Combining thermodynamic laws with multi-agent models:

Velocity of trade phase transitions in economic organization

Draft ˝ not for citation

Douglas R. White March 7, 2000

The physical sciences provide solutions to many of the problems of understanding the principles of self-organizing systems. Thermodynamics and the physical principles of self-organization are among the most difficult concepts for social scientists to understand but are also among the deepest and the most useful physical concepts for social science research.

This paper follows a research strategy advocated by Soodak and Iberall (1987), two physicists of complex systems. As elaborated by these authors, the basic experimental procedures used in the sciences are no different in principle than those for the social sciences, and the underlying physical laws that govern all systems can be translated into theoretical and explanatory frameworks for the social sciences, subject to experimental verification. One thing to pay attention to is how to give a physical description of something not in terms of its attributes but its processes. Typically, processes occur as a function of gradients from point to point in some field, where interaction leads to the equipartition of energies. For example: a moving billiard ball hits a stationary one and the combined momentum is equipartitioned, the moving ball becoming slower, the stationary one absorbs part of its momentum (total momentum is conserved, minus friction). But unlike billiards, in a complex system the process description must include the internal processes of the actors.

Structure Formation

A central task of the social sciences is to understand the structures and processes that are constitutive of society, polity, economics, and culture -- of human life in general. The great contribution of systems physics as developed by Soodak and Iberall (following standard physics applied to stacked or hierarchical systems as actually observed in our universe) is to understand how "structure is laid down, maintained, changed, and degraded by process" and conversely how "process is guided and constrained by structure."

In physical systems, structural changes at a micro level (addition of elements or actors, interactions, energetic inputs, changes in velocities, etc.) are defined as incremental if they lead to no changes in macro behavior, and as phase transitions when they lead to discontinuous macro behavioral change, such as the major change in entropy (see definition below) between a solid and a liquid state. The contribution of Iberall and Soodak (1987), specifically, is a theory of phase transitions as a generalization of the physical theory of energy and material flow dynamics.

The fundamental process in flow dynamics is conservation of energy and matter through motion of particles (atomisms at whatever scale). This follows from the first (Conservation) law of thermodynamics: the sum total of the energy in a system and its surroundings, over time, is constant. Thermodynamics (thermo=energy, dynamics=change) is the study of the patterns of energy change given some boundaries around a "system" that separate "system" from its surroundings, with the following definitions:

Energy coming into a system must be conserved either as heat (increased diffusive motion of particles) or as work (motion in an oriented direction), or dissipated through vibratory propagation (e.g., light, sound). There are only three possible ways for energy or materials to move:

Diffusion and convection correspond to the definitions of the two forms of energy:

Heat (q)=exchange of thermal energy (stored as kinetic energy of atomisms, i.e., movement, where the "heated" or diffusive movements of atomisms are uncorrelated, net undirected)=the capacity to diffuse kinetic movement via collisions from "hot" bodies to "cold" bodies in a unit of time; and

Work (w)=net directed (convective) movement of matter from one location to another = external force x distance moved.

Work requires an external force (and thus energy from the surrounding environment). Heat, on the other hand, tends to equipartition thermal energy via diffusive collision and thus tends to thermal equilibrium: within a system at equilibrium there are no further gradients to do useful work. This follows from the second law of thermodynamics, which is satisfied experimentally in all known physical systems: The sum of energy available for work in a system and its surroundings never increases. Hence a physical system is said to be at equilibrium when all of its internal motions occur by diffusion (random movements into available degrees of freedom).

In physical systems it is experimentally observable that for a kinetic system at equilibrium, in which motion occurs by diffusion (random movements into available degrees of freedom), relatively small energy (e.g., thermal) gradients from the outside will not produce convection but only an increased rate of diffusive motion. This macroscopic behavior of the system can be observed to change, however, when a threshold is reached in terms of the quantity of outside energy to be absorbed. The quantitative change in the phase transition is the emergence of convective flow. As seen at the bottom of a kettle of water put to the stove, the equipartition of thermal energies in the early stage through diffusion gives way to the repartition of energies through convective currents. The reason for this change is that the capacity of individual atomisms (molecules) to repartition energies as they are heated (acquiring higher local velocity) will result, due to the slowness of Brownian motion, in local inhomogeneities that create the internal gradients for convective flows: coherent ensembles or groups of atomisms begin to move convectively to restore thermal homogeneity. These flows are governed by the non-linear Navier-Stokes field equations of extended thermodynamics (the equations of fluid dynamics) that are not fully solvable analytically. They do predict, however, both the threshold value at which convection begins, the rate of convection, and that the convection will be directed or orderly until the next energy threshold is reached, after which turbulent flows will begin. The underlying thermodynamics is thoroughly consistent with experimental observations.

