**Simple behavior from complex micro-level dynamics**

OUTLINE OF THE TALK

PART I : Michael Strevens' project and its application to complex systems

1) Illustration of the method in a gambling device and in an ecosystem

2) Application in Statistical Mechanics: justification of the Maxwell-Boltzmann distribution

PART II : Analysis of the method

3) Defense of the claim that: The method works for any chaotic dynamics

(and also for random dynamics). Analysis of the dispensable role of the dynamics in (2).

4) Philosophical significance of (3): Stable regularities without assuming any particular dynamics (hence without any set of fundamental laws).

The 2 questions

1) Simple regularities in sciences of complex systems

The wheel of fortune

The 'Method of Arbitrary Functions' based on the 'Law of Large Numbers'

**Aldo Filomeno**

2nd Barcelona-Urbino Meeting on History and Philosophy of Science

21-22 Nov. 2013

2nd Barcelona-Urbino Meeting on History and Philosophy of Science

21-22 Nov. 2013

Population Ecology

We do have simple macrolevel laws, e.g. about the rate of rabbits' population increase

(Logistic equation, Lotka-Volterra, Malthus equation, ...)

**Part I**

**Part II**

2) Non-accidental regularities

How is it that population ecology describes simple laws while it is based in a complex underlying level ?

ex: The population of rabbits depends on

- fertility rates,

- weather,

- predators (number, health, ...),

- availability of food resources,

- geographical location,

- ETC.

Strevens [2003],[2005]

Ex: The population of rabbits depends on

- fertility rates,

- weather,

- predators (number, health, ...),

- availability of food resources,

- geographical location,

- ETC.

Other examples: Meteorology, Economy, Sociology, Linguistics, etc.

The wheel of fortune

"The enion probabilities of ecology, statistical physics, economics, sociology, meteorology, and other sciences are microconstant and stochastically independent, in virtue of some of the very properties that are responsible for the complexity of the systems that they describe".

Therefore they exhibit simple laws.

But :

Application in Statistical Mechanics:

Justification of the Maxwell-Boltzmann distribution

The molecular level has to verify the following conditions:

1) Microconstancy in the evolution function of the variables

2) The distribution over the IC angle of impact is smooth

3) Stochastic independence between the collisions

Very simple example:

Px(death per T)=95%

This Px is stochastically independent

x is large

Population Ecology

Law of Large

Numbers

Macrolevel law:

N (T+1) = 0,95 N (T)

If the evolution corresponds to a random walk

it will "wash out" the starting point of the system

The factors the will determine the value will be:

1) the deterministic microlevel dynamics,

2) the outcomes of the microdynamic probabilistic experiments—(foraging decisions, predator/prey interactions, and so on)

Once the walk has gone for long enough there will be no significant correlation between the system's current state and its initial state z.

<<Further justification is needed for, as usual, it is a deep question remaining open as to whether any rerandomizing assumption is even consistent with the underlying deterministic evolution of the phase points>> [Sklar 1993 p.215]

"The properties of the Maxwell-Boltzmann probabilities, then, are the official end points of the inquiry, but as the reader will, I hope, come to see, the journey is more important than the destination: the same considerations used to establish the microconstancy and independence of theMaxwell-Boltzmann probabilities can, I believe, provide an understanding of all the dynamic properties of gases that fall within the ambit of kinetic theory, in particular, the tendency to equilibrium"

Statistical Mechanics

Microconstancy

Extending the scope of the method

Concluding remarks

Failure in explaining the appearance of non-accidental regularities by philosophical accounts of Laws of Nature (Necessitarian, Humean, Antireductionist, Antirealist)

There is a stable 50/50 probability irrespective of the way the croupier launches the ball

(SLIDE 4): almost any probabilistic distribution of the initial conditions will determine approximately

the same final probability at the outcome of the system

Evolution function: The outcome (red or black) in function of the spin speed

I.C. distribution of spin speed for croupier A

I.C. distribution of spin speed for croupier B

What is the role of the dynamics in the obtaining of a stable probability

distribution at the output?

My thesis: It is irrelevant as long as a chaotic trajectory is assumed.

The particular details of (the mathematical form of) the alleged fundamental laws are irrelevant in obtaining the property of microconstancy.

‘The method of arbitrary functions is able to account for the emergence of simple

higher-level regularities in setups exhibiting certain spatial symmetries for any possible

lower-level chaotic dynamics’

The ground for an account of stable regularities without assuming fundamental laws

Any chaotic (or ‘random wandering’) trajectory of a certain subregion of phase space will be microconstant, where this region corresponds to a subregion of the energy hypersurface delimited by a nondynamical feature of the system.

"(...) the precise facts about the frictional forces that slow the wheel do not matter. Only one fact about these forces matters, the rather abstract fact of the rotational symmetry of their combined effect."

"The same is true for a tossed coin: only the symmetrical distribution of mass in the coin matters”

Wheel:

Coin:

”The value of a microconstant probability may come out the same on many different,

competing stories about fundamental physics. The probability of heads on a tossed coin, for

example, is one half in Newtonian physics, quantum physics, and the physics of medieval

impetus theory” (Strevens 2003, 62) (Strevens 1998, xx)

Statistical

Mechanics:

Consideration of random dynamics:

All chaotic trajectories lead to the formation of stable behavior (because they are random-looking).

