that future rates of resource discovery and material longevity will evolve with time. Accounting for such technological change might prove an important consideration for economic and climate forecasters over the coming century.

To this end, the third test is to try to quantify the thermodynamic forces outlined in Eq. () that would enable a more first-principles estimation of the value of ηtech. Methods and data sets for estimating a time series for the sizes of energy reserves, the rate of energy consumption, the rate of raw material consumption, and economic inflation during the period between 1950 and 2010 are described in Appendix C and summarized in Table . Average rates are shown for three successive 20-year periods beginning in 1950, and for the 1950 to 2010 period as a whole.

What stands out in Table 2 is how there was unusually rapid technological change between 1950 and 1970. This period was characterized by rapidly growing access to reserves of oil, gas, and raw materials. It was followed by an abrupt slowdown in 1970 with no clear long-term recovery since. Summing over these forces, and averaged over the entire 1950 to 2010 period, rates of technological change ηtech are estimated to have been a respectable 3.5 % per year. Most of this growth took place in the first 20 years, when it achieved 7.0 % per year. The latest 20-year period averaged just 1.4 % per year.

Improved access to energy reserves and raw materials explains most of the variability in ηtech. Coal power production expanded steadily at a rate of about 2 % per year. Oil reserves, on the other hand, expanded at an average 3.6 % per year between 1950 and 1970 but shrunk at an average 0.7 % per year between 1990 and 2010. The amount of energy required to access key raw materials such as cement, wood, copper, and steel dropped by an average 3.5 % per year between 1950 and 1970, which implies rapid efficiency gains. Since 1970, energy consumption and raw material consumption have grown at nearly equivalent rates, implying no associated force for technological change.

As a check on the first-principles estimate that the average value of ηtech between 1950 and 2010 was 3.5 % per year, Table  shows the 20- and 60-year averages of η and dln⁡η/dt, and it uses these to derive a rate of technological change ηtech from Eq. (). (As a consequence of λ being a constant, calculated rates of return η are similar whether they are calculated from available energy statistics using Eq. () or from GWP statistics using Eq. (). Both have averaged 1.6 % per year overall.)

Innovation rates dln⁡η/dt have been positive overall, meaning rising rates of return. Still, they declined from 3.3 % per year between 1950 and 1970 to just 0.6 % per year between 1970 and 1990. Thus, the estimated average rate of technological change derived from Eq. () (i.e., ηtech = dln⁡η/dt + 2η) is 5.1 % per year, similar to what was derived for the 1950 to 1960 time period as discussed in Sect. 3.2. In comparison, the rate of technological change estimated from the physical parameters described in Table  averages 3.5 % per year, or about one-third lower. Whether the residual 1.6 % per year is due to data uncertainties or theoretical considerations is unknown.

The hindcasts in Sect. 3.2 assumed a constant value for ηtech, whereas the observed rates summarized in Table  point towards much higher variability. Perhaps the reason a constant value nonetheless leads to hindcasts with high skill scores is because there is a timescale of decades for externally forced technological change to diffuse throughout the global economy e.g.,. Assuming a fixed value for ηtech represents this timescale by smoothing the economy-wide impacts of a large impulse of innovative forces that occurred between 1950 and 1970.