The Reynolds number of a fluid system is a dimensionless ratio of observed gradients acting externally on a system from its surrounding (e.g., heat incoming from some point on the system's boundary), to the capacity of the system atomisms to absorb such gradients (energy, matter, momentum, charge, angular momentum) and transmit or dissipate them locally by increased random (Brownian) motion that, through local collisions, will diffuse these gradients omni directionally throughout the system. When Reynolds ratio reaches 1, this capacity to absorb gradients kinetically is exhausted, and a phase transition to convection emerges to transport the gradients in a directed fashion away from those parts of the system boundary where the gradient inhomogeneities are created.

For external gradients of momentum flow (mass x velocity) of a given magnitude at the boundary of a system, the Reynolds number is the ratio of the velocity V of the incoming flow over the velocity vd of local diffusion by which the gradient is kinematically dissipated. Assuming that momentum is equipartitioned between colliding atomisms and not time-delayed by absorption into the internal processes or bulk viscosity (l) of the atomisms (Soodak and Iberall 1987:463), local diffusive velocity is vd = m /r D, where m (mu) is the (shear) viscosity, in units of force/velocity, the mass that can be moved per distance traversed in time in random collisional movement (mass/length D*time), r (rho) is the mass density of the atomism in terms of mass number per volume D3. Hence the Reynolds number,

Re = Velocity V (convection) / Velocity vd (diffusion) = V/(m /Dr )=DVr /m (1)

The threshold effects of the Reynolds number (Iberall and Soodak 1987: 462-464), are such that if

Re < 1, then the diffusion time can handle convective flow, and there is no

phase transition

Re = 1, then a transition is reached from microscopic transport to macroscopic


Re > 1, then the diffusion time is too short to handle convection, and

structural change occurs in the form of convection

Re >> 1, then turbulence begins.

Reynolds equations, as derived from Navier-Stokes, also predict the number and size of the fluid cell organizational units that emerge in the macrostructure of diffusive flows following phase transition. For stability at a Reynolds transition, the number N of fluid cells or atomisms with diameter Do that emerge into a superatomism with diameter D on a surface (where Do = D / Ö N) is computed as follows:

For 1 < Re = DVr /m , and substituting D= DoÖ N,

1 < Re = Do Ö N Vr /m , and solving for N, then

N > (m /r VDo)2, (2)

where N is the predicted number of atomisms in the superatomism.

Momentum may not be fully equipartitioned between colliding atomisms but also time-delayed by absorption into the internal processes or bulk viscosity (l) of the atomisms, so that the local diffusive velocity is vd = (m +l)/r D (Soodak and Iberall 1987:463). The Reynolds number takes this into account in the denominator:

Re = Velocity V (convection) / Velocity vd (diffusion) = DVr /(m +l). (3)

The kinematic viscosity n (nu) = m /r is the area traversed in Brownian motion per unit time. An equivalent Reynolds equation is thus:

Re = DVr /(m +l) = DVr /m (1+l/m ) = DV/ n (1+l/m ). (4)

The predicted number N of atomisms in the emergent superatomism for the case where atomisms have bulk viscosity (equation 5) will be larger than for the case where bulk viscosity is zero (equation 2):

N > ((m +l)/r VDo)2. (5)

This equation implies that as bulk viscosity (l) increases, the increase in number of atomisms in a Reynolds transition will asymptote to a linear function of l as the ratio (m +l)/l approaches unity.

(For a summary see: Physics of Phase Transitions Douglas R. White)



Some basic scientific foundations and research strategies for the study of

generalized phase transitions governed by a Reynolds number

Iberall and Soodak's (1978) claim is that generalized Reynolds laws --the emergence of structure out of dynamics -- govern phase transitions in all systems, and that this can be confirmed by observation. To ground this claim, they begin with observation that systems with persisting macrostructure must obey the laws of thermodynamics, in particular those that are capable of the work involved in maintaining internal structure.

Definition. Thermodynamic engine: A (non isolated) system where energy is transferred from the surroundings to do work. E.g., when heated gas in a chamber moves a piston, the work done is PV (pressure x volume).

The energy in a thermodynamic engine is sometimes described by its enthalpy H= E +PV where -PV is the work done and +PV is the correction for energy used in doing work on the surroundings. H is the capacity of energy differentials to diffuse heat (relative to a surrounding) or to do work. (Because thermodynamic engine processes are often cyclical, and depend on discrete quanta of energy per cycle, H may be quantized).

Energy and enthalpy are state variables of a system. If heat and work are the only forms of energy transferred between a system and its surroundings in a closed system, then the first law states for enthalpy: E2-E1 = D heat + D work = q + w, where D is the symbol for change in a quantity.

Thermodynamic (Carnot-Clausius) Entropy (S @ -w) is the quantitative measure of thermal energy not available to do work (the opposite of w, the energy available to do work). Clausius noticed a certain ratio that was constant in reversible (ideal) heat cycles studied by Carnot, and in 1865 named this ratio "entropy" after deciding that it must correspond to a real physical quantity.