The random trajectory will coincide with some of these trajectories.

IC-variable: Relative angle of impact 'theta'

Outcome: Angle of impact in the next collision

positions and velocities take random walks (4.86)

angle of collision' probabilities are microconstant. (4.42)

angle of collision' probabilities are independent

(follows from microconstancy and independence of impact angle (step3))

Show that: we will have constant partitions of the IC variable space (the initial-angle-of-impact) such that every partition has the same proportion for each possible later-angle-of-impact each outcome

Each partition is associated with collision of one or other molecule.

This division of the IC variable space displays a microsized constant ratio for the later-angle-of-impact (any later-angle-of-impact will be equiprobable)

microsized because a small variation in initial-angle-of-impact already leads to another trajectory hence another target molecule

o any other later colliding molecule?

constantbecause in each partition the proportion for different later-angles-of-impact is the same.

smoothness (4.85)

"the probabilistic behavior of gases will rest not on certain very particular mathematical properties of the laws of fundamental physics, but on very general facts about the nature of collisions" (Strevens, 2003, 217).

1) The evolution function is microconstant (4.84)

2) 'theta' is I.I.D. (step 3, 4.85)

3) S.I.C.

Justification of the Maxwell-Boltzmann distribution

(Maxwell 1860) made 3 assumptions that lead to proposition IV

1) any direction of the velocity is equally likely - spherical symmetry.

Because: "the directions of the coordinates are perfectly arbitrary" (ibid 381). In other words, because the three axes of coordinates are conventional

2) The probability will have the same form for the 3 directions x,y,z.

This follows from (1) and (3) (Truesdell 1975, 37).

3) The velocities of each direction (x,y,z) are stochastically independent of one another.

Because they are mathematically independent.

These 3 constraints entail that the distribution has to be gaussian !

A priori conceptual truths?

The empirical details regarding the form of the dynamics are, among other things, inexistent in the derivation. "Floating free of the particular, contingent physics of our world" (Strevens 2013).

Sufficient for steady-state microconstancy of rabbit's probabilities

(4.84)

washes out the microlevel details

Statistical Mechanics

The reasons for the assumptions are "precarious" (Maxwell 1867, 43).

But they can be justified by other arguments, suggested in the previous propositions (Strevens 2013):

Proposition II. Statistical postulate: any point in the region of collision is equally likely. This entails that any rebound direction is equally likely.

Proposition III. After a certain time positions and velocities will converge to a stable equilibrium distribution.

(Maxwell 1860),

Proposition IV:

1. Micro-constancy of the evolution function:

small change of the IC variable changes the outcome +

constant pattern

2. Smoothness of the IC variable (spin speed)

Two sufficient conditions:

Stable physical probability

of 50% of red outcomes

How to obtain micro-constancy?

High Sensitivity to Initial Conditions +

certain symmetry of the physical setup

The collisions provoke a random adjustement to the direction

micro-constancy + smoothness

Any direction is equally likely (prop. II)

Because of the symmetries of the spheres

Collisions are H.S.I.C. +

symmetry of the spheres

Law of Large Numbers

(Ergodic theorem for Markov Chains)

Proposition III -which states the existence of an equilibrium distribution- is justified through microconstancy and smoothness. Both properties are justified because the H.S.I.C. + equiprobability of the angle of collision. And these in turn are obtained thanks to the spherical shape of the molecules.

The role of the dynamics

The role of the dynamics

For all chaotic dynamics in a setup with stable symmetries the evolution function is microconstant

1. There is a dynamics x s.t. x is chaotic AND does not have a microconstant evolution function ;

2. x is chaotic --> x is H.S.I.C. ;

3. x is H.S.I.C. AND time is large AND the setup displays some e.g. uniform symmetry --> random walk through all the possible values --> equiprobable visiting rates ;

4. Stable symmetries & equiprobable visiting rates --> Stable Probability distribution of outcomes ;

5. Evolution function is not microconstant --> (Not quick variation) Or (Not constant pattern) ;

6. Not quick variation

7. Not constant pattern of outcomes in function of some IC variable

9. There is not a stable equlibrium probability distribution

There is a repeated cycling through every possible value (multiple ergodicity)

and the cycling is smooth (the cycling speed only changes slowly)

P(rabbit's death) is approx. the same for any starting position of the rabbit.

”a smooth density will be approximately flat over any neighboring pair of red and black areas in the evolution function, for which reason the contribution made by that part of the ic-density to the probability of red will be approximately equal to the contribution made to the probability of black” [Strevens, 2003, p.50].

”if the evolution function for an outcome e is microconstant, any smooth ic-density determines the same probability for e, equal to the ratio of red to black”

This is not smooth:

6, 2

Randomization effect --> equiprobable visiting rates

Given the symmetry of the wheel (or of the spheres), the same number of IC (speed, or angle of collision) leads to one outcome than to another.

9, 4 ?

"once we go to the coarse-grained level, the K-system exhibits both product and process randomness" (Frigg 2004, 22).

Non-accidental regularities are "just" strongly stable regularities.