Configurational (Boltzmann-Gibbs) Entropy (S) is a measure of disorder or randomness in a closed system. Boltzmann resolved the apparent contradiction between gas laws at the molecular level, where an elastic collision between molecules would look the same going forward or backward in time, and the second law, which seems to imply irreversibility at the macro level. Processes that occur within a sufficiently short time so that entropy is constant are reversible. His model of how heat is evenly diffused through a gas also showed for the mixing of two gasses that the same processes would lead to thorough mixing. This led Boltzmann to define the "disorder" of a system, such as a solid, liquid or gas, as "the number of ways [W] that the insides can be arranged, so that from the outside it looks the same. The logarithm of that number [ln W] of ways is the entropy" (Feynmann 1963 vol. 3 section 46-5). There are no physical units for this kind of entropy, but Boltzmann defined a constant k for gasses, that relates absolute temperature to the average kinetic energy of a molecule -- experimentally determined to be a universal constant -- so as to define configurational entropy as:

S = k ln W.

Since k is constant for all molecules, if we know thermodynamic entropy S, then Boltzmann's equation can be used to solve for W, the number of microstates to which a system can transform in a given phase or state, and checked experimentally. Thermodynamic entropy and configurational entropy can be equated for physical domains without violating any known physical laws or experimental results.

An extension of the second law is that configurational entropy in a system and its surroundings never decreases. A corollary is that a system can lose configurational entropy (become more ordered or organized, etc.) but only by an increase of configurational entropy in its surroundings.

A system that is at (thermal) equilibrium internally and with its surrounding is incapable of doing work, since there exist no gradients to external forces to operate. Any system that is not at thermal equilibrium has organized gradients of energy, and these gradients necessarily depend on energy inputs from its surroundings.

It is the organization (organized gradients) of energy and matter differentials that determines what gets done within a system: "process is guided and constrained by structure; structure is laid down, maintained, changed, and degraded by process" (Soodak and Iberall 1987: 459-460).

Systems can be stacked, as shown in Figure 1: system S1 exchanges energy with its "surround" S2, and system S2 exchanges energy with its "surround" S3. The universe is full of embedded systems, like Chinese dolls.


example: S3 solar system, S2 earth system, S1 life system

Figure 1: Embedded Systems

Soodak and Iberall (1978) observe that a hierarchical stacking of systems into levels is organizationally consistent with the 2nd law of thermodynamics. In embedded or stacked thermodynamic systems (as illustrated in Figure 1), as the more macroscopic systems dissipate, they release thermal energy (no longer capable of doing work at that level) that represents rising entropy at that level but which at more micro levels within that system can be converted into organized thermodynamic gradients that can be harnessed (by generalized Darwinian selection) into engine processes. Once an engine process begins, it is possible that it will replicate itself for a longer time period (temporal "survival" plus reproduction of number equating in Darwinian terms to "fitness").

The hierarchical stacking of systems is the key to understanding evolution. All kinds of embedded systems evolve, living and nonliving. Life is a special case of "self-organizing" system where it is thermodynamic stacking that is indispensable to self-organization.

Thermodynamic engine processes must be cycled, with energetic "kicks" from the environment, to operate near-equilibrium at various time scales for a system to survive. Because thermodynamic engine processes must be stacked in order to evolve, they will have a spectrum of time scales of internal processes. The "factory day" cycle of repeated processes (Soodak and Iberall 1987:461) at a given level will be associated with an emergent complex atomism.

The complexity of a system or complex atomism is the ratio of internal/external process time (Soodak and Iberall 1987: 461), measured by the ratio of bulk to shear viscosity, l/m . The bulk viscosity (l) of a complex atomism, or transport of action in or out of its interior, is necessarily very high: "Complex systems tend to display a cascade spectrum of many relaxation processes" (Soodak and Iberall 1987: 462). To recap, within complex atomisms there are (typically a cascade of) thermodynamic engine processes driven by external energetic inputs and internal dissipation of gradient energies towards equilibrium interrupted by new external energetic inputs.

Reynolds transitions in the organization of momentum, and analogous transitions in energy, mass, or charge, are ubiquitous in evolutionary processes, and are capable of creating bulk viscosity and stacked complexity out of simple systems. Some such transitions are easily or quickly reversible, depending on which structures of bulk viscosity or complexity are not stabilized by bonding. There is no necessarily linear trend overall towards greater complexity in evolution, but such trends, among others, may be observed locally at various temporal and spatial scalings. Although increases in complexity are not a necessary outcome of evolution they may be identified empirically when they do occur if we have sufficient knowledge of dynamics and of how structure is stabilized.

How structure is stabilized

Two additional conditions for an emergent structure to form a stable association (Soodak and Iberall 1987: 465): (1) rapid transformation of binding energy, eliminated from the structure (given off as energy; energy must be put in to loosen the bonds), and (2) binding energy must be large compared to the energy of interaction between the structure and external agents (i.e., cannot be easily broken apart as incoming energy loosening bonds).

Chemistry involves the study of bond energies (enthalpies) that are released when a bond is are broken and consumed when a bond is made (changes of state). For chemical transformations at a constant (e.g., atmospheric) pressure, the change in enthalpy will be q; typically, some quantum of energy.

Bonding and stacking energies play complementary roles. In a chemical phase transition, bonding energies are taken out of a system, as in changes from gas to liquid to solid; energies that must be put back in to break the bonding. Correlated associations (dynamically emergent macrostructure), on the other hand, emerge from higher energetics and collapse when energies are taken out.

Bonding into stable structures, for the examples of human populations, is a separate problem not treated here, but would predict, for example, the nucleation of a population within a dense nucleus of a territory. The Reynolds model of protourban trade for example, would predict densities for superatomisms, but given bonding these would be nucleated into about 16 independent protourban centers, each of which would have about 1/16th of the population predicted from densities at the larger territory. I am currently stumped, however, in figuring out how to convert densities predicted in the model (per cubic volume) into densities per surface area to make comparisons with empirical data, but we know from archaeological sites (e.g., Catal Huyuk, Mesopotamia; early Uruk, Mesopotamia; Ch'eng-tzu-yai, China) that the earliest protourban settlements had about 6000 people in nuclei of about 50 acres (.08 sq. miles).

Phase transitions governed by a generalized Reynolds number

Thermodynamic laws allow the energy available to do work in a system (and the organization of a system, measured by its complexity, l/m ) to increase, but only because energy is flowing into the system from its surround (example: energy is dissipated from the sun as light that creates energy gradients available to do work on earth). Iberall and Soodak (1978) argue that this is key to understanding organization generally, to understand human society and culture, and to understanding how to develop a social science grounded in scientific experiment and observation. In extending thermodynamics to society, Soodak and Iberall (1987: 467) begin their argument as follows:

1. The organized activities of the society must be described in terms of macroscopic coordinates.

2. That description must be related to the microscopic properties and kinetics of the atomistic units, the individual persons making up the societyÍ.

"[M]acroscopic coordinates and their interactions... are emergent properties, arising from the kinetic behavior; and they represent ... constraints on the kinetic behavior. Thus the micro- and macro-levels are mutually linked." The macro behavior of the field, in turn, "is constrained by boundary conditions from outside the system ... [which] often originate from a higher-level system of which the macroscopic system is simply one of the atomistic units" (pp. 460-461). Micro level kinetics -- what can move where and interact with what in different states (e.g., ice/water/vapor) -- is what gives rise empirically to the dynamic behavior (and entropy) of a field of interacting atomisms.

The study of generalized Reynolds phase transitions for human society is consistent with their requirement that highly coordinated states of systems with macrostructure are not irreversible but depend for stability on dynamic throughputs and/or bonding. As more energy is added to a system, transitions to more coordinated states (departures from dynamic similarity) do not imply greater freedom at lower atomistic levels in the systems-hierarchy, such as at the level of individuals in population systems, but greater coordination of trajectories in convective-cell movement.


Iberall and Soodak (1987) illustrate phase transitions in fluid flow (pp. 503-504), matter condensation (pp. 506-507), chemical patterns (pp. 507-508) and social patterns such as the transition to urban nucleation (p. 508). The fluid flow example follows the standard Reynolds equations discussed above. The chemical phase transition example involves diffusion of autocatalytic chemical reactions that can lead to the type of flow patterns described by Prigogine and chemical engineers. The matter condensation (e.g., gas to liquid to solid) example, however, requires two simultaneous parallel processes, one of matter inflow and the other of energy outflow of the latent heat involved in phase change. They apply Reynolds transitions to the transition to protourban settlements ˝ and to complex systems generally ˝ by describing how system complexity (the ratio of internal/external process time) adds the same term to the denominator of the Reynolds equation as for any system with bulk viscosity (l), "to indicate that it is energy both in translational and long time delayed internal motions that can be absorbed in the local domain" (Iberall and Soodak 1978:18). Hence they start from Reynolds equation (4):

Re = DVr /(m +l) = DVr /m (1+l/m ) = DV/ n (1+l/m ). (4)

where lambda/mu (l/m ) is the complexity ratio of bulk to shear viscosity in the atomisms that calibrates the long time delay of internal compared to external motions. For the motions of human individuals l/m is made up of the product of an energy DEl/Em and time tl/tm bound up in socially involved modes of action:

l/m = (DEl/Em)(tl/tm), (5)

where DEl/Em ¬ 1 (energy tied up and released in internal modes over energy intake, as compared with a minor amount taken up by repair activities), and tl/tm is the time tied up in the maintenance of customs and beliefs, which throughout most human history they estimate to be at the order of a generation, a constant of tl/tm=7000 days/day.

For stability at a Reynolds transition from neolithic villages to protourban settlements, the number N of village atomisms with diameter Do that emerge into a protourban superatomism with diameter D on a surface (where Do = D / Ö N) is computed as follows:

For 1 < Re = DV/ n (1+tl/tm), and substituting D= DoÖ N,

1 < Re = DoÖ N V/ n (1+tl/tm), and solving for N, then

N > (n (1+tl/tm)/VDo)2 (6)

The first row of Table 1 shows stability transition calculations used by Iberall and Soodak (1978:18) to compute constellations of superatomisms ˝ the proto-urban agricultural settlements engaged in trade ˝ emergent in the post-Neolithic. Since tl/tm is sufficiently large that tl/tm ¬ (tl/tm+1), the expression Ö N DoV/n(tl/tm) is the approximate ratio of normalized convective trade velocity V to normalized diffusive trade velocity vd, which cannot exceed Re > 1 without a phase transition. Here, the minimum number of interacting centers following a phase transition is predicted to be N ¬ 16. The kinematic viscosity n (nu) is .11 sq. miles/day, for random moving about in an area of about 8000 sq. feet (the size of a very large house or physical marketplace). At the scale for protourban trade of moving about randomly in a face-to-face marketplace in a day, the diffusive velocity vd is n/Do¬ 1/365 miles/day=14 feet, which seems too little until one considers that this is the consequence of random or Brownian motion, and that the yearly rate of such random movement is about 1 mile/year, or 20 miles/generation, in which case the location of an average "merchant" is spatially dislocated from town of origin to a neighboring town in about 3-4 generations. The predicted diameter of the trading system is 160 miles. Each of these derivations from first principles are plausible estimates, consistent with archaeological findings (Iberall and White 1988). The model scales the underlying phenomena at the proper ratios.


Table 1:

Stability Transitions

Generational relaxation

time tl/tm


Convective velocity V (miles/day)

[trade, husbandry]

Diameter of cell Do (miles)


viscosity n

(miles2/day) [Brownian motion]

Re > 1

¬ Ö N(VDo/n (tl/tm))


N > (n(tl/tm)/DoV)2

minimum number of superatomisms

N Diameter

¬ D ¬ (miles)

Protourban Trade


5 m/d

40 m


=.11 m2/d



16 160 m

Animal Husbandry


8 m/d

200 m


=.33 m2/d



6 500 m

Row 2 of the table repeats these computations for a hypothetical transition to animal husbandry, with larger cell diameters (size of territory occupied) and higher convective velocity and kinematic viscosity n. Here, the predicted minimum number of interacting centers is N ¬ 6 in a trading zone of 500 miles diameter, and the diffusive velocity vd = n/Do for semi-nomadism is approximately 9 feet/day. Given the larger diameters of semi-nomadic society, the average individual is less likely to change membership from one cell to another in a generation, and contacts with other cells are more limited to a first-order zone of contact rather than first and second order zones as for protourban trade.

Economic organization and the velocity of trade

The thesis here is that relative price stability is a precondition for long term (often incremental) increases in the velocity of trade (amount of trade per unit time, over an appropriate spatial scale), and that phase transitions in market systems with pricing mechanism are subject to Reynolds thresholds in which the velocity of international trade exceeds the maximum velocity of local markets, i.e., the capacity of local level market or trading systems to handle a global volume of trade per unit time. The result of surpassing these thresholds is the reorganization of the local market and organization of trade in terms of creating additional levels of organization that can move more goods per unit time.

Figure 1, from Fischer (1996), shows the price of consumables in England between 1200 and 2000, chosen to be roughly descriptive of the general patterns of time-series in prices in Europe over that time period. For our purposes, it is the relatively flat periods of price stability that are of interest, as these are hypothesized to be the periods in which trading relations expand internationally, and the volume of trade increases per unit time, both globally and within the settlements (starting as towns and growing into cities) in which trade is conducted.

Legend: Arrows shown in the web page below represent periods of equilibrium with increasing velocity of exchange, thought to result in phase transitions to more complex structures.

Figure 2: The Price of Consumables in England (adapted from David H. Fischer, 1996. The Great Wave: Price Revolutions and the Rhythm of History. Oxford and New York: Oxford University Press. paperback ISBN: 019512121X.)

The arrows superimposed over the periods of price stability in Figure 1 are intended to suggest that these are the periods in which the average velocities of trade are steadily increasing until, according to our thesis, they reach a Reynolds threshold. We conjecture that at the end of each of these periods of stability, a phase transition occurred that created new levels of organization (macro-structures, such as physical markets, new types of market organization, new types of trading organization, corporations, and so forth). We also posit two earlier equilibrium periods known prior to those of Figure 2, one in the 10th century, and one in the 12th.

The type of correlated events that might be associated with these phase transitions due to the velocity of trade are as follows

The first of these periods is not covered by Fischer (1996) due to lack of adequate price data, but is one in which semiautonomous trading towns developed, and following a period of relative stability and expansion of trade, the institution of Law Merchants was first developed.

The next period in which prices remained comparatively stable for a long period (Fischer 1996:16) was one in which "families, cities, markets, gilds, and fairs multiplied everywhere" and "growth of population and the increase of wealth were roughly in equilibrium." The Űcommunal movementÝ towards town autonomy culminated in development of autonomous Republics formed in the market towns of wealthy merchants, including such towns as Venice and Florence. Paris in this period became a city of 50,000. Trade expanded through new institutions of Market Judges and enhancement of the Merchant Code laws.

In the absence of centralized state system with police power and authority over a sufficiently wide geographical range to enforce commercial contracts, the transition from a reputational system for enforcement of honest behavior among European traders in this early Medieval period, to a system of judges, is an excellent candidate for modeling a Reynolds phase transition. "The role of the judges, far from being substitutes for the reputation mechanism, [was] to make the reputation system more effective as a means of promoting honest trade" (Milgrom, North and Weingast 1990). As trade intensified, "in a large communityÍ it would be too costly to keep everyone informed about what transpires in all trading relations, as a simple reputation system might require." "Intuitively, the system of private judges accomplishes its objectives by bundling the services which are valuable to the community, so that a trader pursuing his individual interest serves the communityÝs interest as well" (NMW p. 3). Those are exactly the kinds of dynamical processes we expect in a Reynolds phase transition. Are the spatial scaling and number of initial units in the emergent organization predicted by the Reynolds transition?

How does this Medieval phase transition compare in scale with that of the earlier protourban transition from Table 1? Table 2 assumes that human beings remain the same, that cities expand their cell size by a factor of 5 (to a 100 mile radius), and that local diffusional or kinematic viscosity n has increased fiftyfold to about 5.5 sq. miles/day (or 2000 square miles per year), for random moving about in an area the size of a small city, or commercial ward of a city. The predicted number of cities in the new trading constellation is on the order of N ¬ 60. At the scale for urban trade of moving about randomly in commercial district in a day, the diffusive velocity, n/Do ¬ .0275 miles/day=145 feet, which again seems like little until one considers that this is the consequence of Brownian motion, and that the yearly rate of such random movement is about 10 miles/year, or 200 miles/generation, in which case again the location of an average merchant is spatially dislocated from city of origin to a neighboring city in about 3-4 generations. These are again plausible scaling numbers, in spite of some lack of precision in the measure of kinematic viscosity n.


Table 2:

Stability Transitions

Generational relaxation

time tl/tm


Convective velocity V (miles/day)


Diameter of cell Do (miles)


viscosity n

(miles2/day) [Brownian motion]

Re > 1

¬ Ö N(VDo/n (tl/tm))


N > (n(tl/tm)/DoV)2

minimum number of superatomisms

N Diameter

¬ D ¬ (miles)

Protourban Trade


5 m/d


40 m


=.11 m2/d



16 160 m


Urban Long Distance Trade

7000 late


25 m/d


200 m


=5.5 m2/d



60 1600 m


Proto Mercantile Trade



75 m/d


500 m


=11 m2/d



4 2000 m

multi-ethnic states (early modernity)

Overseas Colonial Trade



150 m/d


500 m


=27 m2/d



6.3 3150 m

great-city based


Proto Industrial Trade



225 m/d


500 m


=41 m2/d



6.5 3200 m

colonial systems

Globalized Trade


21st C.

1000 m/d


500 m


=137 m2/d



3.7 1850 m


The next period of comparative price stability did not come until the 15th century, and one of the great financial inventions at the end of the Renaissance period of expanding trade was that of finance capital (e.g., in Florence) along with the growth of international banking. The Renaissance expansion of trade sparked a revolution of state building. In Poland, Russia, Hungary, France and England unified national (but initially multi-ethnic) states emerged. A new multinational state was created in Spain, and a new multinational Empire in Turkey. It was the possibility of capital investments in trading expeditions, partly organized by the crowns, but financed by banking capital, that fueled the development of trading companies (another invention) along with their expansion into the spice trade in Asia, and the accidental discovery and development of trading colonies in the New World.

Another period of comparative price stability and stable growth came in the late 17&18th century along with the Enlightenment expansion of trade. Not only did commerce flourish, but also exchange markets in commodities and financial instruments were constructed throughout the Western world. "Labor markets, capital markets, and markets, commodity markets all were made to work more efficiently. Banks multiplied rapidly," and the expansion of roads, canals, bridges and ports was extensive. "The growth of colonies also increased the wealth of Europe and improved its productivity. Marginal returns to capital and labor tended to be higher in the staple industries of colonial economies than in the mother-country" (Fischer 1996:110). The great cities prospered, such as Paris, London, Berlin, and Vienna. A "change of phase" to political stability came to France, England, Germany and Russia (p. 111). Economic policies of the state oscillated between mercantilism, urging an active intervention in economic processes, and laissez faire.

In the Victorian equilibrium "the relative returns to land, labor, and capital were much the same Í as they had been during the Renaissance and the Enlightenment. They were also similar in their social results. In the middle and later stages of every price equilibrium Í the distribution of wealth tended to stabilize, or even to become a little more equal" (Fischer 164). For the first time during such equilibrium, however, population began to rise exponentially while prices, instead of rising, remained stable; an economic revolution had taken place (Fischer 1996:160 citing Wrigley and Schofield (1989:402-412). The invention? The first high growth economic systems, based on a revolution in transportation (railroads, and their connection to shipping), in which rates of growth in population were tuned to price movements in the economy (Fischer 166). "An industrial revolution increased the productivity of labor and capital. A commercial revolution radically improved the efficiency of exchange" (168). "Perhaps the most important factor was the integration of a world market through the nineteenth century, which created vast economies of scale," (168) and included highly newly productive regions of unprecedented growth in commodity production. The transition in scale and scope of modern industrial capitalism described in Chandler (1990, Chapter 1) also has the markings of a Reynolds transition in corporate organization. Are the spatial scaling and number of "initial" emergent units in the corporate revolution predicted by the Reynolds transition formula?

I have only sketched some figures in Table 2 that might be indicative of Reynolds transitions for the Renaissance, Enlightenment, Victorian and 21st century (if any) stable trade expansions, but unlike the revolutionary change from town to city reorganization, the modern period of state formation would seem to be one of greater and greater complexity (higher bulk/shear viscosity ratios) of internal structure rather than one of expanded superatomisms. If anything this suggests a transition from cities to multi-ethnic "nationalizing" states to the breakup of states into more "nationally" homogeneous units, rather than transition to a unitary world order. (Parenthetically, such a revolutionary open parenthesis of early modernity is consistent with Toulmin's (1992) Renaissance thesis, while the closed parenthesis of late modernity might be consistent, in the contemporary era, with the spilling over of urban areas into suburbs less directly involved in long distance trade.) While I am by no means sure of the estimates on which such devolutionary trends might be based, results of this form would be possible if the pace of local kinematic viscosity n (local transports) outstrips that of square root of convective velocity. This suggests the utility of a plot of local transport viscosity against long distance velocities in the evolution of transportation systems.

Multiagent models of the phase transitions

The thermodynamics of phase transitions provides an account of atomistic reorganization into coordinated states with macrostructure, given increases in external energies and materials, in this case velocities of trade. Both the accounts of responses to external pressures and the initial rescalings are remarkable consistent with principles enunciated in organization theory and findings of organizational histories (e.g., Chandler 1990).

What the physics omits is the detailed characterization of how the atomisms ˝ in this case, human beings ˝ carry out the activities of trade within certain temporal and spatial scalings, and how they achieve the reorganizations in response to velocities of trade that are at the limits of their current organizational phase. Consistent with the physics, we look for the gradients that drive individuals, and we find multiagent interaction models, given diverse goals of different agents, to provide a consistent modeling strategy. Tinkering and experimentations with new rules and strategies by individuals or interacting sets of agents provide an additional gradients associated with new organizational forms. In the Champaign Fair example, Milgrom, North and Weingast (1990) use a multiagent game theoretic model (using the prisonersÝ dilemma game) of the Law Merchant System where there is a private judge, the "Law Merchant," who both adjudicates disputes between players and reports whether players are in good standing. They show that individual profit seeking gradients lead towards acceptance of a new form of organization that handles an increased velocity of trade.

Examining the details of their model, in the earlier period prior to the end of the 12th century (a long period of price stability and expanding trade), a simple system of reputations served as a selection mechanism for honest trade (along with informal merchant law regulating contracts but often lacking means of legal enforcement). Reputation was spread by informal diffusion across trading networks in which the velocities of trade gradients was growing. They argue that the formal system of "Law Merchant" judges that evolved in private law, independent of states or state enforcement of contracts (especially in the case of merchants from distant places), "was a natural outcome of the growing extent of trade. "[T]he system of private judges that evolved by trial innovations (and changing gradients for incentives to use these judges under expanding velocities of trade) was able to "transmit just enough information to the right people in the right circumstances to enable the reputation mechanism to function effectively for enforcementÍ. "Briefly, the costs of making queries, providing evidence, adjudicating disputes, and making transfer payments must not be too high relative to the frequency and profitability (italics mind) of trade if the system were to function successfully" (p. 3).

Further, "the system of private judges accomplishes its objectives by bundling the services which are valuable to the individual trader with the services that are valuable to the community, so that a trader pursuing his individual interest serves the communityÝs interests as well." It is by virtue of this physical bundling of activities and information into a vector that was easily convected from site to site ˝ by means of the findings of judges being transported from site to site along with the identity papers of the trader ˝ such that to enter the trade site itself, the trader had to match his identity with a clean legal bill of health already established by the system of judges.


Renormalization is a physical term for taking into account that interaction effects are not continuous in a physical space, and for rescaling a physical theory when interactions at short distances are to be blocked out within short distances because these distances represent the internal diameters of the atomisms. I use the term here because once a macrostructure has emerged, such as a new form of organization; it is the longer-range interactions between these macro units (and not the shorter internal processes that are bound up within them) that need to be renormalized.

After a phase transition to correlated associations, we are left with a new array of dynamically emergent macrostructures that follow a new set of processes of interaction, including the growth of macrostructures into a new distributions of sizes and typical interaction profiles (including chemistries of macro structural bonding and thus "hardening" of structures). Typically, these new heterogeneities entail a time process internal to the new system of interactions until new equilibrium distributions are reached.

In the case of major economic reorganizations produced by phase transitions (exhibiting new macroscopic behaviors), we are left with an uneven distribution of new economic actors, some more powerful, more mobile, more active, and more competitive than others. The "period of renormalization" may be quite destructive, as older forms are eliminated in competition with newer ones, and new alignments of actors struggle to establish stable positions within the new configuration.

Here again, multiagent modeling is useful to study the renormalization process. Basically, what seems to happen following a period of stable price equilibrium such as those described above, once the velocity of trade has risen to surpass the carrying capacities of the current forms of organization, and once emergent macro structural actors have arisen, is that the power differential of these actors gives rise to various types of oligarchies and cartels, the effect of which is to raise prices for scarce goods that are not subject to a competitive market, and as rising costs are passed along, average wages lag further and further behind inflation, economic and political inequalities are magnified, conflict and violent forms of behavior become more common, until the system goes into a state of crisis. The description of these crises, and the eventual establishment of a new renormalization, is the stuff of FischerÝs (1996) book.

Some of the correlated events of long periods of inflationary crisis are, for the various great waves of price-revolutions:

We are currently living "in the late stages of a very long price-revolution, perhaps in the critical stage" after which periods of stable price equilibrium often ensue. As in every such period, the renormalization of economic forms has continued through unabated competition. As seen from Figure 2, the careening rate of inflation in the last century is unique. The industrial process has led to automation, a machine and information age, and the replacement of organizational domains of specialization by replaceable computer software and software engineering. Many of the building blocks of complex organizational forms, which are often cumulative in evolutionary processes, have been drastically replaced in the new global electronic economy. In an era of enormously destructive price-revolution, Fischer (1996:252) notes that the free market as a solution for solving problems is ephemeral. "A free market restores equilibrium only to break it down again, and to set in motion a new sequence of imbalances and instabilities with all the troubles that follow in their train. Further, "The free market in the twentieth century is an economic fiction, much like the state of nature in the political theory of the eighteenth century." Considering the options, Fischer recommends that we learn to think in the long run about collective efforts at institution building that would bring at last the destructive effects of price-revolutions to heel rather than building institutions that exacerbate them.



I have reviewed a precise physical methodology for the study of phase transitions, one that Iberall and Soodak have sought to generalize to social phenomena, and illustrated (as a conjecture) its applicability to major processes in European and world history as outlined by FischerÝs (1996) monumental work on the great waves of price-revolutions that alternated with periods of stability. The conjecture, supported by some initial evidence, but hardly sufficient for a full test of this model, is that it is during these periods of stability that the velocity of trade accelerates beyond the epochal limits of economic organization, and that dynamical instabilities set the conditions for the emergence of new forms of economic organization. The detailed modeling of the exploratory and experimental behavior by which human beings develop new institutional forms under such pressures is entirely compatible with multiagent modeling of adaptive systems. The detailed temporal and spatial scalings provided by the physical model have a surprisingly good fit to the multiple levels of social and economic organization analyzed here. These scalings may prove to be important to multiagent modeling of adaptive systems in which the emergence of institutions and higher order organization can be simulated.

The theoretical conception here is that the conservation laws of thermodynamic physics provide the necessary background for modeling the real-world constraints and material-energetic processes and transformations under which social actors operate, while the multiagent modeling of adaptive systems captures from the bottom up the flavor of the micro behavior of social actors and the mechanisms and emergence properties of their interactions.

Further, the thermodynamic models are crucial to understand the timing of historical transitions. Whereas the timing of transitions in a simulation model may depend on simulation parameters that are unrelated to physical processes, the timing of transitions in the thermodynamic model may be predicted from actual historical time series data and physical properties of the atomisms that are determined empirically.

Hence the combination of both types of models has high potential for relevance to policy analysis. What the results of this paper call for quite clearly is a compilation of European historical data on changes in long distance trade velocities (both in the sense of transportation velocities and volume of trade), intra-urban and regional trade viscosities, and population scalings and bondings (including cultural cell diameters in relation to political boundaries) such as might be relevant to testing hypotheses in the spatial and temporal frame in which there are time-series data like those of David Fischer (1996) on price variations.



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